OPERATIONAL CONSTRAINT AND A SYSTEM THEREFOR FIELD OF INVENTION

This invention for optimizing an operation of a plant having a constraint on its operation is directed to the field of industrial process control and automation and in particular relates to a plant having plurality of devices driven by variable drives.

BACKGROUND Optimizing an operation of a plant having a constraint on its operation, comprising of plurality of devices connected in parallel and/or series configuration and driven by variable drives requires obtaining an optimal solution for an optimization problem constructed based on the constraint on its operation. Output parameters of such plurality of devices are controlled by varying input parameters to each of such devices through variable drives. Parallel/series configuration of plurality of devices is for efficient utilization of power and/or for flexibility of operation by providing alternative arrangement in case of breakdown of any device in the plant comprising of plurality of devices. For instance, a plant comprising of plurality of devices includes a plant comprising of plurality of pumps, and / or a variety of fan assemblies, or other devices having a variable drive. Variable drives include variable frequency drives or variable speed drives; output parameters of plurality of devices include speed, torque or a

combination of these and input parameters of plurality of devices include voltage, frequency, current or a combination of these.

A plant comprising of plurality of motor pumps driven by Variable Frequency Drives (VFD) is described merely as a non-limiting example of a plant comprising of plurality of devices driven by variable drives.

Pumps are widely used for water distribution in irrigation systems and residential and industrial water supply systems. They are also used in refrigeration and air-conditioning systems and fuel transportation pipelines. It is well-known that the power consumption by pumps constitutes the major component of operating cost of a water distribution network. Since the performance and power consumption of a pump is related to its speed, controlling the latter is necessary for achieving desired performance. Pumps with Variable Frequency Drives (VFD) allow changing the pump operational speed, and hence such pumps are also called variable speed pumps (VSP). Advances in technology have brought down the costs of VFDs resulting in their widespread usage. It is also possible to automate the operation of VFDs using programmable logic controllers (PLC). In the diverse applications mentioned above - and especially in the large-scale ones - the normal practice is to connect a number of pumps in series or parallel. This arrangement affords flexibility of operation by providing backup for scheduled and unscheduled repair of any pump(s) in the assembly. In such multi- pump systems, the net hydraulic system head H and flowrate Q is determined by the heads and flowrates of individual pumps and the configuration of the assembly. These relationships are different for parallel and series connection of pumps.

In a multi-pump assembly, it is possible to run each pump at a different speed, thus altering its head and/or flowrate. If the operational requirements on system head and flowrate change, it becomes necessary to change the speed of one or more pumps to meet them. In general, there are multiple possibilities for combination of pump speeds to meet given operational requirements.

As mentioned earlier, the power consumption of each pump depends on its speed and the net power consumption of the multi-pump assembly is the sum of individual power

consumptions. Although the same head and flowrate can be generated by the two different configurations, their net power consumptions can be generally different.

In developed countries, electricity costs for pumping form the largest share of the operational costs of water distribution systems. Hence, it is imperative to operate multipump assemblies in the most efficient manner regarding net power consumption. Since different speed configurations of VFD pump assembly typically result in different net power consumptions, it is necessary to operate the pump assembly at the speed configuration that minimizes the net power consumption while still satisfying operational requirements. Calculating such optimal speeds is a nontrivial task since the pump characteristics that relate head, flowrate, power and speed are highly nonlinear and change over the lifespan of a pump. Although the multi-VFD multi-pump systems afford the possibility of running individual pumps at different speeds, in practice, such systems are generally run using some simple logic that runs on the PLC, which controls the drive operation. For example, one of the drives acts as a master drive and other drives follow it by setting the same speed to respective pumps. Main reason for the use of such elementary logic in systems controlled using PLCs is that the low memory of PLCs puts severe restrictions on implementing any algorithms that involve (even not-too-complex) mathematical operations. However, existing logical procedures of operating multi-pump systems are - in general - not optimal with regard to net power consumption. Pumps in such systems can possibly have different characteristics (different make, capacity etc.). Even when all pumps are identical at the beginning, characteristics of each pump change differently over time depending on usage and other external factors. In such a case, running all pumps at a common speed that meets the head and flowrate requirements of the system is generally not an optimal solution. Ideally, one needs to solve the actual power optimization problem subject to operational constraints by taking into account the (changed) characteristics of the system. If one wanted to use available commercial optimization softwares to solve this problem, one would need a separate computer having sufficient memory for running the chosen software and hardware to communicate the results of the software to the drives. Thus, it would entail an additional cost burden on the pumping system. Since a typical water distribution system has multiple layers of pumping stations from source to various distribution points, the cumulative expenses of these optimization systems would be enormous. To eliminate these additional components and related expenses, it would be desirable to use only the available processing power of the PLCs that control the drives to solve this problem. To devise such an optimization algorithm that runs entirely on a PLC is a challenging task for two reasons:

- the nonlinear nature of the underlying optimization problem, and

- very limited memory available on current PLCs for pump drives. Most practical standard optimization algorithms make use of gradients information that needs significantly more memory than needed for the mere storage of decision variables. Hence, it is not possible to implement such algorithms on the limited memory of PLCs and it is necessary to design algorithms that would work with less memory than is customarily used for such algorithms. Output parameters of individual motor pumps, such as speed, is controlled by varying input parameters of each motor pump, such as input frequency and voltage, through VFD in the plant. Power consumption of each motor pump in the plant depends on speed of such motor pump; and net power consumption of the plant comprising of plurality of motor pumps is the sum of power consumptions of individual motor pumps. Hence, speed of each motor pumps in a plant comprising of plurality of motor pumps influences net power consumption of such plant. Therefore, optimizing the operation of the plant comprising of plurality of motor pumps requires minimizing net power consumption to improve efficiency of working of the plant.

Constraints on operation of a plant comprising of plurality of motor pumps include constraints on system head or flow rate of plurality of motor pumps in the plant. These constraints on operation of such plant necessitate change in speed of one or more pumps in the plant. In a plant comprising of plurality of motor pumps, total power consumption of the plant is regulated by varying the speed of individual motor pumps in such plant through VFDs. A constraint on operation of such plant is satisfied by multiple combinations of speeds of motor pumps in the plant. For instance, for satisfying a given constraint on flow rate and system- head in such plant, one of the possible combinations of pump speed is equal pump speeds for all individual motor pumps, which satisfy the given constraint. Another possible

combination of pump speed, satisfying the same operational constraint of the plant is unequal pump speeds for individual motor pumps. Even though both combinations of pump speed satisfy the same constraint on operation of the plant, the net power consumption of such plant, running on above referred different combinations, is different. A plant comprising of plurality of motor pumps running at equal speeds and satisfying constraint on operation of such plant does not always consume minimum net power, particularly when the

characteristics of individual motor pumps differ. There is a need to optimize the operation of the plant by identifying a combination of speed for motor pumps, which not only satisfies constraint on operation of the plant, but also minimizes net power consumption of the plant. Pump characteristics that relate head, flow rate, power and speed of individual motor pumps are highly non-linear and change over the lifespan of a motor pump. Pump characteristics of each motor pump change differently over time, depending on usage and other external factors, even if all motor pumps are identical initially during their installation. Hence, there is a need to optimize the operation of a plant comprising of plurality of motor pumps considering pump characteristics that change eventually during the life span of a motor pump.

Most of the known arts for optimizing the operation of a plant comprising of plurality of motor pumps, use gradient information and involve complex computations, which require larger memory capacity for storing data computed by these methods. These known methods are not suitable for deployment on Programmable Logic Controllers (PLCs) since PLCs have lesser memory capacity and lesser processing capability. Hence, it is the objective of the present invention to create a method for optimizing an operation of a plant having a constraint on its operation and comprising of plurality of devices driven by variable drives, which can also be deployed on PLCs. It is a further objective of the present invention to implement a method and a system for optimizing an operation of a plant having a constraint on its operation. SUMMARY OF THE INVENTION

The objective of the present invention is achieved by a method according to claim 1 , and by a system according to claim 7.

So according to the invention, a method for optimizing an operation of a plant, said plant having a constraint on the said operation, comprises the steps of: transforming a first optimization problem into a second optimization problem of the said plant based on the said constraint on the operation of the said plant; and computing a solution of the said second optimization problem of the said plant to obtain an optimal solution for optimizing the said operation of the said plant.

In a preferred embodiment of the method according to the invention, the said computing of a solution of the said second optimization problem of the said plant includes: solving the said second optimization problem by Nelder-Mead technique to obtain a solution of the said second optimization problem of the said plant; and estimating an improvement in the said solution of the said second optimization problem of the said plant.

In a further preferred embodiment of the method according to the invention, the said computing of a solution of the said second optimization problem of the said plant further comprises a tuning of the said second optimization problem iteratively to bring the said solution of the said second optimization problem to satisfy the practical tolerance limits.

In a further preferred embodiment of the method according to the invention, the said tuning of the said second optimization problem iteratively comprises of: updating parameters and constants in the said second optimization problem to bring the said solution of the said second optimization problem of the said plant to satisfy practical tolerance limits; and performing one or more iterations of the said computing of a solution of the said second optimization problem of the said plant.

In a further preferred embodiment of the method according to the invention, the said transforming of a constraint on the operation of the said plant into a second optimization problem of the said plant includes steps of: constructing a first optimization problem of the said plant from the said constraint on the said operation; and converting the said first optimization problem of the said plant into a second optimization problem of the said plant.

In a further preferred embodiment of the method according to the invention, the said method is capable of being performed by and in a processing unit like a Programmable Logic

Controller, having lesser memory and / or processing capability thereof.

According to the invention, a system for optimizing an operation of a plant having a constraint on the said operation comprises of: a transforming unit for transforming a constraint on the operation of the said plant into a second optimization problem of the said plant; and a computing unit for computing a solution of the said second optimization problem of the said plant to obtain an optimal solution for optimizing the said operation of the said plant.

In a preferred embodiment of the system according to the invention, the said computing unit for computing a solution of the said second optimization problem of the said plant includes: a solving unit for solving the said second optimization problem by Nelder-Mead technique to obtain a solution of the said second optimization problem of the said plant; and an estimating unit for estimating the improvement of the said solution of the said second optimization problem of the said plant.

In a further preferred embodiment of the system according to the invention, the said solving unit for solving the said second optimization problem outputs a solution of the said second optimization problem of the said plant that is the said optimal solution for optimizing the said operation of the said plant, in which the said solution of the said second optimization problem of the said plant satisfies the practical tolerance limits thereof.

In a further preferred embodiment of the system according to the invention, the said computing unit for computing a solution of the said second optimization problem of the said plant further comprises a tuning unit for tuning the said second optimization problem iteratively, in order to bring the said solution of the said second optimization problem to satisfy the practical tolerance limits.

In a further preferred embodiment of the system according to the invention, the said tuning unit for tuning the said second optimization problem iteratively comprises of: an updating unit for updating parameters and constants in the said second optimization problem to bring the said solution of the said second optimization problem of the said plant to satisfy the practical tolerance limits; and an iteration performing unit for performing one or more iterations of the said computing of a solution of the said second optimization problem of the said plant.

In a further preferred embodiment of the system according to the invention, the said transforming unit for transforming a constraint on the operation of the said plant into a second optimization problem of the said plant includes: a constructing unit for constructing a first optimization problem of the said plant from the said constraint on the said operation; and a converting unit for converting the said first optimization problem of the said plant into a second optimization problem of the said plant. In a further preferred embodiment of the system according to the invention, the said converting unit for converting the said first optimization problem of the said plant includes a separating unit for separating the said constraint on the said operation from the said first optimization problem of the said plant.

In a further preferred embodiment of the system according to the invention, the said plant comprises a plurality of devices driven by variable drives.

In a further preferred embodiment of the system according to the invention, the said system has a processing unit like Programmable Logic Controller or the like, having lesser memory and / or processing capability thereof.

This invention focuses on the issue of designing reliable algorithms for solving such constrained optimization problems while using memory much lesser than is customarily used. It uses the Nelder-Meade (NM) algorithm as the underlying optimization solver since the former does not calculate any gradient information. The method first transforms the

constrained optimization problem into a particular form of unconstrained optimization problem by using certain penalty terms on constraints and their constraint violation. It solves the new - unconstrained - optimization problem using the NM algorithm. Depending on the constraint violation caused by the result of the last step, it updates the penalty parameters in the unconstrained problem formulation and repeats the above procedure. The iterative procedure is stopped when the problem constraints are satisfied to specified tolerance levels. The final outcome of the algorithm is comparable within practical error tolerances to that of the standard optimization algorithms based on use of gradient information. Thus without using gradient calculations or other complex mathematical operations, it calculates the optimal solution for given operational requirements.

This invention for optimizing an operation of a plant having a constraint on its operation more particularly relates to a plant having plurality of devices connected in parallel and/or series configuration and such devices driven by variable drives.

This invention involves constructing a first optimization problem based on the constraint on operation of the plant comprising of plurality of devices; converting the first optimization problem into a second optimization problem and solving the second optimization problem to obtain an optimal solution that optimizes the operation of the plant. The second optimization problem is solved by direct search methods, which do not require gradient information. One such direct search method is Nelder-Mead (NM) Simplex technique. Further, the method involves estimating the improvement in the said solution of the second optimization problem of the plant. The method further involves tuning the second optimization problem by updating parameters and constants in the second optimization problem until the said solution of the second optimization problem satisfies the practical tolerance limits. When the solution of the second optimization problem satisfies the practical tolerance limits, then the solution of the second optimization problem is the optimal solution required for optimizing the operation of the plant having constraint on operation.

BRIEF DESCRIPTION OF THE DRAWINGS The present invention will be more clearly understood from the following description of the preferred embodiments of the invention read in conjunction with the attached drawings, in which:

Fig. 1 shows a system for optimizing operation of a plant having a constraint on its operation, comprising of plurality of devices driven by variable drives. Fig. 2 shows the steps involved in computing an optimal solution for optimizing an operation of a plant having a constraint on total power consumption, comprising of plurality of pumps driven by variable frequency drives.

Figure 3 shows a schematic diagram depicting the operation of the proposed algorithm.

The drawings are merely representative and are not intended to limit the scope of the appended claims.

DETAILED DESCRIPTION

Looking first at figures 1 and 2. A system 1 for optimizing an operation of a plant, having a constraint on operation, comprising of plurality of devices driven by variable drives is described as an embodiment of the invention as shown in Fig. 1 . A plant comprising of m number of pumps, which are not necessarily identical in their pump characteristics, driven by variable frequency drives is described as a non-exhaustive example of a plant comprising of plurality of devices driven by variable drives. Plurality of pumps in the plant is in parallel configuration and hence, system-head H of the plant equals individual heads Hi of the motor- pumps in the plant; and system-flow-rate Q of the plant is the sum of the individual flow-rates Qi of each of the motor pumps in the plant. The interdependency of parameters of the plant such as system-flow-rate Q, system-head H and total power consumed P by the plant are expressed in terms of the individual pump parameters such as individual flow rates Qi, individual heads Hi and power consumed by individual pumps Pi respectively.

The steps involved in a method 20 for optimizing an operation of a plant comprising of plurality of pumps is as shown in Fig. 2.

The constraint on operation of the plant and the pump characteristics are the inputs to the system 1 as shown in step 21 of method 20. The step 22 of method 20 is used to construct a first optimization problem OP1 by a constructing unit 4 from the constraint on operation of the plant. The first optimization problem OP1 for the plant is constructed from the constraint on operation such that the plant power P consumed by the multi-pump is minimum. It is expressed as:

OP1 : Minimize total power consumed by the plant: P = Pi +P ₂ + + P _m - such that: total flow rate of the plant: Q = Qi + Q ₂ + + Q _m ; total system head of the plant: H = Hi = H ₂ = .... = H _m ; and

L H

individual speeds n(i) of each of the motor pump in the plant: n (i) < n(i) < n (i) where i = 1 , 2,..,m;

L H

and n (i) and n (i) denote the lower and upper bounds on pump speeds

respectively.

As shown at step 23 of the method 20, the first optimization problem OP1 of the plant is converted into a second optimization problem OP2 by a converting unit 5 and the resultant second optimization problem OP2 is expressed as:

OP2: Minimize

where, G, denote constraints on the operation of the plant and Gi denote values of their respective deviations of the solution of the second optimization problem from the given constraint on operation of the plant.

Speeds n, of the individual motor-pumps in the multi-pump system are deciding variables since these speeds decide the performance and the total power consumed by the plant. λι, λ ₂ , ... are constants which are independent of the deciding variables η,. μ is also a constant known as the penalty parameter, which is a parameter updated to bring the solution of second optimization problem to satisfy practical tolerance limits.

Hence, a first optimization problem OP1 is transformed into a second optimization problem OP2 on the basis of the given constraint on operation, by a transforming unit 2 comprising of a constructing unit 4 and a converting unit 5.

An optimal solution for optimizing the operation of the plant is computed by a computing unit 3, comprising of a solving unit 6 and an estimating unit 7 and further comprising a tuning unit 8. The second optimization problem OP2 is solved by Nelder-Mead Simplex technique by a solving unit 6 to obtain a solution of the second optimization problem as shown in step 24.

Step 25 of the method 20 is to estimate the improvement of the solution of the second optimization problem, by an estimating unit 7.

Step of 26 of the method 20 is to check if the solution of the second optimization problem satisfies the practical tolerance limits.

When the solution of the second optimization problem satisfies the practical tolerance limits, then the solution of the second optimization problem is the desired optimal solution for optimizing the operation of the plant as shown in step 27. Hence, when the above referred solution of the second optimization problem satisfies the practical tolerance limits, then the desired optimal solution comprises optimal speed of each of the motor pumps such that the plant comprising of plurality of motor pumps, consumes minimum power satisfying the constraint on total power consumption.

If the solution of the second optimization problem does not satisfy the practical tolerance limits, then tuning the second optimization problem OP2 iteratively to bring the solution of the second optimization problem to satisfy the practical tolerance limits is performed by a tuning unit 8, comprising of an updating unit 9 and an iteration performing unit 10.

The penalty parameter and constants in the second optimization problem OP2 are updated as shown in step 28, so as to bring the improvement estimated at step 25 in respect of the solution of the second optimization problem, to satisfy the practical tolerance limits by updating unit 9. One or more iterations of computing an optimal solution for optimizing the operation of the plant is performed by iteration performing unit 10 until the solution of the second optimization problem satisfies the practical tolerance limits.

Now turning to figure 3, which shows the schematic working of the proposed method.

Consider m pumps - possibly non-identical - connected in parallel. Let n, denote the speed of the ith pump, H, its hydraulic head, Q, the flowrate it generates and P, the power it consumes. Since the pumps are connected in parallel, the individual heads H, match the system head H. The interdependency of Q,, H and n, as well as P,, Q,, and n, is expressed in terms of polynomial relations derived using pump characteristics data and pump affinity laws. Hence, the optimization problem formulation for this system becomes: OP1 :

Minimize Total Power: P = Pi + P ₂ + P3 +■■■ + P

s.t. Total Flow: Q = Qi + Q ₂ + . . . + Q ■,m

Head: H = H = H ₂ = ... = H m

Speeds: n ^L (i) <= n(i) <= n ^H (i), i = 1 , ... m where n ^L (i) and n ^H (i) denote respectively the lower and upper bounds on pump speeds. The flowrate constraint is an equality constraint while the bounds on speeds are inequality constraints.

The aim of the proposed methodology is to solve OP1 without calculating any gradient information. The proposed methodology is as follows. The given constrained optimization problem OP1 is transformed into the following unconstrained optimization problem:

OP2: Minimize

where G, denote the values of (equality and inequality) constraints and Gi denotes values of their violations. Parameters μ, λι, λ ₂ ,... are independent of the decision variables ni , ... , n _m . μ is a penalty parameter for constraint violation. Thus, all constraints in OP1 are incorporated in the objective function of OP2.

In the first step, using any suitable choice of μ, λι, λ ₂ ,... ,the unconstrained problem OP2 is solved using the NM method. For an arbitrary choice of μ, λι , λ ₂ ,..., the solution of the transformed problem OP2 obtained using the NM method may not satisfy the constraints of the original problem OP1 to specified tolerances. Hence, depending on the constraint violations at the solution obtained, the values of μ, λι, λ ₂ ,... are updated such that the corresponding terms in the OP2 contribute more to the objective. Then OP2 is re-solved starting from the current solution as initial guess for next iteration of the NM method. This procedure of solving OP2 using NM method and updating μ, λι , λ ₂ ,... depending on the solution obtained is repeated till the solutions obtained at each step converge gradually to the optimal solution.

The proposed method is an iterative procedure to obtain optimal speeds of pumps in the multi-pump assembly. By varying the value of penalty parameter μ and the way it is updated, the rate of convergence of the algorithm to the optimal solution can be varied.

The method does not calculate any gradients, matrix inverses nor does it perform any other complex mathematical operations that require large memory. It is thus capable of running on memory much lesser than is usually used for such type of problems. Hence, the proposed method is applicable for optimization of any systems subject to operational constraints when such systems are controlled by processors having low memory such as PLCs.

Only certain features of the invention have been specifically illustrated and described herein, and many modifications and changes will occur to those skilled in the art. The invention is not restricted by the preferred embodiment described herein in the description. It is to be noted that the invention is explained by way of exemplary embodiment and is neither exhaustive nor limiting. Certain aspects of the invention that have not been elaborated herein in the description are well understood by one skilled in the art. Also, the terms relating to singular form used herein in the description also include its plurality and vice versa, wherever applicable. Any relevant modification or variation, which is not described specifically in the specification are in fact to be construed of being well within the scope of the invention. The appended claims are intended to cover all such modifications and changes which fall within the spirit of the invention.

Thus, it will be appreciated by those skilled in the art that the present invention can be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The presently disclosed embodiments are therefore considered in all respects to be illustrative and not restricted. The scope of the invention is indicated by the appended claims rather than the foregoing description and all changes that come within the meaning and range and equivalence thereof are intended to be embraced therein. List of reference signs

1 System for optimizing an operation of a plant

2 transforming unit

3 computing unit

4 constructing unit

5 converting unit

6 solving unit

7 estimating unit

8 tuning unit

9 updating unit

10 iteration performing unit

20 method for optimizing an operation of a plant

21 input step of method

22 step constructing a first optimization problem

23 step converting the first optimization problem into a second optimization problem

24 step solving second optimization problem by NM technique

25 step estimating improvement of solution

26 step to check if the solution satisfies the practical tolerance limits

27 final step

28 updating step