TITLE: ADAPTIVE CONTROL TECHNIQUE FOR TRANSMISSION LINE CONTROL DEVICES TO HANDLE LARGE DISTURBANCE SCENARIOS

FIELD

[0001] The various embodiments described herein generally relate to devices and associated methods for providing adaptive control for large disturbance scenarios in Transmission Line power systems.

BACKGROUND

[0002] Today's electrical networks are highly interconnected for the economical sharing of resources. However, power generating sites are usually located far from load centers and thus the generated power needs to be transmitted through transmission lines to the load centers. The growth of electric power transmission facilities is restricted despite the fact that bulk power transfers and the use of transmission systems by third parties are increasing. Transmission system expansion is needed, but this is not easily accomplished. Factors that contribute to this situation include a variety of environmental, land-use and regulatory requirements. Furthermore, transmission bottlenecks, non-uniform utilization of facilities and unwanted parallel-path or loop flows are not uncommon. As a result, the utility industry is facing the challenge of the efficient utilization of existing AC transmission lines and transmission systems are being pushed to operate closer to their stability and thermal limits.

[0003] In addition, power systems are complex, dynamic systems that can be subjected to unpredictable disturbances such as a change in load and the occurrence of various types of faults such as, for example, single-line-to- ground faults, three-phase-to-ground faults, and the like. Furthermore, in complex interconnected power systems, lightly damped inter-area modes of oscillations may get excited during disturbances leading to unstable system operation with inter-area oscillation frequencies generally lying in the range of 0.1 to 0.8 Hz.

[0004] In recent years, Flexible AC Transmission Systems (FACTS) devices have been proposed and used as a means to dampen inter-area oscillations. The traditional approach to dampen the inter-area oscillations is by using fixed parameter phase lead-lag supplementary controllers, which are usually tuned for a certain operating condition and therefore may not perform as desired for other operating conditions (in fact performance most often degrades as operating conditions change). Furthermore, fixed parameter lead-lag controllers are tuned using a multiple-run time domain optimization approach for a particular disturbance, which is very time consuming.

[0005] Another approach is to use adaptive control techniques which are useful in achieving optimal operation for a wide range of operating scenarios. Adaptive algorithms use an estimated plant model in order to rapidly and smoothly track changes in the power system in order for there to be uniform control action. This is a challenge depending on the particular adaptive algorithm that is used as well as certain operating conditions.

SUMMARY OF VARIOUS EMBODIMENTS

[0006] In one aspect, in at least one embodiment described herein, there is provided an adaptive controller for a power system. The adaptive controller comprises an input configured to receive a power system output signal; a parameter identification module coupled to the input and being configured to generate an estimated power system output signal based on a power system model, a control signal and a prediction error derived from the power system output signal and the estimated power system output signal, the parameter identification module further being configured to use a constrained parameter identification technique to determine values for parameters of the power system model based on the prediction error and the control signal while reducing the effect of larger prediction errors on parameter identification; an adaptive control module coupled to the input and the parameter identification module and being configured to apply an adaptive pole-shift technique to shift poles of the power system model for stable operation of the power system and to generate the control signal based on the pole-shifted model and the power system output signal; and an output coupled to the adaptive control module and being configured to send the control signal to a power flow control device to alter power flow in the power system during a disturbance.

[0007] In another aspect, in at least one embodiment described herein, there is provided a method for controlling a power system to improve stability. The method comprises receiving a power system output signal and a control signal; generating an estimated power system output signal based on a power system model, the control signal and a prediction error derived from the power system output signal and the estimated power system output signal; determining values for parameters of the power system model based on the prediction error and the control signal using a constrained parameter identification technique while reducing the effect of larger prediction errors on parameter identification; applying an adaptive pole-shift technique to shift poles of the power system model for stable operation of the power system; generating the control signal based on the pole-shifted model and the power system output signal; and sending the control signal to a power flow control device to alter power flow in the power system during a disturbance.

[0008] In another aspect, in at least one embodiment described herein, there is provided a computer readable medium comprising a plurality of instructions that are executable on a microprocessor of a device for adapting the device to implement a method for controlling a power system to improve stability. The method comprises receiving a power system output signal and a control signal; generating an estimated power system output signal based on a power system model, the control signal and a prediction error derived from the power system output signal and the estimated power system output signal; determining values for parameters of the power system model based on the prediction error and the control signal using a constrained parameter identification technique while reducing the effect of larger prediction errors on parameter identification; applying an adaptive pole-shift technique to shift poles of the power system model for stable operation of the power system; generating the control signal based on the pole-shifted model and the power system output signal; and sending the control signal to a power flow control device to alter power flow in the power system during a disturbance.

[0009] In at least one embodiment, the power system model comprises an Autoregressive Moving Average (ARMA) model with a model order of at least 3.

[0010] In at least one embodiment, the constrained parameter identification technique includes an update step for applying a non-linear function to the prediction error signal, the non-linear function being linear for small prediction errors on the order of about ±10% and increasing more slowly compared to a linear rate for larger prediction errors.

[0011] In at least one embodiment, the non-linear function may comprise a sigmoid-like nonlinear function.

[0012] In at least one embodiment, the parameter identification module and the adaptive controller module may be implemented by at least one processor.

[0013] In at least one embodiment, the adaptive control technique may be executed by the at least one processor in first and second sections of instructions to reduce execution time, the first and second sections of instructions being executed in two consecutive interrupts.

[0014] In at least one embodiment, the first section of instructions comprises instructions for performing basic calculations including preparing input matrices and inverting an M matrix.

[0015] In at least one embodiment, the second section of instructions comprises instructions for performing root calculation of a characteristics equation, optimization and calculation of the control signal. [0016] In at least one embodiment, the control signal may be generated by moving open-loop poles of the power system model radially toward a unit circle origin by applying a pole shifting factor a that is determined by minimizing a norm criterion.

[0017] In at least one embodiment, the control signal may be generated by moving open-loop poles of the power system model radially toward a unit circle origin by applying a pole shifting factor a that is determined using a numerical optimization technique.

[0018] In at least one embodiment, a value determined for the pole shifting factor a may be constrained by the stability constraints: [- ^ (l -

A), + (1 - where Λ is a safety factor.

[0019] In at least one embodiment, the power flow control device may comprise a Thyristor Controlled Series Capacitor (TCSC), the power system comprises the TCSC coupled to a transmission network and the adaptive controller is coupled to the TCSC to provide the control signal thereto.

[0020] In at least one embodiment, the power flow control device may comprise a Unified Power Flow Controller (UPFC), the power system comprises the UPFC coupled to a transmission network and the adaptive controller is located in a supplementary control loop to a series voltage sourced converter (Vsc) of a control system of the UPFC.

[0021] In these embodiments, the adaptive controller may receive real power deviations of the power system as an input and generate the control signal to modulate a control input m _se of the series Vsc-

[0022] In another aspect, in at least one embodiment described herein, there is provided a system for determining parameter values for modeling a power system. The system comprises an input configured to receive a power system output signal; a parameter identification module coupled to the input and being configured to generate an estimated power system output signal based on a power system model, a control signal and a prediction error derived from the power system output signal and the estimated power system output signal, the parameter identification module further being configured to use a constrained parameter identification technique to determine the parameter values for the power system model based on the prediction error and the control signal and smooth out parameter variations for disturbances in the power system; and an output coupled to the parameter identification module and being configured to output the parameter values for the power system model.

[0023] In another aspect, in at least one embodiment described herein, there is provided a method for determining parameter values for modeling a power system. The method comprises receiving a power system output signal; generating an estimated power system output signal based on a power system model, a control signal for a power flow control device of the power system and a prediction error derived from the power system output signal and the estimated power system output signal; determining values for parameters of the power system model based on the prediction error and the control signal and smoothing out parameter variations for disturbances in the power system by using a constrained parameter identification technique; and outputting the parameter values for the power system model.

[0024] In another aspect, in at least one embodiment described herein, there is provided a computer readable medium comprising a plurality of instructions that are executable on a microprocessor of a device for adapting the device to implement a method for determining parameter values for modeling a power system. The method comprises receiving a power system output signal; generating an estimated power system output signal based on a power system model, a control signal for a power flow control device of the power system and a prediction error derived from the power system output signal and the estimated power system output signal; determining values for parameters of the power system model based on the prediction error and the control signal and smoothing out parameter variations for disturbances in the power system by using a constrained parameter identification technique; and outputting the parameter values for the power system model.

[0025] In at least one embodiment, the power system model comprises an Autoregressive Moving Average (ARMA) model with a model order of at least 3.

[0026] In at least one embodiment, the constrained parameter identification technique comprises applying a non-linear function to the prediction error signal, the non-linear function being linear for small prediction errors on the order of about ±10% and increasing more slowly than a linear rate for larger prediction errors.

[0027] In at least one embodiment, the non-linear function may be a saturation-type function.

[0028] In at least one embodiment, the non-linear function may comprise one of a sigmoid-like nonlinear function and a tan hyperbolic function.

[0029] In at least one embodiment, a Recursive Least Square (RLS) algorithm that is constrained by the non-linear function is used for parameter identification for the power system model.

[0030] Other features and advantages of the present application will become apparent from the following detailed description taken together with the accompanying drawings. It should be understood, however, that the detailed description and the specific examples, while indicating preferred embodiments of the application, are given by way of illustration only, since various changes and modifications within the spirit and scope of the application will become apparent to those skilled in the art from this detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

[0031] For a better understanding of the various embodiments described herein, and to show more clearly how these various embodiments may be carried into effect, reference will be made, by way of example, to the accompanying drawings which show at least one example embodiment, and which are now briefly described.

[0032] FIG. 1 is a block diagram of an example embodiment of a controller that uses indirect adaptive control.

[0033] FIG. 2 is a plot showing an update value for a Recursive Least Squares (RLS) algorithm based on a penalty function of error (ε) for various values of a parameter a.

[0034] FIG. 3A is a schematic diagram of a 3-area test benchmark with fixed series capacitor compensation.

[0035] FIG. 3B is a diagram of the mode shapes of the 3-area test system of FIG. 3A.

[0036] FIG. 3C is a schematic of an IEEE 12-bus test system.

[0037] FIG. 3D is a schematic of a conventional lead-lag supplementary controller which is used for test comparison purposes.

[0038] FIGS. 4A and 4B show the results of system parameter identification using a robust RLS technique as described herein for the three- area test system, during and after the disturbance in test case 5A.

[0039] FIGS. 4C and 4D show the results of system parameter identification using a conventional RLS technique for the three-area test system, during and after the disturbance in test case 5A.

[0040] FIG. 4E shows open-loop and closed-loop dynamic pole movements as a function of pole-shift factor a, and the projection of the pole- shifting process on the Real-Imaginary axis for the disturbance in test case 5A.

[0041] FIG. 5A is a plot of the time response of the tie-line power flow for the disturbance in test case 5A.

[0042] FIG. 5B is a plot of the time response of the generator load angles G _r G ₄ for the disturbance in test case 5A. [0043] FIG. 5C is a plot of the time response of the generator load angles G5-G4 for the disturbance in test case 5A.

[0044] FIG. 5D to 5E show the time response of the pole-shift factor and TCSC boost factor, respectively, for the disturbance in test case 5A.

[0045] FIG. 6A is a plot of the time response of the tie-line power flow for the disturbance in test case 5B.

[0046] FIGS. 6B and 6C are plots of the time response of the generator load angles G1-G4 and G ₅ -G ₄ , respectively, for the disturbance in test case 5B.

[0047] FIG. 7A is a plot of the time response of the tie-line power flow for the disturbance in test case 5C.

[0048] FIGS. 7B and 7C are plots of the time response of the generator load angles G1-G4 and G5-G4, respectively, for the disturbance in test case 5C.

[0049] FIG. 8A is a plot of the time response of the tie-line power flow for the disturbance in test case 5D.

[0050] FIGS. 8B and 8C show the load angle time responses of the generators Gi and G ₅ , measured with respect to the load angle of G ₄ for the disturbance in test case 5D with negative lead-lag controller gain.

[0051] FIGS. 9A and 9B show the load angle time responses of the generators Gi and G ₅ , measured with respect to the load angle of G ₄ for the disturbance in test case 5D with positive lead-lag controller gain.

[0052] FIGS. 10A and 10B show the relative generator load angle time responses of generators G ₅ and Gi measured with respect to generator G for the disturbance in test case 5E.

[0053] FIG. 10C shows the tie-line power flow time response for the disturbance in test case 5E.

[0054] FIGS. 1 1 A to 11 D show the identified system parameters for test cases 5B to 5E respectively. [0055] FIGS. 12A to 12 D show the control signal AkB for test cases 5B to 5E respectively.

[0056] FIG. 13 shows tie-line power flow time responses in the presence of PSS units for the disturbance in test case 5A.

[0057] FIGS. 14A and 14B show parameter identification using the RLS technique described herein and the conventional RLS technique, respectively, for the disturbance in test case 5A.

[0058] FIG. 14C is a plot comparing the damping that results from using the constrained and conventional RLS algorithms for the disturbance in test case 5A.

[0059] FIG. 14D shows dynamic pole movement as a function of pole- shift factor a for the disturbance in test case 5A, captured at t = 1.18 sec.

[0060] FIG. 15A shows tie-line power flow time responses P7-9 for the disturbance of test case 5A.

[0061] FIGS. 15B and 15C show relative generator rotor angle time responses for the disturbance in test case 5A.

[0062] FIG. 15D and 15E show the control signal and SSSC DC capacitor voltage variations for the disturbance of test case 5A.

[0063] FIGS. 16A and 16B show relative generator load angle time responses for the disturbance in test case 5B.

[0064] FIGS. 17A and 17B show relative generator load angle time responses for the disturbance in test case 5C.

[0065] FIGS. 18A and 18B show tie-line power flow and relative generator rotor angle time responses, respectively, for the disturbance in test case 5D with negative lead-lag gain being used.

[0066] FIGS. 18C and 18D show tie-line power flow and relative generator rotor angle time responses, respectively, for the disturbance in test case 5D with positive lead-lag gain being used. [0067] FIGS. 19A and 19B show the load angle of generators d and G ₆ respectively, measured with respect to the load angle of G ₄ for the disturbance in test case 5E.

[0068] FIGS. 20A and 20B respectively show the time responses of the tie-line power flow (line 7-8) and the relative rotor angle deviation of G ₃ with respect to Gi for the disturbance in test case 5F.

[0069] FIGS. 20C to 20F show the identified system parameters, and control signal generated using the adaptive pole-shift technique and the pole- shift factor for test case 5F, respectively.

[0070] FIGS. 21 A and 21 B respectively show the time responses of the tie-line power flow (line 7-8) and the relative generator rotor angle deviation of generator G3 with respect to Gi .

[0071] FIG. 22A shows the tie-line power flow through line 7-8 for the disturbance in test case 5H.

[0072] FIG. 22B shows the relative generator rotor angle deviation of generator G3 with respect to Gi for the disturbance in test case 5H.

[0073] FIG. 23A shows a block diagram of an example embodiment of a DSK 6713 DSP board that may be used for real-time hardware testing.

[0074] FIG. 23B shows an example embodiment of a test system that can be used for real-time hardware testing of the adaptive control method.

[0075] FIG. 23C-1 shows a first portion of an example embodiment of an adaptive control method that can be used with the hardware of FIG. 23A.

[0076] FIG. 23C-2 shows a second portion of the adaptive control method that can be used with the hardware of FIG. 23A.

[0077] FIGS. 24A and 24B show the identified system parameters for the disturbance in test case 6A for the robust and conventional RLS methods respectively.

[0078] FIG. 24C shows the estimated errors for the robust and conventional RLS methods of FIGS. 24A and 24B respectively. [0079] FIG. 25A shows the tie-line power time response of line 9-10 for the disturbance in test case 6A.

[0080] FIGS. 25B to 25E show the time response of the tie-line power flow (line 7-9) and the relative generator rotor angle deviations between G G ₂ , G4-G1 and G4-G3 respectively for the disturbance in test case 6A.

[0081] FIGS. 26A and 26B show the time responses of the control signal generated by the adaptive pole-shift controller and the TCSC reactance, respectively, for the test case of FIGS. 25B to 25E.

[0082] FIG. 26C shows the corresponding open and closed-loop poles of the identified system for test case 6A.

[0083] FIGS. 27A and 27B show the tie-line power flow and relative generator rotor angle time response for the disturbance in test case 6B.

[0084] FIGS. 28A and 28B shows show the time responses of the tie- line power flow and the relative generator rotor angle deviation 641 respectively for the disturbance of test case 6C.

[0085] FIGS. 29A and 29B show the time responses of the tie-line power flow and relative generator rotor angle deviation δ ₄ ι , respectively, for the disturbance in test case 6D.

[0086] FIGS. 30A and 30B show the time response for tie-line power flow (P ₇₉ ) and rotor angle between G ₄ - Gi respectively for the disturbance of test case 6E.

[0087] FIG. 31 is a block diagram of an example embodiment of a control apparatus that utilizes at least one of the adaptive control methods described herein.

[0088] FIG. 32 is a flowchart of an example embodiment of a control method that can be used with the apparatus of FIG. 31.

[0089] FIG. 33 is a block diagram of an example embodiment of a Unified Power Flow Controller (UPFC) installed in a two-area power system. [0090] FIGS. 34A and 34B are block diagrams of components of the overall control system of FIG. 33 for a damping application for a control system having a series Vsc and a shunt V _S c, respectively.

[0091] FIG. 35 is a block diagram of an example embodiment showing an add-on Self-Tuning controller, according to the teachings herein, that may be used with a power system having a UPFC.

[0092] FIG. 36 is a block diagram of a two-area power system with a UPFC at the sending-end of Area 1 that is used for simulation test purposes of a UPFC without and with an add-on Self-Tuning (ST) controller.

[0093] FIG. 37 is a plot of a simulated tie-line power flow response to the Type 7A disturbance.

[0094] FIGS. 38A and 38B are plots showing the overall modulation of parameter m _se for a PI controller and a PI controller with an add-on ST controller, respectively, for the Type 7A disturbance simulation.

[0095] FIGS. 39A and 39B show pole shift factor a and the ST control signal u(t) for the Type 7A disturbance simulation.

[0096] FIGS. 40A and 40B show the shunt bus voltage fluctuation |Δ¼| and the DC link voltage for the Type 7A disturbance simulation.

[0097] FIG. 41 shows the parameter tracking ability of the CRLS identifier for the Type 7A disturbance simulation.

[0098] FIG. 42 shows a pole-zero plot in the z-plane of open-loop and closed loop poles plotted as a function of the dynamic pole-shift factor a and their movement from 2 s to 5 s for the Type 7A disturbance simulation.

[0099] FIGS. 43A and 43B show the plots for the corresponding tie-line power P ₇₈ and the pole-shift factor or, respectively, for the Type 7B disturbance simulation.

[00100] FIG. 44A shows a plot comparing the power flow responses with and without the ST controller in the supplementary loop for the Type 7C disturbance simulation. [00101] FIG. 44B shows a plot of the control input u(t) for the Type 7C disturbance simulation.

[00102] FIG. 45 shows a plot comparing the power flow responses at the center of tie-line 7-8 with and without the ST controller in the supplementary loop for the Type 7D disturbance simulation.

[00103] FIGS. 46A and 46B show plots comparing the power flow responses with and without the ST controller in the supplementary loop for the Type 7E and 7F disturbance simulations when the tie-length was doubled from about 250 km to 400 km.

[00104] FIG. 47 shows a plot comparing the power flow responses with and without the ST controller in the supplementary loop for the operating condition IV, Type 7G disturbance simulation.

[00105] Further aspects and features of the embodiments described herein will appear from the following description taken together with the accompanying drawings.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[00106] Various apparatuses or processes are described herein to provide an example of at least one embodiment of the claimed subject matter. It should be noted that no embodiment described herein limits any claimed subject matter and any claimed subject matter may cover processes, apparatuses or systems that differ from those described herein. The claimed subject matter is not limited to apparatuses, processes or systems having all of the features of any one apparatus, process or system described herein or to features common to multiple or all of the apparatuses, processes or systems described herein. It is possible that an apparatus, process or system described herein is not an embodiment of any claimed subject matter. Any subject matter that is disclosed in an apparatus, process or system described herein that is not claimed in this document may be the subject matter of another protective instrument, for example, a continuing patent application, and the applicants, inventors or owners do not intend to abandon, disclaim or dedicate to the public any such invention by its disclosure in this document.

[00107] Furthermore, it will be appreciated that for simplicity and clarity of illustration, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements. In addition, numerous specific details are set forth in order to provide a thorough understanding of the embodiments described herein. However, it will be understood by those of ordinary skill in the art that the embodiments described herein may be practiced without these specific details. In other instances, well-known methods, procedures and components have not been described in detail so as not to obscure the embodiments described herein. Also, the description is not to be considered as limiting the scope of the embodiments described herein.

[00108] It should be noted that the term "coupled" or "coupling" used herein indicates that two elements can be directly coupled to one another or coupled to one another through one or more intermediate elements. It should also be noted that the term coupled as used herein can have several different meanings depending on the context in which the term is used. For example, the term coupled may have a mechanical or electrical connotation. For example, in some contexts, the term coupling may indicate that two elements or devices can be physically coupled to one another or coupled to one another through one or more intermediate elements or devices via a physical coupling, such as a wire, cable or transmission line, or an appropriate mechanical coupling such as a fastener, for example.

[00109] In the following passages, different aspects of the embodiments are defined in more detail. Each aspect so defined may be combined with any other aspect or aspects unless clearly indicated to the contrary. In particular, any feature indicated as being preferred or advantageous may be combined with at least one other feature or features indicated as being preferred or advantageous. [00110] It should be noted that terms of degree such as "substantially", "about" and "approximately" as used herein mean a reasonable amount of deviation of the modified term such that the end result is not significantly changed.

[00111] Furthermore, the recitation of numerical ranges by endpoints herein includes all numbers and fractions subsumed within that range (e.g. 1 to 5 includes 1 , 1.5, 2, 2.75, 3, 3.90, 4, and 5). It is also to be understood that all numbers and fractions thereof are presumed to be modified by the term "about." The term "about" means up to plus or minus 2% of the number to which reference is being made.

[00112] As described previously, adaptive control techniques can be used to achieve optimal operation for a wide range of operating scenarios for a power system. One particular adaptive technique is the adaptive pole-shift control technique for FACTS devices. However, with this technique, the start of the estimation process gives poor system response to initial transients when using a normal Recursive Least Square (RLS) estimator. Furthermore, the use of RLS estimators based on a variable forgetting factor causes a large variation in estimated parameters during transients, which leads to noisy control action that takes a longer time to settle down. The RLS-type estimators are optimal if the disturbances are Gaussian such that the error can be modeled using white noise. However, these errors may not be white noise in practice and a direct consequence of the least-squares formulation is that a single large error will have a drastic influence on the result because the errors are squared in the RLS criterion. Accordingly, in power systems, especially during large disturbances, parameter identification using least squares procedures is a real challenge. In general, the parameters that are identified during such conditions have rapid fluctuations due to large estimation errors which lead to poor controller performance.

[00113] The various embodiments described herein are directed to an adaptive control technique for power systems in which a lower-order model is used for identification of the parameters of a model of the power system. Furthermore, in at least some embodiments, the adaptive controller has the ability to adjust its own parameters which has been found to yield good control performance under a variety of operational conditions. Since computational time is important, using a lower order model, as described herein, will require fewer calculations for parameter identification and control operations. Accordingly, a reduced order model is generally used herein that can capture the essential dynamics of the power system but not necessarily represent the exact dynamic behaviour of the power system which is beneficial from a stability and computational viewpoint.

[00114] For example, in at least one example embodiment described herein, a third order model can be used to model a power system. This is beneficial since any dynamic system that depicts an oscillatory response at one frequency can be modeled as a third-order Autoregressive Moving- Average (ARMA) model. For a power system exhibiting an oscillatory behaviour, a third-order model is going to have three roots: a pair of complex conjugate roots and a real root. The complex roots represent the oscillatory part and the real root represents the decaying part of the response. Furthermore, for controlling the dominant mode of an inter-area oscillation in a tie-line, a third-order ARMA model has been found to be sufficient. It should be noted that there may be other inter-area modes of oscillations that are present, along with the dominant mode, but these nondominant modes may be treated as a noise.

[00115] In alternative embodiments, other nonlinear models may be used as long as these models can be represented as a function such as a radial basis function network, a functional link network and the like. In other alternative embodiments, higher order ARMA models can be used such as 5 ^th order, 7 ^th order, or other higher order odd-numbered models. Higher order ARMA models are desirable if there are multiple modes of oscillations present in the power system. However, the order of the model should not be increased if the amount of computational complexity increases so much that the controller would not be effective at dealing with faults and other disturbances.

[00116] At least one embodiment described herein also uses a tracking constrained parameter identification procedure during large disturbance scenarios, such as three-phase faults, for example. This procedure involves an additional calculation step that helps to effectively smooth out or scale down the parameter variations during large or major disturbances (e.g. fault conditions), and thus produce a stable control response without the need of a controller deadzone. Rather, the controller and the parameter identifier are allowed to function all throughout the disturbance condition in unison. The constrained parameter identification procedure described herein also prevents large excursions in the estimated parameters during large disturbance conditions. Also, with the constrained parameter identifier procedure, the controller is able to adapt to sudden disturbances in the power system and still have good performance. It should also be noted that the various embodiments of the constrained parameter identification procedure described herein can be used with any other discrete time linear pole-placement controller. The indirect control principle that is described herein comprises an identifier and a controller. Any of the linear type of adaptive controllers that comprise an identifier and a controller can be used with the constrained parameter identification technique described herein.

[00 17] At least one embodiment described herein also uses a dynamic pole-shift method to find the optimum pole locations without excessive control calculations. For example, the minimum variance principle, pole placement or pole assignment controllers can be used for finding the pole-shift factors efficiently.

[00118] The effectiveness of the control techniques described herein has been demonstrated using (i) a non-linear discrete system, (ii) a three-area, six-machine power test system using balanced three-phase TCSC/SSSC compensation and (iii) an IEEE 12-bus power test system using a balanced three-phase TCSC device (a TCSC is Thyristor Controlled Series Capacitor which is a thyristor switch-based series FACTS device). The results of these tests are described herein.

[00119] An adaptive controller is formed by combining an on-line parameter estimator, which provides estimates of unknown parameters at each sampling instant, with a control law that is formulated to meet certain operating criteria. The controller can be realized using the parameter estimator and the control law in two different approaches. In the first approach, referred to as indirect adaptive control, the plant parameters are estimated on-line and used to calculate the controller parameters. This approach has also been referred to as explicit adaptive control, because the design is based on an explicit plant model. In the second approach, referred to as direct adaptive control, the plant model is parameterized in terms of the controller parameters that are estimated directly without intermediate calculations involving plant parameter estimates. This approach has also been referred to as implicit adaptive control because the design is based on the estimation of an implicit plant model.

[00120] Referring now to FIG. 1 , shown therein is a block diagram of an example embodiment of an adaptive controller 10 that uses indirect adaptive control. The adaptive controller 10 comprises a parameter identifier module 12 and an adaptive control module 14 along with first and second summers 16 and 18. The parameter identifier module 12 tracks the power system dynamics of power system 20 using a model with a particular input-output relation that comprises time-varying coefficients. The coefficients of the model are estimated in real-time. This can be done using a recursive algorithm, for example. The estimated parameters are then used to design a controller to meet specific control requirements. A Pseudo Random Binary Sequence (PRBS) signal was used to excite the power system model. The response of the non-linear power system 20, including a FACTS device (the TCSC block 22) and a transmission network 24, was modelled preferably using a reduced-order or coarse model, although other higher order models may be used in some circumstances where there are multiple oscillating frequencies in the power system 20.

[00121] The PRBS signal is a small white noise signal (having a magnitude less than 2% of the magnitude of average signals in the power system 20) and is added to the adaptive controller 10 to prevent the parameter identification module 4 from going to "sleep" by adding the PRBS signal to the control signal U _c via the summer 18. The addition of the PRBS signal is a standard technique in adaptive control simulation. In actual power systems, there is some random noise that is always present even if the power system is operating at steady-state. Therefore, the PRBS has been added to mimic the behavior of any actual power system but does not have to be used in practice.

[00122J The parameter identifier module 12 and the adaptive control module 14 can be implemented using software instructions that are executed by at least one processor. In an alternative embodiment, at least one of the parameter identifier module 12 and the adaptive control module 14 are implemented using dedicated hardware such as at least one of an Application Specific Integrated Circuit (ASIC), a Field Programmable Gate Array (FPGA), digital signal processing boards, field programmable analog arrays, microprocessors, IBM Power Processors, Intel processors and the like.

[00123] The adaptive controller 10 comprises an input configured to receive a power system output signal Ay(t) and an output that is the control signal U _c which is generated by the adaptive control module 14 The control signal Uc is sent to a power flow control device (e.g. the TCSC 22 in FIG. 1 ) to alter power flow in the power system 20 during a disturbance.

[00124] The parameter identifier module 12 has a first input that is the addition of the PRBS signal and the control signal U _c that is generated by the adaptive control module 14. The parameter identifier module 12 has a first output that is an estimated output Ay(t) for the power system 20. The parameter identifier module 12 uses a plant model (also known as a power system model) to model the power system 20 and generate the estimated output Ay(t). In this example embodiment, the plant model is an Autoregressive Moving Average (ARMA) model. The parameter identifier module 12 also has a second input that is the prediction error ε(ί) results from the subtraction performed by the summer 16 of the estimated output Ay(t) from the actual power system output Ay(t). The parameter identifier module 12 uses the prediction error z(t) to identify the parameters of the power system 20.

[00125] In other words, the parameter identification module 12 is configured to generate the estimated power system output signal Ay(t) based on the power system model, a control signal U _c and a prediction error e(t) derived from the power system output signal Ay(£) and the estimated power system output signal. The parameter identification module 12 is also configured to use a constrained parameter identification technique to determine values for parameters of the power system model based on the prediction error ε(ί) and the control signal U _c while reducing the effect of larger prediction errors on parameter identification.

[00126] The identified parameters {a,, b,} are a second output of the parameter identified module 12 that are sent to the adaptive control module 14. The identified parameters are then used to generate control action using a particular control method which in this example embodiment is an indirect pole-shift control method. In other embodiments, pole assignment algorithms, pole placement algorithms, or eigenvalue placement algorithms may be used. In the pole-shift control method, the control signal U _c is generated by moving the open-loop poles of the model radially toward the origin of the unit circle in the z-plane by applying a shifting factor a using a minimum variance-based optimization criterion which is basically a minimization of the square of the errors (i.e. a Euclidean norm). In other embodiments other norms can be minimized such as, but not limited to, absolute error, the H-infinity norm and the like. The control procedure is computationally simple as there is only one parameter (a) that is determined for the adaptive control module 14. [00127] In other words, the adaptive control module 14 is configured to apply an adaptive pole-shift technique to shift poles of the power system model for stable operation of the power system 20 and to generate the control signal U _c based on the pole-shifted model and the power system output signal Ay(t).

[00128] A challenge with using adaptive pole-shift control as a transmission line control in a power system is based on the fact that the least- square based estimation becomes inaccurate for large disturbance conditions in power systems, such as system faults. Advantageously, the embodiments described herein use a robust method to address the problem associated with the RLS technique in a power system (during large disturbances). This robust method involves using a non-linear function in the parameter update equation so that a large error will have less effect on the parameter update, whereas the parameters will be updated in a linear fashion when the error magnitude is small. An example of such a non-linear function is a sigmoid-like function. This technique effectively smoothes out parameter variations during major disturbances, and therefore a stable control response is produced without the need for parameter freezing or a controller dead-zone. In other embodiments other exponentially decaying or saturation functions can be used for the non- linear function.

[00129] The ARMA model that is used by the parameter identifier module 12 has the form shown in equation 1a. The Recursive Least Square (RLS) algorithm is used for the plant parameter identification because of its simplicity and robustness.

The variables y(t), u(t) and e(t) are system output, system input and noise terms, respectively and (ζ ^_1 ), S(z ^_1 ) and CO ^-1 ) are the polynomials expressed in terms of the backward shift z ^"1 and are defined according to equations 1 b, 1 c and 1d, respectively while n _a , n _b and n _c are the order of the polynomials A(z ^→ , Biz ^'1 ) and C(z ^_1 ) respectively. The variable n _d is a delay term. A(z ^→ ) = 1 + a _x z ^→ + a ₂ ∑- ² + ^■■■ + a _na z ^~n ^ (1 b)

B(z ^→ ) = 1 + b _x z ^~x + b ₂ z ^~2 + - + b _nb z- ⁿ b (1c)

CCz- ¹ ) = 1 + z- ¹ + c ₂ z ^~2 + - + c _nc z~ ⁿ c (1d)

[00130] As the noise term e(t) cannot be directly measured, the model defined by equations 1 a-1d can be obtained by using a suitable approximation, such as that shown in equation 2.

y(t) = ^τ (ί)θ + e(t) (2) where i/>(£) is the measurement variable vector given as:

< t) = [-y(t - 1)] .. -y(t - n _a ) u(t - no) .. u(t - n _d -n _a ) e(t - 1) .. e(t - n _c )f, (3) Θ is the parameter weight vector given as:

0 = a _na b _x .. b _nb c _r .. c _nc f (4) and

e{t) = y{t) - p{t)9{t - l) (5) is the estimation error. The variable e(t) in equation 5 is approximated by the estimation error e{t).

[00131] The system parameter weight vector, 9(t) can be estimated using the following extended recursive least square (RLS) algorithm:

= S(t - 1) + K(t [y(t) - φ ^τ θ(ί - 1)] (6a) K(t\ = P ^(t -iW ⁽ t ⁾ _(6b)

" ^W A(t)-iP{t)Tp{t-imt) ^K ' P(t) = ^ [P{t - 1) - K(tW(t)P(t - 1)] (6c) where (t) is the time varying forgetting factor, P(t) is the covariance matrix and K(t) is the gain vector. The forgetting factor A(t) may be calculated as:

A(t) = λ ₀ λ(ί - 1) + (1 - Ao) (7) where λ ₀ is a positive value between 0 and 1. [00132] During power system transients, the power system may shift from one operating state to another, and the error between the actual plant output Ay(t) and the estimated plant output Ay(t) may vary significantly. Also, it has been found that errors are not purely Gaussian, and therefore a single large error may have a drastic influence on the result. Therefore, it has been determined herein that to limit the effect of such large deviations on the estimation errors, the update algorithm (equation 6a) is modified as given in equation 8.

§(t) = Ht - 1) + K(t)f(s(t)) (8)

[00133] A constant multiplier with a value of less than unity can be used to penalize the gain term in equation 8. However, doing so decreases the sensitivity of the algorithm to smaller errors. The objective is to have an algorithm that is very sensitive to smaller errors, but to make it less sensitive for larger errors. Therefore, the function ί(ε) is selected such that it is linear for small errors (ε) but increases more slowly than linear for large errors (ε). Small errors of less than about ±10% lie in the linear region of the curve. Accordingly, this type of estimation is robust towards large errors. An example of a non-linear function that can be used is the following sigmoid-like nonlinear expression for f(e) shown in equation 9:

/(ε(0) = ^J- (9) where the parameter a is the design constant that determines the parameter update rate. In alternative embodiments, tan hyperbolic or any saturation type function can be used as the non-linear function. For a = 0, the algorithm will become a normal RLS. However, for a > 0, the penalty on the error (ε) increases with an increase in absolute error magnitude as shown in FIG. 2. The value for the parameter "a" may be based on the application and it may be picked offline and does not have to vary dynamically.

[00134] Since the adaptive controller 10 may be implemented on a stand-alone Digital Signal Processor (DSP), the calculations for equation 9 are very attractive, as it provides a very good result with minimum computing resources.

[00135] When the parameters of the plant model are properly estimated, the adaptive control module 14 preferably uses an optimized pole placement control algorithm called pole-shift control to generate the control signal U _c . The transfer function of the adaptive control module 14 is given by equations 10a to 10c:

«(*) _ ^'1 )

y(t) FCz ^"1 ) (10a)

Fiz- ¹ ) = 1 + Az ^"1 + f ₂ z~ ² + - + f _nf z- ⁿ f (10b) Giz ^'1 = g ₀ + giz ^'1 + ₂ z- ² + - + gn _a z ^~n (10c) with rif = n _b - \ and n _g = n _a - l. From equations 1 and 10a-10c, the characteristic equation of the closed loop control can be derived as shown in equation 1.

TO ^"1 ) = Aiz-^Fiz ^{' 1} ) + Biz-^Giz- ¹ ) (1 1 )

[00136] If the characteristic polynomial T(z ^"1 ) is predefined as in pole- assignment control, the controller polynomials F(z ^~1 ) and G(z ^"1 ) can be calculated from equation 1 1. However, in pole-shift control, equation 1 1 takes the form of A(z ^"1 ) with the pole shifted by factor of 'a'. Accordingly, the new characteristics equation can be obtained by replacing 'z ^' in A(z ^"1 ) by 'az ^'1 ' as follows shown in equation 12.

Aiaz ^'1 ) = TCz- ¹ ) = Aiz-^Fiz- ¹ + Biz^C^) (12)

By having the polynomial T(z ^"1 ) take the form of A(az ^"1 ) using equation 12, the control signal is made to be a more stable signal. Rearranging and expressing equation 12 in matrix form gives:

or

Λίω(α¾) = L(a _t ) (14) where

_. 0 0 0 0 bn _b l9n _a

and a _t is the pole-shift factor at time t.

[00137] Equation 13 can be solved for f,- and g, for a known value of a _t . Once the values of f, and g, are obtained, the control signal can be computed using equation 10a. It can be observed that the control signal is a function of a at any time t denoted as a _t . The control signal u(t, a _t ) can be expressed in a Taylor series in terms of factor a _t at any operating point a ₀ .

d ^l u(t,a )

u(t, a _t ) = u(t, a ₀ ) +∑™ ₌ A (0 (a _t - a ₀ (15) equations 10a and 13, the control signal can be calculated as

u(t, a _t ) = X ^T (t)w(a _t ) = X ^T (t)M- ¹ L(a _t ) (16) where X(i) = [-u(t - 1). . - (t - n _f ) - y(t) - y(t - 1). . y t - n _g) ] is the measurement variable vector. The i ^th order differential of u(t, a _t ) with respect to a _t is shown in equation 17.

Equation 15 can be written in simple form as: u(t, a _t ) = u(t, ₀ ) +∑" ^« ₁ s _i a _t ⁱ (18) where s _t = ;X ^T (t)M- ¹ L ⁱⁱ⁾ (a ₀ ). [00138] Once the input is computed at time t, the predicted system output y(t + 1) at time t + 1 can be predicted as follows:

(t + l) = Χ ^τ (ί)β + b _lU (t, _t ) (19) where β = ... -b _nb a _x a ₂ ... a _n J is an identified parameter vector.

[00139] For a fixed value of a _t , the control algorithm becomes a special case of pole-assignment control. However, in the pole-shift control algorithm, the value of a _t can be selected to satisfy some optimized performance indices. An example of one such performance index is the minimization of the one time-step ahead system output prediction error, i.e.:

7(t + 1, a _t ) = E[y(t + 1) - y _r (t + l)] ² (20a) where J(t + l, a _t ) defines the performance index, E is the expectation operator, y(t + 1) is next time-step predicted output and y _r (t + 1) is the reference output for the next time step. By minimizing the performance index of equation 20a, the controlled system output y(t) follows the pre-specified reference y _r (t) as closely as possible. In alternative embodiments, other performance indices that can be used include, but are not limited to, the absolute value norm, the quadratic norm, the cubic norm, the H-infinity norm, and the like. Minimization of the objective function defined in equation 20a yields the optimal value of a _t . The value of _t is also kept in the range of [- 1/A _t < a _t < 1/AJ to satisfy stability constraints, where A _t represents the largest absolute value of the roots of the characteristics equation T(z ^"1 ). Furthermore, the control signal also preferably lies within the control constraint:

(20b) where u _min and u _max are minimum and maximum control signal boundaries.

[00140] Since _t reflects the stability of the closed-loop system, it is desirable to assign the pole-shifting factor a _t to a specified value under steady state, such as 'a _ss '. To accommodate this ability, the cost function given in equation 20a can be modified as: m J(t + 1, a _t ) = E[y(t + 1) - y _r (t + l)] ² + μ(α, - a _ss ) ² (21 ) where μ is a weighting coefficient and a _ss is the steady-state pole shift factor. Equation 21 is second order in nature and modern optimization routines can easily find the exact roots in a few iterations provided that the starting point is the previous sampling time root. Minimization of the objective function (equation 21 ) gives the optimal pole-shift factor a _opt . This information is used in equation 16 to generate an optimal control signal so that the next time-step system output (t + l) follows reference signal y _r (t + 1 ) with minimum variance.

[00141] To demonstrate the effectiveness of the adaptive controller 10 in damping inter-area oscillations, two multi-area test benchmark systems were used. The first test system 30 was a three-area, six-machine system which consists of two machines in each area: Gi and G ₂ in area 1 , G ₃ and G ₄ in area 2, and, G ₅ and G ₆ in area 3, respectively as shown in FIG. 3A. Although this is a test system, it serves well to illustrate the concept, which remains the same for large power systems. Area 2 is connected to Areas 1 and 3 through a 400 km long double-circuit transmission line. The compensation degree is defined as the ratio (X _c + XTCSC)/XL)*100, where X _L is the line inductance, and Xc and XTSCS are the fixed capacitor and TCSC capacitive reactances respectively. Shunt capacitors are installed at buses 7, 9 and 14 to support voltage between 1 ± 0.03 p.u. of rated value. Further, the two transmission lines between bus 7 and 9 are replaced by an equivalent single transmission line to keep the focus of the studies on adaptive control design.

[00142] The electromagnetic transient simulation software EMTP-RV™ was used to simulate the test system. For time domain simulation studies, the electromagnetic transient simulation software PSCAD/EMTDCTM was used. The synchronous generators are represented in the d-q-0 reference frame using 7 ^th order differential equations. The complete TCSC and SSSC models described in Hingorani (Understanding FACTS: concepts and technology of flexible AC transmission systems, L. Gyugyi, Ed. New York: IEEE Press, 2000, pp. 315-319) were used in the simulation studies. The transmission lines were modelled as lumped impedances. Dynamics of the generator excitations and governors were included in the simulation to mimic the real power system operation. The inclusion of governor dynamics stabilized the plant for simulation initialization and improved stability during large disturbances. Machine electrical parameters were taken from Klein et al. (M. Klein, G. Rogers, and P. Kundur, "A fundamental study of inter-area oscillations in power systems", IEEE Transactions on Power Systems, vol. 6, no. 3, pp. 914-921 , 1991 ) and are given in Appendix A.

[00143] The rotor angle oscillation modes for the three-area test system were obtained using eigenvalue analysis. The results are shown in Table 1. There are five different oscillations modes: three local modes in each area, and two inter-area modes. Inter-area Mode 1 is characterized by having a slightly higher frequency (0.78 Hz) than Mode 2 (0.46 Hz). The inter-area oscillations mode values are shown in FIG. 3B.

[00144] Five different cases listed in Table 2 were considered for the verification of the adaptive controller 10 on the three-area test system 30. These cases cover a wide range of power system operating conditions and various disturbance scenarios. System loading for the test cases is shown in Table 3. TABLE 1 : Rotor Angle Oscillation Modes of the Three Area Test System

Mode type Real part Imaginary Frequency Damping ratio

(σ) part (o) _ri ) (Hz) (ς, %)

-0.5840 7.6616 1.2194 7.60

Local -0.5767 6.8333 1.0876 8.41

-0.7099 6.6681 1.0613 10.59

Inter-area -0.1730 4.9093 0.7813 3.52

-0.8353 2.9503 0.4696 27.24

Table 2: Three-area system: inter-area oscillation damping case studies

Approximate power Disturbance description Disturbance flow on line 7-9 type

415 MW 3 cycle, three-phase fault at bus 7 5A

415 MW 3 cycle, 2-phase-to-ground (a,b-ground) 5B

fault at bus 7

220 MW 3 cycle, three-phase fault at bus 7 5C

-180 MW 3 cycle, three-phase fault at bus 7 5D

415 MW 600 MW load rejection at bus 9, at f = 1 sec 5E Table 3: Load Data (MVA, MVAR) for different study cases

Case Bus 7 (L ₇ , C ₇ ) Bus 9 (L ₉ , C ₉ ) Bus 14 (L ₁₄ C ₁₄ )

A, B, E 1400 + ylOO, -/350 1800 + ylOO, -y500 1200 + ylOO, -;220

C 1400 + jlOO, -y ^' 260 1600 + j ^' 100, - ^' 350 1450 + ylOO, -/220

D 1755 + jlOO, -j ^' 260 1200 + ylOO, -)350 1200 + ylOO, -;240

[00145] The second test system 40 that was considered was an IEEE 12-bus power system that is suitable for inter-area oscillation studies using FACTS devices as shown in FIG. 3C. The test system 40 covers three areas: area 1 is mainly for generation, area 3 is a load center and area 2 has limited generation. The generators G ₂ and G ₃ are hydro generators and the generator G ₄ is thermal generator. These generators were modelled using 7 ^th order differential equations (P.C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, 2 ^nd ed. Wiley-lnterscience, 2002). The constant voltage and frequency source model Gi at infinite bus in Jiang et al. (S. Jiang, U. Annakkage, and A. Gole, "A platform for validation of FACTS models", IEEE Transactions on Power Delivery, vol. 21 , no. 1 , pp. 484-491 , Jan. 2006) was replaced by a classical generator with a high inertia of 20s. The test system 40 exhibits three lightly damped inter-area modes of oscillation frequencies at 1 .12 Hz, 0.85 Hz, and 0.75 Hz, respectively as shown in Table 4. The oscillation modes have damping factors of 3.30%, 1 .07%, and 7.17%, respectively, which are very close to each other. The size of the system is such that it is large enough to demonstrate electromechanical oscillation modes and to exhibit phenomena such as congestion in power corridors and inter-area oscillations, but yet is small enough to be modelled in detail with an electromagnetic transient simulation program. The test system 40 is suitable for various FACTS device related studies. The details of the test system, machine parameters, line data, and system loading conditions are given in the Appendix B.

[00146] Three cases were studied as shown in Table 5 for the validation of the adaptive controller 10 with respect to the IEEE 12-bus test system 40.

Table 4: Inter-area modes of the IEEE 12-bus test system

Modes Real part Imaginary part Frequency Damping ratio

(σ) (ω _ά ) (Hz) (ς, %)

1 -0.232 7.027 1.12 3.30

2 -0.058 5.332 0.85 1.07

3 -0.339 4.710 0.75 7.17

Table 5: Type of studies on the IEEE 12-bus test system

Disturbance description Disturbance type

1-phase-to-ground (a-ground) 200 ms fault (bus 3) 5F

Three-phase, 200 ms fault (bus 3 and 6) ^' 5G

Three-phase fault at 100 km from bus 4 on line 4-5, 5H

fault cleared by disconnecting line 4-5 after 9 cycles

[00147] An optimally tuned fixed parameter lead-lag supplementary controller 50, as shown in FIG. 3D, was used to compare with the performance of the adaptive controller 10. This lead-lag controller 50 consists of a washout filter 52, two lead-lag phase compensation blocks 54 and 56, and a gain compensation block 58 shown respectively as four separate blocks from left to right in FIG. 3D.

[00148] The time constants and gain of the controller were optimized using a multiple time-domain simulation based simplex algorithm. The multiple time-domain simulation based simplex optimization technique exploits the advantage of a multiple-run enabled electromagnetic transient computer simulation tool to optimize the non-linear controller parameters. This tuning procedure is used conventionally for tuning controller parameters. More details on the simplex algorithm can be found in Gole et al. (A. Gole, S. Filizadeh, and P. Wilson, "Inclusion of robustness into design using optimization-enabled transient simulation", Power Delivery, IEEE Transactions on, vol. 20, no. 3, pp. 1991-1997, July, 2005).

[00149] The objective function used for tuning the conventional lead-lag supplementary controller 50 is shown in equation 22. The symbols T _c and T _s represent fault clearing time and simulation end time, respectively. Similarly, P _L is the tie-line power flow, and P _Lo is the steady state pre-disturbance tie- line power flow from line 7-9. A fixed parameter controller that is tuned to satisfy the objective function expressed in equation 22 will damp the tie-line power flow deviation effectively. F _obj {T _r *, K _P ) = ff _c dt (22)

[00150] For testing, a user defined component was created in PSCAD/EMTDC to implement the parameter identifier module 12 and the adaptive control module 14. The methods used by the parameter identifier module 12 and the adaptive control module 14 were written in FORTRAN programming language using Intel Visual Fortran and Microsoft Visual Studio Integrated Development Environment. This component received deviation in power flow as input and generated an optimal supplementary control signal for TCSC or SSSC devices. Also for testing, the simplex optimization procedure took 150 simulation runs to optimize parameters T-i , T ₂ , T ₃ , T ₄ and K _p for the disturbance in test case A. [00151] A third-order ARMA model was used to approximate the power system dynamics. An Intel Visual Fortran code was developed for implementing identification, control and optimization algorithms and interfaced to the PSCAD/EMTDC simulation environment via the user defined component. A single variable Nelder-Mead simplex optimization algorithm was used to find the optimum pole-shift factor a _opt by minimizing the cost function expressed in equation 21. This minimized the deviation between the predicted next time-step plant output AP _L (t + 1 ) and the reference plant output ΔΡι,,τθί (t + 1 ). In alternative embodiments, other optimization techniques can be used, such as but not limited to, Newton's method.

[00152] The test system had the highest inter-area mode of oscillation at 0.7913Hz. Accordingly, the sampling time for the simulated adaptive controller was set to 10 Hz (i.e. the identifier and controller subroutines were allowed to run at 100 ms intervals). The power systems under investigation had maximum inter-area oscillation frequencies of 0.78 Hz (for the three-area, six-machine system) and 1.12 Hz (for the IEEE 12-bus system), and the sampling time of 100 ms was found to be sufficient for both cases.

[00153] The extended recursive least square identification method given in equations 6a to 6c was able to find the plant coefficients effectively by including 2 ^nd order noise terms. The maximum number of iterations for optimization was found to be less than 7 for the test scenarios presented herein. The modern computing hardware needed a very small fraction of the total sampling time for this kind of optimization.

[00154] In time domain simulation studies, disturbances were applied at 1 sec for 3 cycles. The performance of the adaptive controller 10 in each test case listed in Table 2 was compared to the: i) corresponding TCSC case without a supplementary controller, and ii) corresponding case with the optimally tuned lead-lag supplementary controller. Moreover, in the following figures illustrating the system time responses, no supplementary controller, traditional lead-lag supplementary controller, and adaptive pole-shift controller are denoted as "No suppl." in dotted lines, "Lead-lag" in thin lines with triangles, and "Pole-shift" in thick lines, respectively.

[00155] The performance of the adaptive pole-shift controller technique described herein with a TCSC compensated three-area, six-machine system is now discussed. The TCSC is assumed to be installed in the test system 30 of FIG. 3A near bus 9 between buses 7 and 9 replacing a portion of the fixed series capacitive compensation of the tie-line. The total series compensation is 0.5 p.u. of the total line reactance and the TCSC contribution in the total compensation is 0.25 p.u. The deviation in power flow through line 7-9 is considered as the plant output. The supplementary control signal AkB is considered as the plant input.

[00156] The estimated system parameters using two RLS estimation algorithms are shown in FIGS. 4A to 4D for the disturbance test case 5A. FIGS. 4A and 4B show the identified plant parameters using the robust RLS algorithm described herein while FIGS. 4C and 4D show the identified plant parameters for a commonly used RLS procedure. It is evident that the parameter identification method proposed herein helps to identify system parameters smoothly even during the large disturbances (FIGS. 4B and 4D). In the case of the conventional RLS technique, a drastic change in parameter values occurs during and after the clearing of the disturbance. This sudden change causes an undesirable controller output which leads to poorer damping performance.

[00157] The dynamic pole-shifting process that takes place during and after the disturbance for the test case of FIGS 4A to 4D is shown in FIG. 4E. The closed-loop poles and open-loop poles are captured for the duration of 1 sec to 4.4 sec and plotted as a function of the pole-shift factor a. The closed- loop and open-loop poles of the system at time t = 1.18 sec are indicated using triangles (open loop) and a square (closed loop), and, at that time, the pole-shift factor was found to be a = 0.149. It can be observed that, during the transient, there is significant pole-shifting. The adaptive controller 10 is generating an optimal control signal by moving the poles toward the origin. Moreover, it can be noted from the projected plot that the open-loop poles are almost stationary, while the closed-loop poles are being shifted dynamically by the adaptive controller 10. The steady state pole-shift factor for this case was found to be a = 0.5952 and the identified plant coefficients were a = [- 1.374, 0.457, 0.096] and b = [0.025, 0.031 , -0.076] respectively. Following equation 12, the transfer function of the controller can be derived as:— =

y(t)

3.501-4.614z~ ¹ -0.738z~ ²

l+0.644z- ¹ +0.585z ^"2

[00158] The test case 5A corresponds to a 415 MW tie-line power flow through bus 7 to bus 9 and was considered as a base operating case for the rest of the test cases described herein. The fixed parameter lead-lag controller was tuned for this operating condition and the parameters are given in Appendix D.

[00159] FIGS. 5A-5C show the time response of tie-line power flow (line 7-9) for the disturbance in test case 5A. The identified plant parameters and the dynamic pole movement with respect to pole-shift factor for the case of FIG. 5A are shown in FIGS. 4A and 4B and 4E respectively.

[00160] It is evident from the responses that the system was exhibiting poorly damped oscillations and both supplementary controllers (i.e. the adaptive controller 10 and the fixed parameter lead-lag controller) helped to damp out the inter-area oscillations significantly faster than was the case without a supplementary controller. The performance of the adaptive controller 10 in this operating condition was slightly better than the simplex- tuned lead-lag controller. The real advantage of the adaptive controller 10 is that the algorithm needs very little information of the plant, and the control response was consistent for a wide range of operating conditions, whereas in the case of the conventional lead-lag controller, a detailed model of the system is required to properly tune the parameters for a particular operating condition.

[00161] FIGS. 5D to 5E show the time response of the pole-shift factor and TCSC boost factor, respectively, for the disturbance in test case 5A. It is evident from these responses that the algorithm tracks the change in power deviation and adjusted pole-shift factor, which eventually triggers the controller output which is evident on the TCSC boost factor plot. The adaptive controller achieved steady-state within 5 seconds of disturbance clearing.

[00162] Test case 5B is a two phase-to-ground (a,b-ground) unsymmetrical fault at bus 7, and the power flow is similar to that of test case 5A. FIGS. 6A to 6C show the time response of tie-line power flow (line 7-9) and the generator rotor angle deviations of Gi and G ₅ with respect to G ₄ . From the time response plots in FIGS. 6A to 6C, it is evident that the adaptive controller 10 was able to suppress the first swing and settle the subsequent swings within 8-10 sec after the clearance of the disturbance. Furthermore, the performance of the adaptive controller 10 was relatively better than the lead-lag controller.

[00163] Test case 5C corresponds to the lower tie-line power flow scenario. The operating condition is achieved by reducing the load on bus 9. This type of operation is very common in a complex power system. In this operating condition, the generator oscillation will have less amplitude. The conventional controller gave a small damping effort leading to a longer settling time, whereas the adaptive controller 10 adapted to the new operating conditions automatically and yielded optimal performance. FIG. 7A shows the time response for the tie-line power flow for the disturbance in test case 5C. FIGS. 7B to 7C shows the time response of the generator rotor angle deviations of d and G ₅ with respect to G ₄ . It should be noted that the generator angle deviation has a smaller magnitude than for the test case 5A (i.e. FIG. 5C). The adaptive pole-shift controller 10 provided lower oscillation amplitude and faster settling time compared to the lead-lag controller, as expected.

[00164] The test case 5D was used to test the performance of the adaptive controller 10 to tie-line power flow reversal. The tie-line power flow on line 7-9 (i.e. from bus 9 to bus 7) was set to 180 MW by increasing the load on bus 7 and decreasing the load on bus 9. This type of operation is very likely to occur in a highly inter-connected power system. The adaptive controller 10, without any external interference, adjusted to a correct set of identified parameters, and generated a stable control behavior.

[00165] FIG. 8A shows the time response for the tie-line power flow for the disturbance in test case 5D. FIGS. 8B and 8C show the time response of the generator rotor angle deviations of Gi and G5 with respect to G ₄ respectively. The power flow reversal introduced a 180° phase shift to the input signal, and so a negative gain was used. However, in practice, the lead- lag controller would need a logical element, such as a power reversal function (for example, a power reversal relay), or in the context of modern power systems, a Phasor Measurement Unit (PMU), to detect the power flow reversal and adjust the sign of the gain for the controller. In contrast, the adaptive controller 10, without any additional logical element, very effectively dampened the inter-area oscillations for the reversed power flow condition.

[00166] Similarly, FIGS. 9A and 9B show similar time responses as was shown in FIGS. 8B and 8C except for a positive gain. In this case, the power system becomes unstable after the clearing of the disturbance for the lead-lag controller. The reason behind this is that the conventional lead-lag controller was tuned for test case 5A, and the stabilizing signal reversed its polarity in this operating condition. The conventional controller should be retuned at this new operating condition for the proper operation. However, it should be noted that in the case of the adaptive controller 10, the identifier module 12 properly identified parameters for the new operating conditions and the adaptive controller 10 generated a stable supplementary control signal.

[00167] In the test case 5E, a 300 MW load was disconnected from bus 9 at t = 1 sec keeping the operating condition of test case 5A. This caused a huge disturbance on the network. The load angle of generator G ₅ and measured with respect to generator G ₄ at these operating conditions are shown in FIGS. 10A and 10B. FIG. 10C shows the tie-line power flow time response at these operating conditions. [00168] Comparing the responses shown in FIGS. 5A to 10B, the positive contribution of the adaptive controller 10 to the damping of the inter- area oscillations is very clear. As it can be observed from FIGS. 5B, 5C, 6B, 6C, 7A to 7C, 8A to 8C, 10A and 10B, the adaptive controller 10 reduced the first swing and effectively dampened the subsequent swings during the different operating conditions.

[00169] The identified parameters and the pole-shift control signal for the various test cases are shown in FIGS. 11A to 11D and 12A to 12D respectively (and are described in the Brief Description of the Figures section herein). It can be seen that the parameter values have been constrained effectively in the various test cases.

[00170] To verify the effectiveness of the adaptive pole-shift based controller 10 in the presence of Power System Stabilizers (PSSs) in the system, three PSSs were connected to generators Gi, G ₃ , and G5 of areas 1 , 2 and 3, respectively, to dampen the respective local modes of oscillation in areas 1 , 2 and 3. The PSSs that were used are of the single input lead-lag variety (IEEE PSS1A model). The operating condition of the power system was the same as in test case 5A. A set of optimal PSSs parameters that were used in this test were obtained using a multiple time-domain simulation based simplex optimization procedure and those values are given in Appendix C.

[00171] FIG. 13 shows the time response for the tie-line (i.e. line 7-9) power flow during and after the disturbance in test case 5A. Prony analysis reveals that the approximate damping factors are 1.96, 7.31 and 15.23% for (i) in absence of PSSs and the supplementary pole-shift controller for TCSC, (ii) only PSSs, and (iii) in the presence of PSS and the supplementary pole- shift controller for the TCSC, respectively. The result shows that even with PSSs in the power system, TCSC, aided by the adaptive controller 10, provides better damping and reduces the settling time of the power system oscillatory modes.

[00172] The effectiveness of the adaptive controller 10 for damping inter- area oscillation on the three-area, six-machine test system 30 (see FIG. 3A) consisting of an SSSC compensated transmission line was also tested. The SSSC was installed on the test system 30 near bus 9 between buses 7 and 9, replacing the portion of fixed series capacitive compensation of the tie-line. The total series compensation was 0.5 p.u. of the total line reactance and the SSSC contribution in the total compensation was 0.25 p.u. The deviation in power flow through line 7-9 was also considered as plant output. The supplementary control signal U _s was considered as the plant input. The studies were carried out for the test cases presented in Table 2.

[00173] The parameter tracking capability of the two RLS estimation algorithms that were tested is shown in FIGS. 14A and 14B for the disturbance in test case 5A. FIG. 14A shows the identified plant parameters using the robust RLS algorithm described herein for the parameter identification module 12. Similarly, FIG. 14B shows the estimated plant parameters for the same test case of FIG. 14A, using the conventional RLS algorithm. It is evident from these tracking responses that the parameter constrained identification method described herein helps to identify the system parameters more smoothly, even during a large disturbance. In contrast, the conventional RLS algorithm exhibited a large deviation in parameter estimation during and after the clearance of the disturbance. This large deviation in parameter value caused an undesirable controller output, which led to poorly damped performance, as shown in FIG. 14C.

[00174] FIG. 14D shows the dynamic pole-shifting process for the test case of FIG. 14A. The closed-loop poles and open-loop poles are captured for the time stamp of t = 1.18 sec and the corresponding pole-shift factor was a = 0.59. It can be observed from the result that there was significant pole- shifting taking place during transients. The adaptive controller 10 generated the optimal control signal by moving the poles closer to the origin.

[00175] The test case 5A was considered as a base operating case for the rest of the test cases. The fixed parameter lead-lag controller was tuned for this operating condition, and the parameters are given in Appendix C. [00176] FIGS. 15A to 5C show the time response of the tie-line power flow (line 7-9) and relative generator rotor angle plot for the disturbance in test case 5A. The effectiveness of the adaptive controller 10 in damping inter- area oscillations is demonstrated through relative generator rotor angle response time shown in FIG. 15A. The response showed that the adaptive controller 10 dampened the tie-line power oscillations relatively faster than the conventional lead-lag controller. The identified plant parameters and dynamic pole movement with respect to pole-shift factor for the case of FIGS. 15B and 15C are shown in FIGS. 14A and 14D respectively. Furthermore, FIGS. 15D and 15E show the control signal generated and the SSSC DC capacitor voltage variations for the disturbance of test case 5A.

[00177] The test case 5B was a two phase-to-ground (a,b-ground) unsymmetrical fault at bus 7, and the power flow was similar to that seen in test case 5A. FIGS. 16A and 16B show the time response of the generator rotor angle deviations of Gi and G ₅ with respect to G ₄ , which clearly demonstrate the effectiveness of the adaptive controller 10 in dealing with power oscillations. The adaptive controller 10 was able to suppress the first swing and dampen the subsequent swings within 8-10 sec after the clearance of the disturbance. The performance of the adaptive controller 10 was seen as being relatively better than that of the conventional lead-lag controller.

[00178] The test case 5C was used to study performance of the adaptive controller 10 during lower tie-line power flow scenarios. FIGS. 17A and 17B show the time response of the relative generator rotor angle deviations G ₄ -Gi and G4-G5. It should be noted that the generator angle deviation has a smaller magnitude than in test case 5A (i.e. FIGS. 15B and 15C). The adaptive controller 10 resulted in a smaller oscillation amplitude and a faster settling time compared to the conventional lead-lag controller, as expected.

[00179] The test case 5D was used to test performance of the adaptive controller 10 to the tie-line power flow reversal. The tie-line power flow on line 7-9 (i.e. from bus 9 to bus 7) was set to 180 MW by increasing the load on bus 7 and decreasing the load on bus 9. This type of operation is very likely to occur in a highly inter-connected power system. Although the power flow changed direction, the input to the supplementary controllers (i.e. the lead-lag and adaptive pole-shift controller 10) are kept the same as in the previous cases.

[00180] FIGS. 18A and 18B shows the d and G ₆ load angles, measured with respect to the G ₄ load angle for the disturbance in test case 5D. It should be noted that, in the case of the conventional lead-lag supplementary controller, a negative gain was again used, because of the reversal in power flow through tie-line 7-9. It can be seen from the responses again that the adaptive controller 10 self-adjusted without the need for any external logical element to effectively dampen the inter-area oscillations when the power flow reverses.

[00181] FIGS. 18C and 18D show the same time responses of FIGS. 18A and 18B for a positive gain. The time response of the conventional lead- lag controller case shows sustained oscillation when the feedback signal is as it is (i.e. with no modification). The conventional controller would need a separate triggering signal to detect the reversal of the power flow so that the gain can be adjusted to a negative value in order to achieve the proper operation for the controller. On the other hand, the adaptive controller 10, without any external interference, adjusted to the correct set of identified parameters, and showed stable control behaviour.

[00182] The test case 5E was used to study the performance of the adaptive controller 10 during a significant change in operating conditions. A significant change in operating point was achieved by disconnecting a 300 MW load on bus 9 at t = 0 sec, connecting the 300 MW load on bus 9 at t = 10 sec and again disconnecting it at t = 35 sec. The time response of the relative generator rotor angles G ₄ -Gi and G ₄ -G ₆ are shown in FIGS. 19A and 19B and demonstrate that the adaptive controller 10 provided better damping and faster settling time than the conventional lead-lag controller. It should be noted that the conventional lead-lag controller was optimized for test case 5A, and its performance deteriorated as the operating point changed to the new values used in test case 5E.

[00183] From the responses shown in FIGS. 15A to 19B, it can be seen that the adaptive controller 10 was very effective in damping the inter-area oscillations. As seen from FIGS. 15B, 15C, 16A, 16B, 17A, 17B, 18C, 18D, 19A and 19B, the adaptive controller 10 reduced the first swing and also effectively dampened the subsequent swings for various operating conditions.

[00184] The approximate damping factor for the three-area, six-machine test system was calculated using the Prony analysis tool provided in the commercial power system analysis software tool DSATools™. For this purpose, the tie-line power flow between buses 7-9 was taken under consideration.

[00185] The damping factors were calculated for the most dominant mode and are given in Table 6. The damping provided in the presence of the adaptive controller 10 was relatively higher than that of the no supplementary controller case and the lead-lag controller supplementary case. Furthermore, Table 6 shows that the adaptive controller 10 was very effective in damping the oscillations in the presence of the power system stabilizers.

Table 6: A roximate damping factor (in %) - Balanced TCSC

5D 2.44 -2.291 13.957

5E : 1.586 6.436 13.404

In presence of No PSSs PSSs PSSs+ Pole-shift

PSS, Case 5A 1.96 7.31 15.23 [00186] For the purpose of three-phase SSSC compensation studies, tie-line power flow through SSSC, between buses 7-8, was taken under consideration. The damping factors are presented in Table 7.

Table 7: A roximate damping factor (in %) - Balanced SSSC

[00187] It can be observed that the damping provided in presence of the adaptive controller 10 was significantly higher than that of the no supplementary controller case and the lead-lag supplementary controller case, which indicates the effectiveness of the adaptive controller designs taught herein.

[00188] For the time domain simulation studies that used the IEEE 12- bus system, a TCSC unit was placed near bus 7 between lines 7-8 of the test system 40 shown in FIG. 3C. The steady-state series capacitive compensation provided by the TCSC was 0.15 p.u. of the total line impedance. The power flowing through line 7-8 was taken as the input to the supplementary controller that was being tested. Three test cases (see Table 5) were studied to validate the performance of the adaptive controller 10. The disturbances were applied at t = 1 sec and the responses were plotted for a 0 to 10 sec time window. Moreover, in the following figures illustrating various time responses in the power system, the test cases of no supplementary controller, traditional lead-lag supplementary controller and adaptive pole-shift controller are denoted as "No suppl." in dotted lines, "Lead-lag" in lines with triangles, and "Pole-shift" in thick lines, respectively. [00189] The test case 5F represents a single line-to-ground fault at bus 3. This type of disturbance is most common in power systems. FIGS. 20A and 20B show the time responses of the tie-line power flow (line 7-8) and relative rotor angle deviation of G ₃ with respect to Gi for the disturbance in test case 5F. These simulation results demonstrate that the adaptive controller 10 provided better damping and faster settling time compared to the conventional lead-lag controller.

[00190] The identified system parameters, control signal generated using the adaptive pole-shift technique and the pole-shift factor for test case 5F are shown in FIGS 20C to 20F. The response plots of FIGS. 20C and 20D show the smooth parameter variation during and after the disturbance in test case 5F.

[00191] The test case 5G was used to study the performance of the adaptive controller 10 during a severe system disturbance. The disturbance was created by applying simultaneous three-phase fault at buses 3 and 6. These two simultaneous faults were applied to excite multi-mode oscillations in the power system. The lead-lag supplementary controller was tuned for this operating condition and the tuned parameters are given in Appendix C.

[00192] FIGS. 21 A and 21 B show the time response of the tie-line power flow (line 7-8) and the relative generator rotor angle deviation of generator G ₃ with respect to generator G It is evident from the responses that the power system exhibited poorly damped oscillations and both supplementary controllers helped to damp out the inter-area oscillations significantly faster than the case of not using a supplementary controller. The performance of the adaptive controller 10 in this operating condition was relatively better than simplex-tuned lead-lag controller.

[00193] The test case 5H corresponds to a three-phase system fault followed by a line (4-5) trip after 9 cycles. This test case was used to test the performance of the adaptive controller 10 during a delayed breaker operation (9 cycles) for clearing the faulted line. FIG. 22A shows the tie-line power flow through line 7-8 for the disturbance in test case 5H. FIG. 22B shows the relative generator rotor angle deviation of generator G ₃ with respect to G-i for the same disturbance. The responses plotted in FIGS. 22A and 22B demonstrate that even under such a severe situation, the adaptive controller 10 was able to damp the oscillations very effectively.

[00194] It is evident from the response plots in FIGS. 20A, 20B, 21A, 21 B, 22A and 22B, that the adaptive controller 10 provided better damping and faster settling time compared to the conventional lead-lag controller. Furthermore, the lead-lag controller was tuned for the disturbance in test case 5G, and hence its performance is better for this case but its performance deteriorated for other disturbances, whereas the operating condition did not affect the performance of the adaptive controller 10.

[00195] The severe disturbances in test cases 5G and 5H excited multi- mode oscillations. Prony analysis on the power flow on line 7-8 for test case 5G revealed the presence of 3 modes, as shown in Table 8. The approximate damping factors from Table 8 show that the adaptive controller 10 improved the multi-mode oscillation damping. The responses also showed that the third-order model used by the parameter identification module 12 and the adaptive control module 14 provided better damping compared to the lead-lag supplementary controller, even when inter-area modes were closely spaced.

Table 8: Prony analysis of power flow on line L7-8 (disturbance in case 5F)

Controllers

Modes

No suppl. Lead-lag Pole-shift

Mode l - f(Hz), 1.209 Hz, 1.185 Hz, 1.221 Hz,

Damping (%) 2.18% 4.10% 9.38%

Mode 2 - f(Hz}, 0.891 Hz, 0.901 Hz, 0.870 Hz,

Damping (%) 1.32% 1.81% 11.69%

Mode 3 - f(Hz), 0.762 Hz, 0.764 Hz, 0.660 Hz,

Damping (%) 4.14% 5.01% 32.74%

[00196] With these simulation tests, the effectiveness of the controller 10 in damping inter-area oscillations was seen on a three-area, six-machine test system as well as on the IEEE 12-bus test system. These simulation tests also showed that the robust recursive least square identification technique used by the parameter identification module 12 minimized parameter deviation during large disturbances (for example, faults, line switching, etc.) in a power system. This identification technique also helped to achieve effective control action along with the use of the adaptive pole-shift type control technique, and also overcame the problems encountered with conventional lead-lag controllers during the fault conditions that were tested. Prony analysis also showed that the controller 10 was more effective than a fixed parameter lead-lag controller in damping inter-area oscillations for a wide range of operating conditions for the test cases.

[00197] In an actual system, the controller 10 is implemented using a combination of hardware and software. The controller 10 also faces real-time implementation issues that have to be addressed such as the un-modelled noise that is present in the power system 20, proper communication protocols, communication delays between the controller 10 and the power system 20, as well as implementing the control technique to operate within the computational capacity of the processor that is used for implementation. Therefore, hardware-in-the-loop (HIL) tests of the controller 10 were conducted to further assess performance. In particular, a hardware realization of the controller was tested by developing a prototype controller 70 on a Texas Instruments DSP board (TMS320C6713 DSP Starter Kit DSK 6713) and using a Real-Time Digital Simulator (RTDS) platform to test the controller 70 performance in real-time. The RTDS provides very accurate models of the power system 20 and simulates the power system 20 on a realtime basis. The testing demonstrated that the adaptive controller 10 can be realized in real-time and can be advantageously used for a wide range of power system operation.

[00198] The RTDS unit from RTDS Technologies® is a fully digital, real- time power system simulator that is widely used in the industry. The system is capable of providing continuous real-time electromagnetic transient simulations at time steps of 50 ps. The RTDS unit can be used for development and dynamic performance testing of various control and protection devices as well as studying the impact of a controller on a power network without endangering real power system operation. It also helps to conserve resources since prime-mover energy is not required for testing devices.

[00199] The RTDS unit that was used had two 3 PC cards, two GPC cards, a Gigabit Transceiver Analog Input (GTAI) card, a Gigabit Transceiver Front Panel Digital input/output interface card (GTFPI), and a Workstation Interface Card (WIF). The WIF card facilitates communication between the RTDS platform and a host computer via an Ethernet link. Each of the 3PC cards and GPS cards are equipped with twenty-four 12-bit Digital to Analog (D/A) output ports.

[00200] The GTAI analog input interface has 12 differential channels in which the individual input channels are equipped with a first order anti-aliasing filter. The anti-aliasing filter has two manually selective cut-off frequencies of either 10.1 kHz or 84.2 kHz. The GTAI card was used to send an input signal to the RTDS unit. The analogue output channels available in each 3PC processor were used to obtain signals from the RTDS unit.

[00201] The DSP starter kit (DSK 6713) has input ports 72 including MIC IN and LINE IN, output ports 74 including LINE OUT and HP OUT and a 225 MHz TMS320C6713 Floating Point DSP 76 from Texas Instruments. The DSP 76 is capable of fetching eight 32-bit instructions per cycle or every 4.44 ns and is very well suited for numerically intensive algorithms. The DSK 6713 has a built-in stereo audio codec AIC23 86 that is capable of handling a stereo audio signal (e.g. two inputs 72 for left and right channels) and a stereo audio output (e.g. two outputs 74 for left and right channels). The codec 86 uses 16-bit ADCs and DACs to convert an analog signal to a digital signal and vice versa, and has a sampling frequency that can be selected to be 8, 16, 24 or 48 kHz. To avoid possible DC biases, the audio codec is equipped with a DC blocking capacitor (not shown) having a capacitance of 470 nF. [00202] FIG. 23A shows a block diagram of the DSK 6713 DSP board (the source is Digital Spectrum Incorporated, "TMS320C6713 DSK Technical Reference", 2003). It should be understood that this is one example of an implementation and other implementations may also be possible. Accordingly, some of the components of the controller 70 may be optional or replaceable as is known by those skilled in the art.

[00203] In this example embodiment, the controller 70 further comprises a CPLD 78, flash memory 80, and SDRAM 82 that are all coupled to the DSP 76 and a Memory Exp 84 that is coupled to the SDRAM 82. The controller 70 further comprises a multiplexer MUX that is coupled to the codec 86, the DSP 76 and the peripheral Exp 102 for routing signals therebetween.

[00204] The controller 70 also comprises a USB port 94, an embedded JTAG 96, a second multiplexer MUX 98 and an Ext. J TAG port 100. The MUX 98 coupled the embedded JTAG 96 and the Ext. JTAG to the DSP 76 for routing signals therebetween.

[00205] The controller 70 also comprises a power input PWR 90 for receiving power from an external source and a voltage regulator 92 that receives the external power and provides a power signal for use by the various components of the controller 70.

[00206] The controller 70 also comprises an LED 104, a DIP 106 and a Host Port Interface 108.

[00207] The DSK 6713 can be operated in a hardware interrupt mode for real-time applications. In this mode, the DSP 76 responds to hardware interrupts generated by the audio codec 86 for every received sample. The higher the sampling frequency that was selected, the smaller was the available time for signal processing between two consecutive interrupts. In order to maximize the time available between two consecutive samples, the lowest sampling rate (i.e. 8 kHz) was selected. At this sampling rate, the time available for the computation was 0.125 ms. If the adaptive controller method described herein takes more than 0.125 ms to run, the next sample data will be lost. Therefore, to limit the time consumed by a piece of software code to 0.125 ms, some longer pieces of codes (i.e. instructions) were divided into smaller parts.

[00208] The DSK 6713 board analog input/output interface having inputs 72 and outputs 74 is designed for audio processing applications and so it only has one stereo analog input and one stereo analog output channel. Therefore in this example embodiment, the number of available input/output channels imposes a restriction on the number of electrical signals that can be exchanged between the RTDS and the DSP board. To overcome this limitation, the supplementary signal (i.e. power deviation) is passed to the DSP 76 from the RTDS instead of calculating the power deviation inside the DSP controller. This simplified the signal exchange between the RTDS and the DSP 76 since this involved passing only a signal necessary for the supplementary controller, i.e. power deviation, and receiving only the controller output. The supplementary control signal was then added to the main TCSC controller inside the RTDS.

[00209] Furthermore, the DSP codec I/O signal bandwidth was between 20 Hz to Fs/2, where Fs is the sampling frequency of the codec. This bandwidth imposed another restriction on communication between the DSP 76 and the RTDS. The inter-area frequency was in the range of 0.1 to 0.8 Hz and since the audio codec 86 attenuated signals below 20 Hz, communication between the RTDS and the DSP 76 at the frequency of interest was not directly possible. This problem was solved using an amplitude modulation technique known as Double Sideband-Suppressed Carrier (DSB-SC) modulation. The DSB-SC modulation scheme was selected because of its simplicity in hardware implementation. The low frequency oscillation signal was modulated with a carrier frequency of 0.5 kHz and sent to the DSP. Similarly, the output of the DSP 76 was modulated by the same carrier and sent back to the RTDS.

[00210] A supplementary signal of 0.8 Hz was modulated using the DSB-SC modulation technique. To mimic the oscillation in the power system 20, the magnitude of the signal was selected to be similar to that of an exponential decaying signal. The carrier signal was chosen to be a 0.5 kHz sinusoidal signal. The supplementary signal and the carrier signal are expressed in equations 23 and 24 respectively.

y(t) = e ^_a8t cos(2 x π x 0.8 x t) (23) v _c (t) = cos(2 x π x 500 x t) (24)

For the demodulation process used in the DSB-SC technique, a second-order Butterworth filter having a cut-off frequency of 5 Hz was used. The original supplementary signal had a maximum amplitude of 1 .0 unit and the demodulated signal had a maximum amplitude of 0.5 units.

[00211] The simulation facility processing capability that was used allowed for modelling a two-area test system 109 with a TCSC device. The test system 109 is shown in FIG. 23B. It includes two generating areas interconnected through a 210 km long transmission line. Each area consists of two generators. The exciter and governor models were used to regulate the terminal voltage and the input mechanical power. The tie-line was partially series compensated (capacitive) using a three-phase TCSC module. The steady state series capacitive compensation of the TCSC was 0.2 p.u. of the total line reactance of 1 1 1 .1 Ω. The test system 109 had one inter-area mode of oscillation at 0.5 Hz. The test system 109 is a fairly-well sized system for the experimental studies that needed to be conducted for HIL testing.

[00212] Five test scenarios listed in Table 9 were considered for the real-time hardware testing of the adaptive controller 10. These case studies cover a wide range of operating conditions and different type of disturbances. Table 9: Test cases for the two-area system studies in RTDS

Approximate power Disturbance description Disturbance type flow on line 7-9

315 MW 3 cycle, three-phase fault at bus 7 6A

315 MW 6 cycle, three-phase fault at bus 7 6B

150 MW 3 cycle, three-phase fault at bus 7 6C

-230 MW 3 cycle, three-phase fault at bus 7 6D

167 MW Load rejection at bus 9 (Load flow 6E

of case A), at t=lsec. [00213] The two area system 109 including the TCSC was developed using RSCAD software. The RSCAD software provided the ability to set up simulations as well as to control and modify system parameters during simulation, data acquisition, and analysis of the results. The DRAFT module of the RSCAD software suite was used for developing power system models. The developed model was then uploaded to the RTDS unit using the RunTime module. The RunTime module also allows a user to run and control the simulation in real time.

[00214] The synchronous generators were modelled in a d-q-0 reference frame. The generators were equipped with IEEE type ST1 exciters and IEEE type 1 turbine-governors to mimic real power system operation. The transmission line was represented as a lumped impedance model.

[00215] Three units of a single-phase TCSC model were used on each phase for series compensation. The closed loop impedance control method was implemented for the steady state TCSC impedance control (R. Mathur and R. Varma, Thyristor-Based FACTS Controllers for Electrical Transmission Systems, 1 ^st ed. IEEE Press, New York, 2002). The Discrete Fourier Transform (DFT) technique was used to estimate line current and voltage across the TCSC. The estimated components were used to calculate the TCSC impedance. A proportional-integral controller was used to regulate the firing angle of the TCSC to achieve the desired TCSC impedance.

[00216] For inter-area oscillation damping control, the power deviation through line 7-9 was used as a supplementary signal. This supplementary signal was modulated using a sinusoidal carrier signal having a frequency of 0.5 kHz. The machine and line parameters used for the study are given in Appendix E. In the experimental setup, the RTDS provided the signals P _mod and Vcarner to the DSK 6713 and the DSK 6713 provided the signals U _s , _m0 d and Vcarner to the RTDS. The model of the power system was downloaded to the RTDS using the RSCAD program.

[00217] For the controller implementation on the DSP 76, Code Composer Studio (CCStudiol ) version 3.1 was used to develop the controller algorithm. The CCStudiol provides a facility to write source code in machine assembly language and/or C/C++ language.

[00218] The AIC23 audio codec 86 uses 16-bit signed integer I/O channels with a maximum allowable input signal level of 1 V rms. However, the DSK C6713 contains a potential divider with a gain of 0.5 between the Line IN input ports which increases the maximum allowable input signal level to 2 V rms. Accordingly, a safe input voltage level of ±1 V peak-to-peak was used for input/output from the DSP board. The input level gain was set-up such that a peak-to-peak input of ±1 V to the DSP 76 corresponds to 0.5 p.u. of the power deviation, whereas an output level of ±1 V peak-to-peak corresponds to a ±1 p.u. TCSC impedance change. The input samples received from the codec had a 16-bit signed integer format and was converted to voltage values using ADC conversion for the processing inside the DSP 76.

[00219] Two analog input signals (the modulated power deviation and the carrier signals) from the RTDS unit were received via the audio codec 86 at a sampling frequency of 8 kHz. The signals were processed by a demodulation algorithm in order to extract a low frequency power oscillation. Since the frequency of the inter-area oscillation was around 0.5 Hz, the input samples received from the codec 86 were further downsampled to about 10 Hz. This means that only one input data sample out of 800 (8 kHz/10 Hz = 800) received from the input codec was processed. To minimize the effect of aliasing, a second order anti-aliasing Butterworth filter with a cut-off frequency of 10 Hz was used. [00220] The output of the control algorithm was modulated using a carrier signal and converted back to an integer count using DAC conversion. The controller output was demodulated inside the RTDS unit and used as a supplementary control signal for the TCSC.

[00221] It was found that it was better to execute the software code for the adaptive pole-shift control technique before the software code for the constrained parameter identification technique in order to minimize the delay that was introduced by the adaptation process because the parameter update and controller design were done after sending the control signal to the plant. The code execution time was maintained within the 0.125 ms limit by executing only a certain portion of code (i.e. instructions) in each interrupt, as is described in further detail below. It is a valid approach, since the next set of data will arrive only after 799 interrupts for the sampling rate of 8 kHz and the 10 Hz downsampling that was considered in the real-time hardware tests.

[00222] Referring now to FIGS. 23C-1 and 23C-2, shown therein collectively is an example embodiment of an adaptive control method 1 10 that can be used with the hardware of FIG. 23A. Once the method 1 10 starts at 1 12, the DSP hardware is then booted up (i.e. activated) at 1 14. The method 1 10 then proceeds to 1 16.

[00223] At 116, the method 50 determines whether there is an interrupt. If not, the method 110 goes to 144 at which point it determines if it should stop operation. If this is true, the method 1 10 ends its operation at 146. If the comparison at 144 is not true, then the method 1 10 goes to 148 and 150 to determine if the counter interrupt_no should be reset to 0. The counter interrupt_no is rest to 0 if the counter interrupt_no is larger than a threshold related to the sampling rate, which is 799 in this example because the data is being downsampled to a sampling rate of 10 Hz from a sampling rate of 800 Hz and therefore skips about 800 data points. Afterwards, the method 110 goes to 1 16 to check if there is an interrupt.

[00224] At 1 16, if there is an interrupt, the method 1 10 goes to 1 18 to determine whether various parameters of the control method 1 10 have been initialized. If the comparison at 1 18 is false, the method 1 10 goes to 120 at which point various variables, only a portion of which are shown for ease of illustration, are initialized. The method 1 10 goes to 122. If the method 1 10 determines that the variables have been initialized at 1 18, then the method 1 10 goes to 122.

[00225] At 122, the method 1 10 performs demodulation and anti-aliasing filtering on the input data and control signal as previously explained. The counter interrupt_no is also incremented by 1. The method 1 10 then proceeds to 124.

[00226] At 124, the method 1 10 checks to see if the counter interrupt_no is equal to 1. If this comparison is true, then the method 1 10 goes to 126 at which point the demodulated input data and control signal data is read. If the comparison at 124 is false, then the method 1 10 goes to 128.

[00227] At 128, the method 1 10 determines whether the counter interrupt_no is equal to 2. If this comparison is true, then the method 1 10 proceeds to 130 where a first portion of the adaptive pole shift control technique is performed. The first portion of the adaptive pole shift control technique involves performing basic calculations such as preparing input matrices and inverting the M matrix, for example. The method 1 10 then proceeds to 132.

[00228] If the comparison at 128 is not true, the method 110 goes to 132 to and determines if the counter interrupt_no is equal to 3. If this comparison is true, the method 1 10 proceeds to 134 where a second portion of the pole shift control technique is performed. The second portion of the adaptive pole shift control technique involves performing operations such as root calculation of the characteristics equation, optimization and control signal calculation, for example. The method 1 10 then proceeds to 136.

[00229] If the comparison at 132 is false, then the method 1 10 proceeds to 136 and determines if the counter interrupt_no is equal to 4. If this comparison is true, the method 1 10 proceeds to 138 where the robust (i.e. constrained parameter) RLS estimation is performed to determine new values for the model parameters. The method 1 10 then proceeds to 140. If the comparison at 136 is false, the method 1 10 proceeds to 140 without performing the robust RLS estimation technique.

[00230] At 140, the control signal data is modulated and then output at 142 as the control signal U _c . The method 1 10 then proceeds to 144 where it checks whether it should stop execution as was previously described.

[00231] It should be noted that while the example embodiment of the method 1 10 used particular values for comparisons with the counter interrupt_no, in alternative embodiments, other values may be used as is known by those skilled in the art.

[00232] In testing, the software code (i.e. instructions) for the constrained parameter identification technique was found to take about 0.10 ms to execute on the DSP 76. To improve efficiency, the identifier coefficient values were frozen if the difference between the maximum and minimum input values in the last Ns samples, such as the last 25 samples for example, was below a threshold value. The threshold can be selected to be about 1 % of the peak-to-peak value of the input signal. The number of samples Ns used for calculating the power deviation (e.g. 25 samples) was selected such that it included slightly more than 1 cycle of data for a 0.5 Hz sampled signal. The frozen coefficients were used as the initial value for the next disturbance which helped parameter estimation to converge faster. If there was a disturbance in the system, the supplementary signal deviation will be higher than the threshold value, which activates the software code (i.e. instructions) that implements the constrained parameter identification technique.

[00233] The software code (i.e. instructions) for the adaptive pole-shift control method 1 10 was implemented in C programming language for the real-time hardware testing. Optimization of the cost function expressed in equation 21 was carried out using a Brent's optimization method although other optimization methods can be used in other embodiments. Minimization of the cost function of equation 21 yielded an optimal pole-shift factor a _opt which was used to compute the optimal control signal required to minimize the deviation between the predicted next time-step plant output AP _L (t + 1 ) and the reference plant output AP _L , _r ef(t ^{+ 1} )·

[00234] Brent's optimization method was selected because of its simplicity in implementation. It is very good for predefined boundaries of the roots, which in this case are the stability constraints f- - (l + - (1 - A)],

L A A

where Λ is the safety factor. The safety factor Λ = 0.05 was considered for these HIL tests. Also, the weighting coefficient (μ) and steady-state pole-shift factor (a _ss ) (please refer to equation 21 ) were selected for the HIL tests as 0.5 and 0.7, respectively.

[00235] The weighting coefficient was set as μ = 0.5 to give half as much weight to the pole-shift factor deviation ( - a _ss ) compared to the output deviation y - y _re f in the optimization cost function. This signifies that the oscillation damping is more important in this application. The value of μ can be selected differently for different applications since the value of μ determines the performance for the power control equipment (e.g. the FACTS device). If μ is large then the control signal u(t) has more impact on the output y(t) whereas if μ is small then the control signal u(t) has less impact on the output y(t). Some utilities permit full control whereas other utilities only permit partial control. Accordingly, the value for μ is chosen beforehand according to the needs of the utility or power company that is operating the power system.

[00236] The steady-state pole-shift factor was set as a _ss = 0.7. If _ss is chosen too small, the effort of the adaptive controller 70 will be higher to achieve significant damping and if a _ss is set to be close to unity, the adaptive controller 70 merely moves the closed loop poles within the unit circle and the damping improvement will be very little. The selected value for a _ss was chosen based on offline simulation studies that showed it gave the best performance overall for the various test cases.

[00237] In the real-time hardware testing, the complete pole-shift controller algorithm took about 0.18 ms to execute, which is larger than the sampling interval of the input codec. So the controller code (i.e. instructions) was broken down into two sections. The first section of the instructions performed basic calculations such as preparing input matrices and inverting the M matrix. The second section of instructions performed root calculation of the characteristics equation, optimization and control signal calculation. As described, the two sections of instructions are allowed to run in two consecutive interrupts. This process reduced the code execution time well below the sampling time interval, i.e. 0.092 ms for the first section, and 0.088 ms for the second section.

[00238] The effectiveness of the adaptive controller 70 for damping inter- area oscillation was demonstrated on a two-area test system 109 consisting of a TCSC compensated transmission line as shown in FIG. 23B. The deviation in power flow through line 7-9 was considered as the plant output. The supplementary control signal U _s was considered as the plant input. The performance of the robust (i.e. constrained) RLS estimation and adaptive pole-shift control methods were evaluated for the various disturbance scenarios listed in Table 9.

[00239] The identified system parameters for the disturbance in test case 6A are shown in FIGS. 24A to 24C. FIG. 24A shows the identified plant parameters using the robust RLS algorithm described herein whereas FIG. 24B shows the estimated plant parameters for the same disturbance using the conventional RLS algorithm. The estimated errors for these two algorithms are plotted in FIG. 24C. It is evident that the robust identification method helps to identify system parameters smoothly even during large disturbances (i.e. large errors). In the case of the conventional RLS algorithm, the large deviation in estimation error caused significant deviation in the estimated parameters. This rapid change in parameter values caused undesirable controller output, which led to poor damping as seen in FIG. 25A.

[00240] For each case listed in Table 9, two sets of studies were carried out to evaluate the effectiveness of the adaptive controller 70 for damping inter-area oscillations. The first set of studies was done for when the adaptive controller 70 was disabled, denoted by "No suppl.", and the second set of studies was done for when the pole-shift controller was in service, denoted by "Pole-shift".

[00241] The test case 6A represents a 400 MW tie-line power flow through line 7-9 and was considered as a base operating condition for the rest of the test cases. FIGS. 25B to 25E show the time response of the tie-line power flow (line 7-9) and the relative generator rotor angles for the disturbance in test case 6A, respectively. The rotor angle deviations between Gi-G ₂ (FIG. 25C) and G4-G3 (FIG. 25E) represent the local oscillations at area 1 and area 2, and the rotor angle deviation between G^-G^ (FIG. 25D) represents the inter-area oscillation between area 1 and area 2, respectively. The responses without and with the adaptive controller 70 are presented in dotted and thick lines respectively. It should be noted that for the inter-area oscillation it takes more than 20 sec to settle the oscillations when the adaptive controller 70 is disabled. The introduction of the adaptive controller 70 significantly reduced the settling time since the oscillations were settled within 8 sec of the fault being cleared.

[00242] The time responses of the control signal generated by the adaptive controller 70 and the TCSC reactance for the case of FIGS. 25B to 25E are shown in FIGS. 26A and 26B, respectively. The corresponding open and closed-loop poles of the identified system are plotted in FIG. 26C. The open loop and closed loop poles are located at [0.8 ± j0.2966, -0.3058] and [0.56 ± j0.2076, -0.2140] respectively. The steady-state pole-shift factor was found to be σ = 0.69. It was evident from FIG. 26C that the adaptive controller 70 moved the open loop poles closer to the origin and improved the stability and the damping of the power system.

[00243] To study the real-time hardware performance of the adaptive controller 70 under a severe disturbance, a 6 cycle three-phase fault was applied at bus 7 as the operating condition of test case 6B. The longer duration fault caused a large oscillation on the power network. The effectiveness of the adaptive controller 70 for such a fault was demonstrated in FIGS. 27A and 27B. It should be noted that the magnitude of the oscillation in δ ₄ ι reached as high as 80° as compared to 50° for a 3 cycle fault of the disturbance of test case 6A.

[00244] Test case 6C was used for real-time hardware testing of the adaptive controller 70 for lower tie-line power flow than the designed value. Accordingly, the tie-line power flow from bus 7 to bus 9 was set to 150 MW by decreasing the load on bus 9. This kind of operating condition is very common on a power system. In this operating condition, the tie-line power oscillation (i.e. feedback signal) may have a lower magnitude than that of a 400 MW power flow. Even in such a condition, the adaptive controller 70 will adjust the gain automatically yielding optimal damping.

[00245] FIGS 28A and 28B show the tie-line power flow and relative generator rotor angle deviation δ ₄ ι time responses for the disturbance of test case 6C. The performance of the adaptive controller 70 was found to be similar to the performance seen for test case 6A.

[00246] Test case 6D was used for real-time hardware testing of the adaptive controller 70 for a change in reverse power flow condition through the tie-line. This type of operation is very likely to occur in a highly interconnected power system. For this test case, the tie-line power flow on line 7-9 (from bus 9 to bus 7) was set to 230 MW by increasing the load on bus 7 and decreasing the load on bus 9. The input to the adaptive controller 10, i.e. deviation of the power flow through line 7-9, was kept the same as in the earlier cases.

[00247] FIGS. 29A and 29B show the tie-line power flow and relative generator rotor angle deviation δ ₄ ι time responses for the disturbance in test case 6D. The responses show that the adaptive controller 70 significantly improved the settling time. Accordingly, the performance of the adaptive controller 70 was not affected by the change in operating condition and the reversal of the feedback signal. [00248] Test case 6E was used for real-time hardware testing of the adaptive controller 70 for severe disturbances in the power system. In this case, a 167 MW load was disconnected from bus 9 at t = 1 sec which caused a severe disturbance on the network and shifted the steady-state generator operation to a new equilibrium point. The tie-line power flow and the relative generator rotor angle (G ₄ -Gi) time responses are shown in FIGS. 30A and 30B respectively. It can be observed from these responses that the tie-line power flow decreased significantly and the relative generator rotor angle also decreased in a similar fashion. However, even in such severe situations, the adaptive controller 70 was able to dampen the oscillations effectively.

[00249] A comparison of the time responses shown in FIGS. 25B through 30B clearly demonstrates the benefit of the adaptive controller 70 for damping inter-area oscillations. The hardware test results show that the adaptive controller 70 reduced the first swing oscillation and also effectively dampened the subsequent swings in all test cases 6A to 6E.

[00250] The calculation of the damping coefficients of the power flow on Line 7-9 gives an approximate representation of the degree of damping provided by the adaptive controller 70 to the inter-area oscillation mode. For this purpose, the approximate damping factor of the dominant mode in the power flow on Line 7-9 was calculated for all of the study cases using Prony analysis. The analyzed data of the tie-line power time response lies within a 1.1 to 19 second time window.

[00251] The approximate damping factors obtained using Prony analysis are presented in Table 10 for all the test cases of Table 9. Table 10 summarizes the results for the test cases when: (i) the adaptive controller 70 was disabled, and (ii) the adaptive controller 70 was enabled. It can be seen that the system exhibited poor damping of around 5% when the adaptive controller 70 was disabled. However, when the adaptive controller 70 was enabled, the system exhibited significantly improved damping of around 12%. It should also be noted that the damping provided by the adaptive controller 70 was very consistent for all operating conditions. Table 10: Approximate damping for the inter-area oscillation mode

Pole-shift disabled Pole-shift enabled

Cases

Freq. (Hz) Damp. (%) Freq. (Hz) Damp. (%)

6A 0.497 5.430 0.471 13.661

6B 0.496 5.119 ¹ 0.458 11.054

6C 0.545 3.180 0.465 13.249

6D I 0.480 5.181 0.512 13.552

6E 0.500 5.123 0.540 12.721

[00252] The real-time execution of the identification and controller codes in the DSP was achieved using hardware interrupt mode. The interrupt was generated every 0.125 ms and therefore imposed a maximum limit on the code execution time. To limit the code execution time, longer code (i.e. instructions) segments were divided into smaller sections and each section was sequentially evaluated in consecutive interrupts. For example, the constrained parameter identification instructions took 0.1 ms to execute whereas the pole-shift control instructions took about 0.18 ms to execute. As the controller instructions execution time was higher than the interrupt interval, the controller instructions was further divided into two sections, with each section taking 0.092 ms and 0.088 ms, respectively.

[00253] The Prony analysis revealed that the approximate damping contribution by the adaptive controller 70 was very consistent (at around 12%) for all the operating conditions under study. The experimental results of the studies that were conducted demonstrate the effectiveness of the adaptive controller 70 in damping inter-area oscillations for a wide range of operating conditions as well as the practical applicability of the adaptive controller 70 in power systems as the adaptive controller 70 was computationally efficient and comparably simple to implement in DSP hardware compared to other conventional techniques that use a higher order model for the plant or more sophisticated signal processing techniques. However, a controller has to be finally implemented on a piece of hardware and thus hardware implementation issues such as the code (i.e. instructions) execution time limits and communication between the RTDS and the DSK 6713 should be studied. Furthermore, it was found that the adaptive controller techniques described herein that use a robust recursive least-square identification algorithm provided consistent damping for a wide range of operating conditions.

[00254] Referring now to FIG. 31 , shown therein is a block diagram of an example embodiment of a controller 150 that utilizes the adaptive control techniques described herein. The controller 150 can output the control signal U _c to the FACTS device and receive the system output Ay(t) as was shown for the controller 10 in FIG. 1.

[00255] The controller 150 comprises a processing unit 152, an interface unit 154, an isolation and preprocessing stage 156, a user interface 158, a display 160, a power unit 162, a wireless unit 164, an isolation and post-processing stage 166 and a memory unit 168. Some of these components can be optional in various embodiments such as, for example, the wireless unit 164, the display 160 and the user interface 158. The memory unit 168 stores software code (i.e. instructions) for implementing an operating system 170, various programs 172, a control module 174 and databases 176, which may be optional. The controller 150 is usually a standalone device with dedicated hardware and associated software and firmware that is configured to provide adaptive control for the power system 20 as was described herein for the controller 10 or the controller 70. However, in some cases it can be incorporated into a power control center for the power system 20.

[00256] The processing unit 152 controls the operation of the controller 150 and can be any suitable microprocessor, microcontroller or digital signal processor that can provide sufficient processing power. For example, the processing unit 152 may be a high performance general processor. In alternative embodiments, the processing unit 152 can include more than one processor with each processor being configured to perform different dedicated tasks. In alternative embodiments, specialized hardware can be used to provide some of the functions provided by the processing unit 152 such as filtering, for example.

[00257] The interface unit 154 can be any interface or communication port that allows the controller 150 to communicate with other devices or computers. In some cases, the interface unit 154 can include at least one of a serial port, a parallel port or a USB port that provides USB connectivity. The interface unit 154 may also include at least one of an Internet or local area network connection through an Ethernet, Firewire or modem connection or through a digital subscriber line. For example, the interface unit 154 may be used to implement a communication protocol to allow the controller 150 to be part of a SCADA system or any other communication system. Examples of communication ports include, but are not limited to, a MODBUS interface, an RS232/RS485 interface, a TCP/IP interface, an IEC 61850 interface and the like. Various combinations of these elements can be incorporated within the interface unit 154 in various embodiments.

[00258] The isolation and preprocessing stage 156 is used to protect the internal processing components of the controller 150 from any voltages or currents in the power system 20 that may damage these components. The isolation and preprocessing stage 156 also includes components that are used to preprocess the input data so that it can be processed by the processing unit 152. The isolation and preprocessing stage 156 generally includes isolation elements such as voltage and current transformers, voltage and current reduction elements (such as a potentiometer), filters and at least one Analog to Digital Converter (ADC). Some of these elements may be provided in data channels and the controller 150 may have several of these data channels. In some embodiments, the ADC may be part of the processing unit 152.

[00259] The controller 150 receives various types of information such as current and voltage information about the portion of the power system 20 that it is controlling to counteract against particular disturbances. The controller 150 typically receives this information via the interface unit 154; however, it may also receive some input data via the wireless unit 164. Since the controller 150 is connected to the power system 20 to receive this information, the isolation elements are used to provide isolation from dangerous voltages or currents from the power system 20. For example, a voltage transformer can be used as an isolation element in order to reduce input voltage levels as well as provide electrical isolation to the rest of the controller 150. Other elements can also be used for isolation such as a metal oxide varistor at an input of an isolation transformer to provide protection against large transients in the power system 20.

[00260] Since the current and voltage information are provided by instrument transformers, this information typically has a large magnitude that is not suitable for the processing unit 152. Accordingly, the isolation and preprocessing stage 156 includes components that can be used to scale down the input voltage and current information to a level that is suitable for use with the processing unit 152. In some cases, the isolation elements, such as transformers, can be used to provide at least a portion of this scaling. Further scaling can be provided by various other components such as a potentiometer, for example. In some embodiments, a current transformer can also be used to reduce the magnitude of input currents, and a resistor can be coupled to the secondary winding of the current transformer to convert the input current to a corresponding input voltage.

[00261] Filtering is used after isolation and scaling in order to avoid aliasing when the input data is sampled by an ADC. Low pass filters are used to accomplish this task as well as limit the effects of unwanted noise at higher frequencies. The amount of filtering that is needed depends on the nature of the tasks being performed by the controller 150 and the type of data which is being sensed.

[00262] In some embodiments, the isolation and preprocessing stage 156 may also downsample the input data before sending it to the processing unit 152 for processing as was described for the real-time hardware tests that were performed. In an alternative embodiment, the isolation and preprocessing stage 156 may not provide a downsampling function while the processing unit 152 may downsample the input data prior to processing it. In another alternative embodiment, downsampling may not be used, depending on the sampling rate that is used in ADC.

[00263] The user interface 158 allows an individual, such as a technician, to set certain parameters for the controller 150. Accordingly, the user interface 158 can include at least one of rotary dials, switches, buttons, slide switches, a keyboard, a touch screen, a thumbwheel, and the like depending on the particular implementation of the controller 150. In some cases, the user interface 158 may be optional if input to the controller 150 can be provided in some other fashion, such as through a data port.

[00264] The display 160 can be any suitable display that provides visual information depending on the configuration of the controller 150. For instance, the display 160 can be a monitor, an LCD display, a series of lights, instrumentation outputs and the like. The display 160 may be used to show settings, and real-time measurements of current/voltage in the power system protection zone corresponding to the controller 150.

[00265] The power unit 162 can be any suitable device that provides power to the various components of the controller 150. For example, the power unit 162 may be a power supply, a power adaptor that is connected to a power supply, a rechargeable battery pack and the like depending on the implementation of the controller 150 as is known by those skilled in the art.

[00266] The wireless unit 164 is optional and can be a radio that communicates utilizing the CDMA, GSM, GPRS or Bluetooth protocol according to standards such as IEEE 802.1 1a, 802.1 1 b, or 802.1 1 g. The wireless unit 164 can therefore be used by the controller 150 to communicate with other controllers, protective relays or a control center for the power system 20, which can be used in various tasks, such as providing coordinated action when dealing with a large disturbance in the power system 20. [00267] The isolation and post-processing stage 166 typically comprises a Digital to Analog Controller (DAC), scaling elements and isolation elements. Accordingly, the isolation and post-processing stage 166 receives output data from the processing unit 152, post-processes the data and then sends the post-processed data to the interface unit 154 or the wireless unit 164 depending on the nature of the output data. Scaling elements that can be used in the isolation and post-processing stage 166 include amplifiers and transformers. Isolation elements that can be used in the isolation and postprocessing stage 166 include voltage transformers and current transformers.

[00268] The memory unit 168 can include RAM and flash memory elements as well as other storage elements such as disk drives, hard drives and ROM. The memory unit 168 is used to store the operating system 170 and the various programs 172 as is commonly known by those skilled in the art. For instance, the operating system 170 and some of the programs 172 provide various basic operational processes for the controller 150. The programs 172 include various utility programs as well as user programs so that a user can interact with the controller 150 including running various diagnostic programs as well as downloading and sending data.

[00269] The memory unit 168 also stores the control module 174 and one or more databases 176. The control module 174 is generally used to send a control signal to hardware device, such as a FACTS device, that is used to stabilize a power system 20 when a disturbance occurs in the power system 20. The control module 174 generally implements the functionality of the parameter identification module 12 and the adaptive control module 14. Accordingly, the controller 150 can be integrated to work with a power system 20 as was shown in FIG. 1 for the adaptive controller 10. The one or more databases 176 can also store other information required for the operation of the programs 172 or the operating system 170, such as dynamically linked libraries and the like.

[00270] Referring now to FIG. 32, shown therein is a flowchart of an example embodiment of a control method 200 that can be used with the controller 150 or another controller that operates according to one of various embodiments of the adaptive control and constrained parameter identification techniques described herein. The control method 200 implements the functionality that was described for the parameter identification module 12 and the adaptive controller module 14.

[00271] The method 200 is executed at 202 and initializes its parameters at 204 including those used to model the plant and to provide feedback control in the case of disturbances. The method 200 then monitors the power system 20 at 206 which involves generating an estimated output Ay(t) and comparing the estimated output Ay(t) to the actual system output Ay(t) to determine the prediction error ε(ί). The estimated output A (t) is generated based on the plant model and the control signal Uc as was previously described for the parameter identification module 12.

[00272] At 208, the method 200 determines if the prediction error e(t) is greater than a threshold. If this comparison is false, then the method 200 goes to 206 and continues to monitor the power system 20 for disturbances. If the comparison at 208 is true, then the method 200 goes to 210.

[00273] At 210, the method 200 determines new values for the parameters of the model based on the prediction error e(t) and the control signal U _c . A constrained parameter identification technique is used so that the values of the parameters do not change rapidly and do not become too large too fast. For example, the constrained RLS estimation method described earlier for the parameter identification module 12 may be used.

[00274] At 212, the parameters of the model (which were just updated at 210) are then used to determine the new value of the control signal U _c . This can be done using the adaptive pole-shifting control technique that was described for the adaptive control module 14.

[00275] At 214, the new control signal value is sent to a control device, such as but not limited to a FACTS device for example, that is used to stabilize the operation of the power system based on the control signal U _c when disturbances occur.

[00276] At 216, the method 210 determines whether it should stop monitoring the power system 20. If this determination is true, then the method 210 proceeds to 218 and ends. If the determination at 216 is false, then the method 210 proceeds to 206 to continue to monitor the power system 20.

[00277] The adaptive control techniques described herein can be used as a control system for generation, transmission and distribution control devices (e.g. FACTS devices and the like) that are installed in a power system 20.

[00278] The adaptive control techniques described herein are a new type of series compensation scheme that are capable of improving power system dynamics by damping sub-synchronous resonance and low frequency oscillations. The techniques are economical compared to full three-phase FACTS devices and use lower numbers of switching components. Furthermore, these techniques utilize a robust parameter constrained identification method and an adaptive control method which allows for enhanced system damping over a wide range of the operating conditions of the power system 20. In particular, based on test data, the robust parameter constrained identification method has the ability to handle large disturbance conditions such as three-phase faults as well as smooth the out the parameter variations that are fed to the pole-shift controller. Also, the adaptive control technique described herein overcomes the need for designing fixed parameter type controllers and has the ability to adjust its own parameters on-line in real- time to yield satisfactory control performance over wide operating conditions of the power system 20. Furthermore, the pole-shifting technique moves the poles inside the unit circle in the z-plane so as to achieve damping of oscillations in the power system 20 during faults and other major disturbances in the shortest period of time. The effectiveness of the techniques described herein has been demonstrated using: (1 ) a non-linear discrete system, (2) a three-area six-machine power system with a TCSC, and (3) an IEEE 12 bus power system configuration with a TCSC.

[00279] The adaptive control techniques described herein have also been described by authors Dipendra Rai, Ramakrishna Gokaraju and Sherif O. Faried in the article: "Adaptive Control Using Constrained RLS and Dynamic Pole-Shift Technique for TCSCs", IEEE Transactions on Power Delivery, Vol. 29, No. 1 , February 2014, the contents of which are hereby incorporated by reference.

[00280] It should be understood that the adaptive control techniques described herein may also be used in a Self- Tuning (ST) control scheme for a Unified Power Flow Controller (UPFC). In particular, a conventional UPFC can be augmented by supplementing its PI controller with an ST feedback comprising an identifier, such as an adaptive Constrained Recursive Least Squares (CRLS) parameter identifier module and a Pole-Shift (PS) controller, which are similar to the parameter identifier module 12 and the adaptive control module 14 of FIG. 1 that have been described herein. Such a control system for use with a UPFC has been described by authors Uvri Malhotra and Ramakrishna Gokaraju in the article: "An Add-On Self Tuning Control System for a UPFC Application", IEEE Transactions on Industrial Electronics, Vol. 61 , No. 5, May 2014, the contents of which are hereby incorporated by reference.

[00281] For the UPFC, the dynamic oscillations in the power system may be modeled using a lower order model, such as a third-order ARMA model. The CRLS parameter identifier module smooths out parameter variations efficiently during large disturbances (i.e., large power swings) following a fault condition.

[00282] Referring now to FIG. 33, shown therein is an example embodiment of a UPFC 250 installed at the sending-end (s) of a bus that serves as an inter-tie of a two area transmission network. In modeling this network with the UPFC 250, it has been realized herein that it is important to include the switching characteristics especially when studying the interaction of the converters and the power system in response to system faults. Thus, according to the teachings herein, the UPFC converter 250 may be modeled in PSCAD-EMTDC using a two-level, six-pulse voltage sourced converter (Vsc) topology with the thyristor firing pulses generated through a pulse width modulation (PWM) switching technique.

[00283] A UPFC consists of two identical converters, the shunt V _S c and the series V _S c connected in shunt and series with the AC transmission network, respectively. Each converter is connected to the AC system through corresponding coupling transformers, T _sh and T _se , respectively, and are operated from a common DC link supported by a DC capacitor C.

[00284] In general, the operation of a UPFC is governed by four control inputs m _S h, S _S h, m _se , <5 _se that are provided by a control system 252. The four control inputs m _S h, ¾/?, m _se , <5 _se are used to provide firing commands that are given to thyristor switches where m and δ denote converter modulation index and phase angle, respectively.

[00285] The primary function of the UPFC 250 is provided by the series Vsc that generates a voltage V _in f = \ V _ini \ sin(ai _f - 0 _/n y) at fundamental frequency (ω) with variable amplitude (0 < \ V _inj \≤ \ V _inj \ _max ) and phase angle (0 < 0jnj≤ 277) that is added to the AC system by the series coupling transformer, T _se . The shunt Vsc serves two purposes. The shunt Vsc provides the real power demanded by the series V _S c from the AC system via the common DC link. This is made possible by keeping the DC link capacitor C charged at all times, replicating an energy source for real power exchange. The shunt V _S c also acts as an independent reactive power compensator thereby maintaining the system bus voltage at a specified value. With its four control inputs, the UPFC 250 can be commanded to force an appropriately varying line power to effectively damp power oscillations.

[00286] The control of the UPFC 250 can be divided into two parts: the control of the shunt V _S c and the series V _S c where each V _S c is controlled by two control inputs (modulation index m and phase angle δ). The shunt V _S c and the series V _S c can be operated in different control modes where a particular mode is chosen depending upon the application objective. For example, for damping of inter-tie oscillations, the series V _S c may be operated in Automatic Power Flow Control mode 260 as shown in FIG. 34A. The d-q control method suggested by K. Sen and M. L. Sen ("Introduction to FACTS Controller: Theory, Modeling, and Applications", New York, NY, USA: IEEE Press, 2009, p. 298) may be adopted, for example, wherein the desired Pdesired and Qdesired are compared with the measured line flows, P _mea s and Qmeas and the corresponding deviations in real and reactive power (ΔΡ and AQ) are used to drive the d-q components ( Vd and V _q ) of series injected voltage, V _inj . Note here that the control inputs m _se , m _sh and 5 _se and 5 _sh correspond to steady state converter settings that can be easily obtained depending upon the desired inter-tie real and reactive power flows.

[00287] For the purpose of tightly regulating the connected bus voltage (< 5%), the shunt Vsc may be operated in Automatic Voltage Control mode 270 as shown in FIG. 34B. The bus voltage regulation is accomplished by varying the shunt modulation index m _sh while the DC link voltage regulation is achieved by varying the phase angle order 6 _S h.

[00288] Conventionally, the UPFC control of FIGS. 34A and 34B is based on PI control law that is inadequate in tracking system changes and becomes increasingly less effective in damping power oscillations with a set of non-optimal PI parameters. On the contrary, the proposed ST controller according to the teachings herein has been found to do well with system changes and has good tracking capability even for large disturbances such as three phase faults. Further, in a UPFC, the series V _S c can directly influence the power flow and hence the oscillatory modes. Keeping the aforementioned points in mind, a proposed ST controller 264 may be added in a supplementary control loop 262 to the series V _S c to modulate the control input m _se as shown in FIG. 34A with the real power deviations AP as the control input. Furthermore, since the proposed ST controller 264 is adaptive it is independent of tuning. It has been found in tests, that the adaptive ST controller 264 in the feedback loop 262 should improve the damping capability offered by the overall UPFC control system 252. [00289] FIG. 35 represents a schematic overview of an example embodiment of the adaptive ST controller 282. The adaptive ST controller 282 comprises a pole shift control module 284 and a CRLS identifier module 286. The measured output y(t) from the power system 294 is the oscillations in real power AP that are provided as inputs to the pole shift control module 284 and the CRLS identifier module 286 while the controller output u(t), generated by adaptive ST controller 282, is the damping input m _se that supplied to the series Vsc of the UPFC as is shown in FIG. 34A.

[00290] The equations used to design the operation of the CRLS identifier module 286 (see section lll-A of the Malhotra and Gokaraju article) are similar to equations 1 a to 8 used to develop the parameter identifier module 10. However, in the case of the UPFC application, the equation 8 can be rewritten as shown in equation 22:

= S t - 1) + K(t) [y(t) - § f - 1)φ(ί)]β(ί) (22) where ?(t) may be calculated at each sampling instant as follows:

where N _t = ||S(t)|| ₂ , N ₂ =

of the corresponding vector and β ₀ is a positive constant which determines the rate of parameter update. At the time of fault inception (and the duration shortly thereafter) on a power system 290, the ratio ^ progresses quickly to be greater than β ₀ . During this time period, /3(f) takes the value ^ reducing the estimation rate and therefore contributing to a smooth controller action. After fault removal, when the power system 290 is moving to a stable operating point, the ratio — gradually progresses to be less than β ₀ making _j 8(f) = 1. At this time, the constrained-RLS algorithm may switch back to the standard-RLS algorithm. A larger /¾ (i.e. β ₀ > 0.75) slows down the estimation rate which is undesirable, while choosing a smaller /¾ (i-e. /¾ < 0.5) can cause the CRLS identifier to respond to even minor disturbances. Thus, the /3(f) may be chosen as 0.5, for example, for optimal tracking capability. The variable 3(f) in equation 22 initiates a smooth controller response.

[00291] When a two-area system is subject to a disturbance it exhibits inter-area mode of oscillations in the range of 0.1 to 0.8 Hz, and using a third- order ARMA model in the identification process has been found to be sufficient to represent this single mode of oscillation as discussed previously. In alternative embodiments, 5 ^th , 7 ^th , 9 ^th or higher odd numbered models may be used if they result in a damping improvement and can be implemented efficiently from a computational/hardware standpoint such that the response time is sufficiently quick.

[00292] Once the system parameters are identified by the CRLS identifier module 286, the control signal may be calculated based on a discrete version of the ARMA model by assuming the transfer function of the feedback loop is of the form given in equation 10a. The Pole Shift (PS) control module 284 may be defined as given in equations 10b to 21 previously. In a UPFC application, the value of m _se may be chosen to be less than 1.0 to prevent overmodulation of the series-Vsc-

[00293] Testing of the performance of the ST control method along with the CRLS identifier was investigated using a Single Input Single Output (SISO) discrete system as is discussed in section lll-C of the Malhotra and Gokaraju article. In this testing, the control signal may be determined using equation 18 based on the identified plant coefficients using the CRLS identifier of equation 22.

[00294] The proposed ST controller 282 may be used with a UPFC as an add-on to an existing PI control system as is shown in FIGS. 34A and 35. To test the damping enhancement capability of the ST controller 282 operating in unison with the non-optimal PI controller, Kundur's two-area four- machine test system 300 (see the Hingorani reference noted previously) was used which is shown in FIG. 36. The system loading resulted in net power being exported from Area 1 to Area 2. Three oscillatory modes were present in this system; two intra-area modes (1.04 and 1.13 Hz), one in each area, and one inter-area mode (0.52 Hz) where damping of the inter-area mode was of prime concern. To improve inter-area mode damping, the UPFC 292 was installed on one of the tie-line connecting buses 7 and 8 (i.e. the sending- end of Area 1). The UPFC 292 was equipped with the supplementary ST controller 282 proposed herein to damp power oscillations with real power deviation from bus 7 as the feedback signal.

[00295] The electromagnetic transient simulation software PSCAD/EMTDC was used for simulations. The synchronous generators of FIG. 36 were represented in the d - q - 0 reference frame by detailed 7 ^th order differential equations, transmission lines were modeled as lumped impedances and system loads as constant impedances. The dynamics of the synchronous machine exciters and governors were also included in the simulations with their details along with the system, UPFC and PI parameters which are given in Appendix F.

[00296] Several fault types imposed at four different operating conditions listed in Table 1 1 were studied. Note here that P ₇ s refers to the inter-area line power flow with the UPFC at the sending end. The UPFC PI parameters were optimized using the multiple time domain simulation based simplex algorithm (see A. Gole, S. Filizadeh, R. Menzies, and P. Wilson, Optimization enabled electromagnetic transient simulation," IEEE Trans. Power Del. , vol. 20, no. 1 , pp. 512-518, Jan. 2005) at operating condition II in Table 1 1 . A sampling time of 20 ms (50 Hz) was chosen for the ST controller 282 and was found to be adequate for damping low-frequency (e.g. 0.1 - 0.8 Hz) oscillations.

[00297] FIG. 37 compares the inter-area power flow response between the PI control and the PI control assisted with ST control for a 6 cycle, 3- phase fault (Type 7A) applied at bus 8. The effectiveness of ST control is evident with the effective damping of power oscillations by the add-on controller as opposed to the standalone-PI controllers. The corresponding series modulation indices (Am _se - _P i and m _se -ps) responsible for providing damped power oscillations on tie-line 7-8 are shown in FIGS. 38A and 38B where the steady-state value {m _se ) was 0.475. It can be easily seen that the proposed ST scheme greatly impacted power flow and hence power oscillation damping through larger variations in m _se .

[00298] The ST control signal u(t) under the control limits of ±0.55 and the varying pole-shift factor or are plotted in FIGS. 39A and 39B. The behavior of a at post fault occurrence was such that it progressed to a lower value of a = 0.6025 indicating maximum control effort and slowly settled at a = 1.0 once the subsequent oscillatory swings were adequately damped out. Also, during fault, the shunt Vsc maintained the shunt bus voltage fluctuation |AV ₇ | within 0.05 p.u. along with regulating the DC link voltage {V _D dref) = 42.4 kV for all test cases) that facilitated real power exchange between the shunt V _S c and the serial V _S c as shown in FIGS. 40A and 40B. Further, the CRLS identifier equipped with the tracking constraint /3(f) of equation 23 provided smooth parameter tracking even during a large disturbance as shown in FIG. 41.

[00299] On the other hand, the improved power oscillation damping of FIG. 37 due to the supplementary pole-shift algorithm can be verified from the corresponding pole-zero plot (in the z-plane) which is shown in FIG. 42. The open-loop and closed loop poles were plotted as a function of a and their movement was captured from 2 s to 5 s. This snapshot was taken at r = 2.56 s when a pole-shift factor of or = 0.485 shifted the open-loop poles indicated by a "*" radially inwards to the closed-loop pole locations shown by a "x". Thus, the post-fault condition imposed a significant pole shifting process. For the same power condition, a disturbance of Type 7B was simulated with the system response and the corresponding tie-line power P ₇₈ and the pole-shift factor a are shown in FIGS. 43A and 43B, respectively.

[00300] The robustness of the proposed add-on ST controller 282 in damping power system oscillations was validated (see FIGS. 44A-47) for different inter-area power flows and system configurations highlighted in Table 1 1. FIGS. 44A and 44B depicts a comparison of power flow responses with and without the ST controller 282 in the supplementary loop as well as the control input u(t) when the transmission network 294 was subjected to a Type 7C disturbance. From the responses, it can be ascertained that the oscillations were damped very quickly by the supplementary controller 282 proving its superiority over the conventional PI-UPFC control system.

[00301] A similar observation is true when the test system was subjected to a Type 7D disturbance at the center of tie-line 7-8 with the damped inter- area power flow response shown in FIG. 45. However, the damping improvement by the ST controller 282 was not significant when compared with other operating scenarios since the PI controllers were optimally tuned for those particular operating points. In another test, the tie-length was increased from 250 km to 400 km and the performance of the overall UPFC damping model was tested for Type 7E and Type 7F disturbances. The corresponding dampened responses are shown in FIGS. 46A and 46B, respectively.

[00302] The positive impact of the add-on control scheme in damping inter-area oscillations was also analyzed for the operating condition IV, Type 7G fault. The corresponding powerflow oscillation after the application of the fault was plotted in FIG. 47. It is apparently evident that the ST controller 282 damps the oscillation within 10 s owing to its ability to adapt to different operating conditions without requiring retuning. [00303] The damping improvement shown in Table 1 1 was investigated using the Prony analysis tool. Table 12 shows the study results for the base case (without any damping controller) and the four test cases with standalone- Pi controllers and PI controllers assisted with the proposed ST controller 282.

TABLE 12: Inter-Area Oscillation Damping Case Studies

[00304] The Prony analysis tool was applied on the short circuit response of PJS to find the frequency and damping of the system's inter-area mode. Table 12 shows that the inter-area oscillatory mode frequency lies between 0.43 and 0.58 Hz. Also, the corresponding damping ratio was about 2.4% - 5.3% and being less than about 5% is considered to be significantly under-damped. The overall damping improvement seen with PI controllers was typically between 6% and 10% whereas with the add-on ST controller it was between 17% and 22%. However, the damping improvement by the addon ST controller 282 observed for operating condition II was the lowest since the PI controllers were optimally tuned for this test. Nevertheless, the add-on ST controller scheme, according to the teachings herein, appears to provide improved damping performance.

[00305] It should also be understood that at least some of the elements of the controllers 10, 70 and 150 that are implemented via software may be written in a high-level procedural language such as object oriented programming or a scripting language. Accordingly, the program code (i.e. instructions) may be written in at least one of C, C ⁺⁺ , SQL or any other suitable programming language and may comprise modules or classes, as is known to those skilled in object oriented programming. In alternative embodiments, at least some of the elements of the controllers 10, 70 and 150 that are implemented via software may be written in at least one of assembly language, machine language or firmware as needed. In either case, the program code (i.e. instructions) can be stored on a storage media or on a computer readable medium that is readable by a general or special purpose programmable computing device having at least one processor, an operating system and the associated hardware and software that is necessary to implement the functionality of at least one of the embodiments described herein. The program code (i.e. instructions), when read by the computing device, configures at least one processor to operate in a new, specific and predefined manner in order to perform at least one of the methods described herein.

[00306] Furthermore, the computer readable medium may be provided in various non-transitory forms such as, but not limited to, one or more diskettes, compact disks, tapes, chips, USB keys, magnetic and electronic storage media and external hard drives or in various transitory forms such as, but not limited to, wire-line transmissions, satellite transmissions, internet transmissions or downloads, digital and analog signals, and the like. The computer useable instructions may also be in various forms, including compiled and non-compiled code (i.e. instructions).

[00307] While the applicant's teachings described herein are in conjunction with various embodiments for illustrative purposes, it is not intended that the applicant's teachings be limited to such embodiments. On the contrary, the applicant's teachings described and illustrated herein encompass various alternatives, modifications, and equivalents, without departing from the embodiments, the general scope of which is defined in the appended claims. APPENDIX A: Three-Area Six Machine System

Generator data: 900 MV A,20kV, r _a = 0.0025, x _l = 0.2, x _d = 1.8, x _q = 1.7, x _d ' = 0.3, x _q ' = 0.55, x _d '' = 0.25, x _q '' = 0.25, T _d0 = 8 s, ¾ = 0.035, T _q0 = OAs, T _q '' ₀ = 0.05s, iV(C ₁ &G ₂ ) = 6.5s, H(G ₃ &G ₄ = 6.175s, tf(G ₅ ) = 5.5 s, (C ₆ ) = 5 s.

Table A.1 Three area system load data (MVA, MVAR) for different test cases

Case Bus 7 (L ₇ , C ₇ ) Bus 9 (Lg, Cg) Bus 14 (L ₁₄ ,C ₁₄ )

Α,Β,Ε 1400 + ylOO, -;350 1800 + ylOO, -)500 1200 + yiOO, -y ^' 220

C 1400 + yi00, -y ^' 260 1600 + yiOO, -7 ^' 350 1450 + ylOO, -y ^' 220

D 1755 + i00, -7 ^' 260 1200 + ylOO, -7 ^' 350 1200 + ylOO, -7 ^' 240

Generator steady state data: G^. V = 1.034-0° p.u.,G ₂ : 700 MW, V = 1.01 p.u., G ₃ :720 MW, V = 1.01 p.u., G ₄ : 700 MW, V = 1.01 p.u., G ₅ :800MW, V = 1.02 p.u., G ₆ : 780 MW, V = 1.01 p.u.

Transmission line data: r = 0.053 d/km, X _L = 0.53 il/km Exciter data: IEEE type ST1A exciter, T _r = 0.01 s, T _c = 1 s, T _B = 10 s, K _A = SO.VMAX = 9p-u.,V _MLN = -9 p.u..

Steam governor data: GE mechanical - hydraulic controls, Droop (ft) = 0.04 p. u,

Speed relay lag time constant (TC) (Tj = 0.1 s, Gate servo TC (T ₃ ) = 0.25 s.

Steam turbine data (in p.u.): IEEE type 2 thermal turbine, = 0.0, K ₃ = 0.25, K ₅ = 0.0, K ₇ = 0.0, K ₂ = 0.25, K ₄ = 0.5, K ₆ = 0.0, K ₈ = 0.0,

Steam chest TC (T ₄ ) = 0.42 s,

Reheater TC (T ₅ ) = 4.25 s, Reheater/ cross - over TC (T ₆ ) = 0.72 s.

TCSC Parameters: kB _ref = 1.75, C _rcsc = 87.76 μΨ, L _TCSC = 12.8 mH, PI controller: K _P = 10,T _C = 0.05 s. APPENDIX B: IEEE 12-bus System

Table B.1 : IEEE 12-bus system: Bus data

Bus Nominal Specified Load, Shunt capacitor, Generation, voltage, kV voltage, kV MVA MVAr MW

1 230

2 230 280+j200

3 230 320+J240

4 230 320+J240 160

5 230 100+160 80

6 230 440+βΟΟ 180

7 345

8 345

9 22 1.04

10 22 1.02 500

11 22 1.01 200

12 22 1.02 300

Table B.2: IEEE 12-bus system: Branch data, system base: 100 MVA

Line Voltage, Length, R, p.u. X, p.u. B, p.u. Rating, kV km MVA

1-2 230 100 0.01144 0.091 11 0.18261 250

1-6 230 300 0.03356 0.26656 0.55477 250

2-5 230 300 0.03356 0.26656 0.55477 250

3-4(1 ) 230 100 0.01144 0.091 1 1 0.18261 250

3-4(2) 230 100 0.03356 0.091 1 1 0. 8261 250

4-5 230 300 0.03356 0.26656 0.55477 250

4-6 345 300 0.03356 0.26656 0.55477 250

7-8 345 600 0.01595 0.17214 3.28530 500

APPENDIX C: Lead-lag Controller Parameters for the three-area, six- machine test system

For Inter-area Oscillation Damping Studies:

Simplex-optimized lead-lap supplementary controller (Balanced TCSC): T _w = 1 s, T _t = T ₃ = 3.2925 s, T ₂ = T ₄ = 0.17072 s, K _P = 0.0114.

Simplex-optimized lead-lap supplementary controller (Balanced SSSC): T _w = 1 s, T _x = T ₃ = 0.20 s, T ₂ = T ₄ = 0.4090 s, K _P = 0.5.

Power system stabilizer parameters:

IEEE type PSS1A power system stabilizer, T _w = Is, T _x = 4.597s, T ₃ = 3.131s, T ₂ = 2.082s, T ₄ = 3.173s, K _P = 0.21.

APPENDIX D: IEEE 12 bus system

Table D.1: IEEE 12-bus system: Bus data.

Bus Nominal Specified Load, MVA Shunt Generation, voltage, kV voltage, kV capacitor, MW

MVAr

1 230

2 230 280+j200

3 230 320+1240

4 230 320+j240 160

5 230 100+j60 80

6 230 440+1300 180

7 345

8 345

9 22 1.04

10 22 1.02 500

11 22 1.01 200

12 22 1.02 300

FIG. D.1 : IEEE 12-bus test system used for inter-area oscillation damping studies Table D.2: IEEE 12-bus system: Branch data, system base: 100 MVA.

Line Voltage, Length, , p.u. X, p.u. B, p.u. Rating, kV km MVA

1-2 230 100 0.01144 0.09111 0.18261 250

1-6 230 300 0.03356 0.26656 0.55477 250

2-5 230 300 0.03356 0.26656 0.55477 250-4(1) 230 100 0.01144 0.09111 0.18261 250-4(2) 230 100 0.03356 0.09111 0.18261 250

4-5 230 300 0.03356 0.26656 0.55477 250

4-6 345 300 0.03356 0.26656 0.55477 250

7-8 345 600 0.01595 0.17214 3.28530 500

APPENDIX E: System Data - Experimental Studies

Generator data: 700 MV A, 20 kV, r _a = 0.0025, xi= 0.2, x _d = 1.8, %,= 1.7, x< = 0.3, x _q '= 0.55, _d "= 0.25, "= 0.25, T _d0 = 8 s, 7V'= 0.03 s, 7 ' = 0.4 s, 7V'= 0.05 s, iV(Cl& G2) = 6.5 s, //(G3& G4) = 6.175 s, H(G5)= 5.5 s,

(C6) = 55.

Transmission line data: x _{ = 0.001 p. u./km , ^ = 0.0001 p. u./km Transformer data: 900 iW A, V _x /V ₂ = 230/20 fcV, r = 0, x = 0.15 p. u.

Exc/ter cteta: /£ ^■ £ ^■ £ ^■ type exciter, T _r = 0.01 s, T _c = 1 s, Γ _Β = 10 s, tf„ = 50, V _MAX = 5.7 p. u., V _MIN = -4.9 p. u., tf _e = 0.175, K _f = 0.0, 7} = 1.0 s. Governor and Turbine data: IEEE type 1 governor /turbine, K = 20, 7 = 15, Γ ₂ = 15, r ₃ = 0.25 s, T ₄ = 0.20 s, ΛΊ = 0.30, ff ₂ = 0, T ₅ = 5.0, K ₃ = 0.705.

ICSC Parameters: X _TCSC = 22.2 Ω, C _TC5C = 176 /.F, L _TCSC = 9.0 m/i, P/ controller: K _P = 4.0, r _c = 0.0085.

APPENDIX F: Machine and System Data and UPFC Design Parameters

Table F.1: Machine and System Data

Table F.2: UPFC Design Parameters