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On Multiobjective Volt-VAR Optimization in Power Systems

Miroslav M. Begovic, Branislav Radibratovic, Frank C. Lambert School of Electrical and Computer Engineering, Georgia Institute of Technology

Atlanta GA 30332-0250 [email protected]

Abstract

The need for simultaneous optimization of reactive resources for the transmission and distribution system

has long been recognized. If investment resources are

limited, various combinations of solutions (voltage level,

amount and type of reactive support, etc.) may impact a number of objectives (distribution losses, distribution

feeder power factor, voltage profile for conservative

voltage reduction, transmission losses, transmission

capacity, voltage stability, etc.). While solutions have been proposed for subsets of the above problems, few

algorithms have undertaken the simultaneous

optimization of reactive resources in the transmission

and distribution network. This paper attempts to address various issues that need to be solved: decomposition of

the transmission model from the distribution model,

design of an interface suitable for simultaneous

optimization, and development of the methodology (multiobjective optimization based on building Pareto

fronts with the help of custom-tailored genetic

algorithms). Some of the issues discussed in this paper

are illustrated on suitable examples and guidelines proposed for building a practical model that would

incorporate all of the concerns with the modeling issues.

Index Terms—volt/var optimization in power systems, multiobjective optimization.

1. Introduction

Modern electric utility companies are faced with the problem of constant load growth together with a strict limitation of investment resources, which severely limits the growth of the infrastructure. One method for increasing transmission capacity is investment in reactive resources, which are used in both transmission and distribution networks. While locating and sizing reactive support, different objectives can be chosen. Design goals in distribution networks are usually optimization of distribution losses, distribution feeder power factor and voltage profile. Reactive power

planning in transmission networks often addresses transmission losses, transmission capacity and voltage stability as main objectives.

Various algorithms have been proposed to solve the capacitor placement problem, either on the transmission network or on the distribution feeders, when only one of the above objectives is optimized. These types of problems are often referred to as single-objective optimization problems. In recent years, multi-objective problems have arisen in many engineering applications. Two or more objectives (usually confronted) need to be simultaneously optimized. The problem of simultaneous optimization of reactive resources on the transmission and distribution system represents a further step in the generalization of the problem. Only a few algorithms are applicable for simultaneous optimization of reactive resources in the transmission and distribution network.

This paper addresses various issues that need to be solved: Decomposition of the transmission model from the distribution model, design of the interface suitable for simultaneous optimization and development of the methodology (multiobjective optimization based the on building of Pareto optimal solution fronts through the use of custom-tailored genetic algorithms).

2. Concept of multi-objective optimization

In many practical problems, several optimization criteria need to be satisfied simultaneously. Moreover, it is often not advisable to combine them into a single objective. While it may sometimes happen that a single solution optimizes all of the criteria, the more likely scenario is when one solution is optimal with respect to a single criterion while other solutions are best with respect to the other criteria. The increase of the “goodness” of the solution with respect to one objective will produce a decrease of its “goodness” with respect to the others. While there are no problems in understanding the notion of optimality in single objective problems, multiobjective optimization requires the concept of Pareto-optimality.

Proceedings of the 37th Hawaii International Conference on System Sciences - 2004

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The solution is said to be Pareto-optimal (belongs to the Pareto-optimal front, or set of solutions) if, with its change not one objective function can be improved without degrading all of the others. All of the solutions that make up a Pareto-optimal front are said to be non-

dominated (by other solutions). Concepts of the Pareto-optimal front, non-dominated and dominated solutions are further explained in Fig. 1. The axes on Fig. 1 (F1 and F2) are two objective functions. Possible solutions for minimization are presented in the F1-F2 plane. Solutions marked with triangles are called non-dominated and they make up the Pareto-optimal front. Those marked with circles are the dominated (non-Pareto optimal) solutions.

Fig. 1. Pareto-optimality, non-dominated and dominated solutions, bi-objective case

A solution x is dominated if there exists a solution ysuch that for all objective functions Fi stands:

Fi(x) ≤ Fi(y) for all i ∈ {1,2,…, n}

If the solution is not dominated by any other feasible

solution, we call it a non-dominated (Pareto-optimal)

solution. If the domination operator is “ ”:

• x1 x3 and x2 x3 (x3 is dominated)

• x1 x2 and x2 x1 (x1, x2 are non-dominated)

3. Test model

Any attempt to solve the multiobjective Volt-Var optimization for the entire power system will present a monumental challenge due to the system size and complexity of the solution. To that end, it is proposed to decouple the transmission model from the distribution model. The transmission system and each distribution feeder can be investigated separately. The Pareto-optimal front of the solutions is found for every distribution feeder and the transmission system as well. These solutions then provide input data to a suitably designed interface algorithm. The design of the interface algorithm capable of finding the globally optimal solution is the main goal of this research. The algorithm should solve for the Pareto optima while simultaneously taking into consideration both transmission system and distribution feeder solutions.

As an illustration of the system decomposition, transmission and distribution system models are shown in Fig. 2 and Fig. 3, respectively. These models will be used as an instructional example for the proposed algorithm. The transmission model is derived from the IEEE 5-bus system by removing the second generator originally connected to bus 3. This alteration is made to enable two possible alternatives for capacitor placement on the transmission network, namely bus 1 and bus 3.

The distribution feeder model is derived from the IEEE 13-node test feeder. This feeder is very small but relatively highly loaded, which enables various possibilities for capacitor placement. The following modifications are performed on the original feeder: The existing switch and low voltage transformer are removed from the model, the distributed load is neglected, and all loads have been balanced. Table 1 summarizes the necessary feeder data.

Fig. 2. Transmission system model

Non-dominated solution

Dominated solution

x1

x2

x3

Pareto-optimal

front

F1

F2

P3,

Q3

4 2 3

1

G

P1, Q1


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Fig. 3. Distribution feeder model

Table 1. Feeder load data

Node P (kW) Q (kVAr)

2 400 290

3 170 125

4 230 132

5 1155 660

6 1013 613

9 126 86

10 170 80

To link the distribution feeder models with the transmission network, all of the feeder loads are doubled and it is assumed that ten identical feeders are connected to each transmission load bus (bus 1 and bus 3). The set of feeders on each transmission bus represents a model of the distribution system.

4. Optimization algorithm

The general approach for solution of the multi-objective Volt-Var problem requires that reactive resources be divided between the transmission and distribution system. The list of optimization objectives may include distribution losses, distribution feeder power factor, voltage profile, transmission losses, transmission capacity, voltage stability, etc.

We propose that the power system model be separated into transmission and distribution subsystems. These systems are to be solved separately. Families of solutions for capacitor placement are found for each subsystem. These solutions should then be combined with a suitably designed interface algorithm to filter the unique Pareto-optimal solution front. The optimization problem for both systems, treated separately, can be formulated as:

Optimize: F

Subject to: G( , V, Qc) = 0 (1)

Qci = k Qc0 k = 0, 1, 2 …

where: F - vector of objectives G - set of power flow equations

- vector of voltage angles

V - vector of voltage magnitudes Qc - vector of reactive support Qci - reactive support applied to the bus “i” Qc0 - incremental reactive support step

4.1. Distribution system

This section presents Pareto-optimal solutions for the feeder shown in Fig. 3. Solving for any Pareto-optimal front assumes more than one objective. Two (confronted) objectives chosen here are: 1) the investment in reactive resources (assumed to be directly proportional to the amount of reactive support); and 2) feeder losses. The optimization problem (1) therefore can then be reformulated as:

Minimize: Ploss = Ploss ( , V, Qc)

Minimize: 11

1

ci F

i

Investment Q P=

= ⋅

Subject to: G( , V, Qc) = 0;

Qci = k Qc0 k = 0, 1, 2 …

where: PF - price of capacitor on the distribution feeder

4.1.1. Genetic algorithm as an optimization tool.

There are several ways to solve the optimization problem (2). Genetic algorithms (GAs) are the natural tool for solving the problem, even more so when other objectives are also included in the optimization. As described in [1], “Genetic algorithms are based on the mechanics of natural selection and natural genetics”. GA differs from traditional (calculus-based) optimization and search procedures in following ways: • It uses probabilistic transition rules rather than the

deterministic ones. • It does not need the knowledge of gradients or any

other auxiliary knowledge of the objective function. It uses only the objective function values, evaluated at a number of points.

• It works with a population of solutions rather than with a single solution.

For the solution of the problem defined in (2), bi-objective GA is applied. The cost of the feeder reactive support is assumed to be Pf = 15 $/kVA. The incremental reactive support step is assumed to be Qc0 = 100

0

3 24 1

8 610 5

9 7

TS

(2)


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kVA/phase. The optimization results are presented in Fig. 5 and Table 2. Figure 5 shows the Pareto-optimal front of the solutions, while Table 2 provides numerical explanations for only a few of the possible solutions. Due to the linear relationship between investment and the amount of reactive support, the latter is shown as an objective in Fig. 5.

270 290 310 330 350 370 390 410 430 450 4700

1

2

3

4

5

6

D istribution losses (kW )

Re

act

ive

su

pp

ort

(M

VA

)

Fig. 5. Distribution feeder, Pareto-optimal front of solutions (every point represents a different

capacitor allocation on the feeder)

Table 2. A few of the distribution feeder Pareto-optimal solutions

C1 MVA

C2 MVA

C3 MVA

C4 MVA

C5 MVA

C6 MVA

C9 MVA

ΣCMVA

Ploss

kW

0 0 0 0 0 0.3 0 0.3 466

0 0 0 0 0 1.5 0.3 1.8 367

0 0 0 0 1.2 1.5 0.3 3.0 315

0 0.6 0.3 0.3 1.8 1.5 0.3 4.8 280

0.9 0.6 0.3 0.3 1.8 1.5 0.3 5.7 277

4.2. Transmission system

The transmission system should be solved separately from the distribution system. Transmission losses and the amount of reactive support are the two selected minimization criteria. The optimization problem (1), applied on transmission system model shown in Fig. 2, can be reformulated as:

Minimize: Ploss = Ploss ( , V, Qc)

Minimize: 2

1

ci T

i

Investment Q P=

= ⋅


Qci = k Qc0 k = 0, 1, 2 …

This problem can be solved using the same genetic algorithm as the distribution feeder. The cost of the transmission reactive support is assumed to be PT =

10 $/kVA. The incremental reactive support step is assumed to be Qc0 = 1 MVA/phase. The optimization results are presented in Fig. 6 and Table 3. Figure 6 shows the transmission system Pareto-optimal front of solutions, while Table 3 provides numerical explanations for a few of the possible solutions. Due to the linear dependence between investment and the amount of reactive support, the latter is shown as an objective in Fig. 6.

6 7 8 9 10 11 12 13 140

10

20

30

40

50

60

70

80

90

Transmission Losses (MW)

Re

act

ive

sup

po

rt (

MV

A)

Fig. 6. Transmission system, Pareto-optimal front

Table 3. Structure of transmission system Pareto-optimal solutions

C1 (MVA) 0 2 11 18 25 31 42

C3 (MVA) 10 28 29 32 35 39 44

ΣC (MVA) 10 30 40 50 60 70 86

Ploss (MW) 12.3 9.7 8.69 7.86 7.20 6.73 6.43

4.3. Interface algorithm

The main challenge is how to optimize reactive resources, with respect to multiple criteria, for the entire transmission and distribution system. Defining system losses as the optimization objective, the optimization problem can be cast as:

Minimize: ΣPloss = Ploss,TS + ΣPloss,F


Qci = k Qc0T

Qcj = k Qc0F

. .T ci F cj

i j

P Q P Q I R⋅ + ⋅ ≤

(3) (4)


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where:

ΣPloss,F - sum of losses of all feeders in system k - non-negative integer Qci - reactive support applied at transmission bus

“i” Qcj - reactive support applied at feeder node “j” Qc0T - transmission incremental reactive support

step Qc0F - feeder incremental reactive support step I.R. - limitation of investment resources

In the optimization problem (4), the questions to be answered are: • How to allocate resources in both the transmission

and distribution networks.

• How to divide the resources between networks.

• How to divide the support between feeders

connected to different transmission buses.

The first question is already answered with the Pareto fronts of Fig. 5 and Fig. 6. The answers to the subsequent questions will be provided by the interface algorithm. The interface algorithm combines both Pareto fronts and finds particular solutions that represent a global optimum.

4.3.1. Algorithm structure.

Step 1. Choose an initial solution. Find a transmission Pareto solution (Fig. 6) that corresponds to a certain amount of investment (reactive support). If the investment resources are higher than the investment corresponding to the top point in the Pareto front, choose the top point as the starting point (keeping in mind that additional investment should be available for the distribution system). Step 2. Outer loop starts. Starting from the initial solution, go down the transmission Pareto front. Repeat the following steps for each transmission solution. Step 3. Calculate the available feeder support (reactive resources). Step 4. Inner loop starts. Compensate the feeders at each transmission bus with the available reactive support. Step 5. Find the optimal capacitor allocation for the feeder(s). Use sensitivity analysis of losses with respect to the active and reactive load on the transmission buses with compensated feeders and find the optimal feeder schedule (number of compensated feeders on a particular transmission bus). Reduce the reactive support from the transmission bus that is the most insensitive to the losses and transfer that support to the most sensitive bus. Step 6. Inner loop ends. Step 7. Outer loop starts.

Step 8. Extract optimal solution.

4.3.2. Application of Interface Algorithm to the Test

System. In the following example, the limit for investment resources is chosen to correspond to the top point of the transmission system Pareto front. It amounts to $860,000 (10$/kVA*86MVA). To further reduce the complexity of the problem, it is assumed that the voltage on the source end of the feeders (node 0) is kept constant, which further decouples the problem. In this way, the feeder consumption (P and Q loads at transmissions buses), together with feeder losses, is kept dependent only on the amount of reactive support applied on the feeder. The feeder consumption is not dependent on voltages at transmission buses; i.e. it is not dependent on transmission capacitor allocation.

The above algorithm produces the set of solutions depicted in Fig. 7. It represents overall losses as a function of the transmission reactive support (in MVA or in dollar terms). Investment resources are kept constant. The difference in reactive support between the initial solution (86MVA) and any subsequent point is transferred as support to the distribution feeders. The graph in Fig. 7 contains 19 sets of solutions corresponding to 19 possible scenarios of feeder compensation (19 members of the feeder Pareto-optimal front).

14 14.5 15 15.5 16 16.50

10

20

30

40

50

60

70

80

90

Overall losses (MW)

Tra

ns

mis

sio

n r

ea

cti

ve s

up

po

rt (

MV

A)

Fig. 7. Overall losses vs. level of transmission reactive support (investment resources limited

at $860,000)

The minimal overall losses correspond to the solution when only 14 MVA is applied in the transmission system. The following capacitor placement scenario yields the minimal losses:

Transmission system: C1 = 0 C2 = 14MVA

Distribution feeder: C2=0.6MVA C3=0.3MVA C4=0.3MVA C5=1.8MVA C6=1.5MVA C9=0.3MVA


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Optimal feeder schedule: BUS1 - 50%, BUS3 - 50%

The transmission system was supported with 14MVA, while the overall feeder support was 48MVA (ten feeders compensated with 4.8MVA each). The fact that the distribution feeders require 77% of the overall reactive support is because of their high load and extremely bad power factor. The best solution yields an

overall loss ΣPloss= 4.29MVA, which is 12% less than the initial solution (16.21MVA), and still requires the same investment. (14MVA*$10/kVA + 10feeders*4.8MVA*$15/kVA = $860,000)

5. Impact on voltage stability margin

The influence of the capacitor allocations discussed in the above example on the voltage stability margin is investigated. Continuation power flow, applied on the system model without reactive support, yields some alarming data. The critical loading factor of the system

is λ = 1.108 (10.8 percent load increase before voltage collapse). Applying reactive support to the system is expected to be beneficial, but it is not known whether the transfer of reactive support from the transmission to the distribution portion of the system would worsen the voltage stability loading margin. The answer appears to be negative. Transferring reactive support to the distribution network decreases the reactive load of the transmission system and increases the system voltage stability loading margin. Figure 8 illustrates this observation. The following PV curves are shown in Fig.8:

• No reactive support to system (λ = 1.108) • Entire ($860k) reactive support applied to the

transmission network (critical loading margin

increased to λ=1.294) • Minimal loss scenario (critical loading margin

further increased to λ=1.508)

6. Conclusions

The problem of simultaneous optimization of reactive resources on the transmission and distribution system is solved by decoupling the analysis of the transmission and distribution networks. The investment resources are assumed to be limited and known. Under this constraint, a number of optimization objectives can be chosen (distribution losses, distribution feeder power factor, voltage profile for conservative voltage reduction, transmission losses, transmission capacity, voltage stability, etc.). This paper addresses the various issues that need to be resolved to solve this problem.

1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Loading factor (p.u.)

Vo

ltag

e a

t b

us

3 (

p.u

.)

initial solution (86MVA support to transmission system) no reactive support solution that minimizes loses

Fig. 8. System’s PV curves for different capacitor scenarios

Decomposition of the transmission model from the distribution feeders is proposed as a necessary step to reduce the complexity of the problem. After the decoupled parts of the system are solved independently, a suitable interface is designed for simultaneous optimization. Custom designed genetic algorithms are used as multiobjective optimization tools that rely on sensitivity analysis to reduce the search space and allow implementation for large system models.

7. Acknowledgment

Financial support of the National Electric Energy Testing, Research and Applications Center (NEETRAC) used for part of the work presented in this paper is gratefully acknowledged. The authors also would like to acknowledge Dr. Damir Novosel, with whom they had many fruitful discussions about multi-objective optimizations.

8. References

[1] D Goldberg, Genetic Algorithm in Search, Optimization

and Machine Learning, New York: Addison Wesley, 1989.

[2] B. Baran, J. Vallejos, R. Ramos, U. Fernandez "Reactive Power Compensation using a Multi-objective Evolutionary Algorithm" IEEE Porto Power Tech Conference, Porto, Portugal September, 2001

[3] J.T. Ma, L.L. Lai “Evolutionary Programming Approach to Reactive Power Planning” IEE Proceedings – Generation, Transmission and Distribution, Vol 143, No. 4, July 1996


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