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Reactive Power Optimization by using Sequential Quadratic Programming

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Reactive Power Optimization by using Sequential Quadratic Programming

Suresh. V,

Department of EEE, Parisutham Institute of Technology and Science, Thanjavur, India.

Varadarajan. M

Department of EEE, Parisutham Institute of Technology and Science, Thanjavur, India.

Abstract Electrical power transmission losses in a power system are minimized by adjusting the reactive power control variables like transformer tap ratio, generator voltage magnitude and reactive power compensation devices such as capacitor banks. Also it results in improved voltage profiles, system security and power transfer capability. In this article, reactive power optimization, including comprehensive concern of the practical constraints was considered. The reactive power optimization problem was formulated and solved by using a Sequential Quadratic Programming method. To study the application of this method, a simulation test was carried out on a 3-bus & 6-bus system. It was found that the optimization problem was solved while the satisfying all the constraints imposed.

KeytermsReactive Power Optimization, Optimal Power Flow, Active Power Loss, Sequential Quadratic Programming.

I INTRODUCTION

The OPF problem was defined in the early 1960s [1] as an extension of conventional economic dispatch to determine the optimal control settings in a power system networks with various constraints [2]. Today any problem that involves the

determination of the instantaneous optimal steady state of an electric power system is referred to as an optimal power flow problem. The OPF function can be designed to operate in real time or study mode: to schedule active /or reactive- power controls and to optimize a defined operational objective function [3].

OPF is a static constrained non-linear optimization problem, whose development has closely followed advances in numerical optimization techniques and computer technology. It has been a traditional optimization problem in power system control/planning. It schedules the power system so as to optimize a given objective function subjected to a set of non-linear equality and inequality constraints [4]. In general, the equality constraints are the real and reactive power balance at each bus of the system; the inequality constraints are the physical and operational limits on the control and state variables of the system.

Reactive power optimization (RPO) is one of the important optimization in power system optimization. The RPO is a large-scale highly constrained nonlinear optimization problem. The main task here is to minimize the power losses of transmission and maintain the bus voltage

magnitudes within the designated limits for secure and efficient operation of power system.

Gradient-based search algorithms and various mathematical programming methods have been proposed to deal with Optimal Power Flow (OPF) problems. A number of techniques such as dynamic programming [5], linear programming [6] and interior point methods [7] has been proposed and used to solve reactive power optimization problem. However these techniques have several limitations in handling nonlinear, discontinuous objective functions and constraints [8]. The further discussions on these techniques will be discussed in [9].

II PROBLEM FORMULATION

The objective of reactive power optimization is to minimize system active power losses, to improve the voltage profile, and to determine optimal reactive power compensation placement keeping the values of various operational parameters within their limits. To achieve these objectives a set of control variables such as generator voltages magnitude, shunt capacitor banks, transformer tap ratio are chosen. The real power loss is a nonlinear function of bus voltage magnitude and phase angles which are the functions of control variables. The power loss function can be expressed as

where,

is the total real power loss

is the number of branches in the system is the conductance of branch k

is the voltage magnitude at bus is the voltage magnitude at bus j

is the voltage angle difference between i and j

The real power loss function, given by equation (1), is subjected to equality and inequality constraints. The equality constraints are the real and reactive power balance at each node of power system.

  1. Equality constraints

    In power system, the total active power balance is expressed as

    Similarly, the total reactive power balance is expressed as

    where,

    is the real power injected into network at bus i

    is the reactive power injected into network t bus

    i

    is the mutual conductance between bus i and j

    is the self-conductance of bus i

    is the mutual susceptance between bus i and j

    is the self susceptance of bus i

    is the total number of buses

    is the total number of buses excluding slack bus is the number of PQ buses

    The inequality constraints are as follows:

  2. Voltage Constraints

    Generator voltages outputs are restricted by their lower and upper limits as follows:

    is the generator minimum voltage magnitude. is the generator maximum voltage magnitude.

  3. Generator reactive power capability limit

    Generator reactive power outputs are restricted by their lower and upper limits are as follows:

    is the minimum generator reactive power. is the maximum generator reactive power.

    is the number of generator buses.

  4. Reactive power compensation limits

    Capacitor reactive power outputs are restricted by their lower and upper limits are as follows:

    is the reactive power generated by capacitor bank is the number of capacitor banks

  5. Transformer tap ratio

Transformer tap ratios are bounded as follows:

is the number of tap ratio transformer branches is the tap ratio of transformer at branch k

III. SEQUENTIAL QUADRATIC PROGRAMMING Sequential Quadratic Programming (SQP) [10] is a form of

Non Linear Programming (NLP). The objective function of

SQP optimization model is quadratic and the constraints are in linear form. SQP has higher accuracy than Linear Programming (LP)-based approaches. The NLP having lot of practical importance and is referred to as quadratic optimization. Derivation of the sensitivity method is aimed at solving the NLP on the computer. SQP is also very important because many of the problems are often solved as series SQP problems.

ALGORITHM

A quadratic programming is a special case of NLP problem wherein the objective function is quadratic and the constraints are linear. Both the quadratic approximation of the constraints are based on Taylor series expansion of the nonlinear functions around the current iterate

Step 1: The objective function is replaced by a quadratic approximation as

Step 2: The step is calculated by solving the following quadratic programming sub problem.

Step 3: Subject to

whereJ and I are the jacobian matrices corresponding to the constraint vectors G and H, respectively.

Step 4: The Hessian of the that appears in the objective function,equation (9), is computed using a quasi-Newton approximation. Once is computed by solving equations (9) to(11),X is updated using

where is the step length.

IV. RESULTS AND DISCUSSION

To evaluate the effectiveness and efficiency of SQP based reactive power optimization approach, the numerical experiments are made in the standard 3 bus and standard IEEE 6-bus systems.

Case (1) 3-Bus System

In this section performance of SQP for reactive power optimizaton was evaluated on standard IEEE 6-bus system with simulation parameter in table III and IV.

The system has two generator at buses 1 (slack bus) and 2 (PV bus). The lower voltage magnitude limits at all buses are

0.90 pu and the upper limits are 1.10pu.

The SQP based optimization procedure was tested on the 3-bus system and results provided in the Table I and Table II. The control variables, for this problem, before and after optimization is given in the Table I. The optimized case the voltage magnitude is maintained within the limits only. The optimized voltage magnitude values are minimized when compared to the base case results.

TABLE I

CONTROL VARIABLES OF 3-BUS SYSTEM BEFORE AND AFTER OPTIMIZATION

Control variables

Vmin

Vmax

Base Case

Optimized Case

0.90

1.10

1.10

1.0337

0.90

1.10

0.90

1.1000

The state variables of the 3-bus system results are provided in Table II. The results are compared with the base case studies. It is seen that the constraints on the control is enforced while the objective function is minimized.

TABLE II

STATE VARIABLE OF 3-BUS SYSTEM BEFORE AND AFTER OPTIMIZATION

satisfying equality and inequality constraints. The reduction is about 34.36 MW and is significant. Table III gives the active power losses in system at base case and after optimization

Fig.1 voltage profile of 3-bus system

In Fig 1 shows the voltage profile of the test system before and after optimization. It is observed that the voltage profile improved after the optimization.

State variables Base Case

Optimized Value

21.308° 9°

0.828° -0.83°

0.857 0.9241

93.53 59.19

273.17 10.782

-64.10 34.79

Fig.2 voltage profile of 3-bus system

.

.

TABLE III

ACTIVE POWER LOSS BEFORE AND AFTER OPTIMIZATION FOR STANDARD 3-BUS SYSTEM

Active Power Loss PL(MW) Base Case 63.528

Optimized Value 29.17

The transmission real power losses are reduced from a base case value of 63.528 MW to 29.17 MW while

In Fig 2 shows the voltage profile of sample 3-bus system. It shows the variations of bus voltage magnitude before and after optimization along with minimum and maximum limit on the variation. From Fig 2 it is clear that the bus voltage magnitudes are maintained within the limit while reducing transmission loss.

The bus voltage angle profile of 3-bus system in base case and after optimization is shown in the Fig 3. The voltage angle at bus-1 is reduced to great extent and thereby reducing the angular displacement between bus-1 and bus- 2. The reduction in angle implies reduced real power flow and hence the power loss.

variables i.e., state variables of the system before and after optimization is given in the Table V.

TABLE V

Fig 3 Voltage Angle Profile of 3 bus system

Case (2) Study of 6 bus system

In this section performance of SQP for reactive power optimization was evaluated on 6-bus system with simulation parameter in Table IV and V. The system has 2 generators at buses 1, 2 and two transformers with off nominal tap ratio and one switchable VAR source. The capacitor will be added at the 4th bus. The lower voltage magnitude limits at all buses are 0.95p.u and the upper limits are 1.05p.u for all buses.

TABLE IV

CONTROL VARIABLES OF 6-BUS SYSTEM BEFORE AND AFTER OPTIMIZATION

STATE VARIABLES OF 6-BUS SYSTEM BEFORE AND AFTER OPTIMIZATION

State variables Base Case Optimized

Value

-7.181° -3.26

0.881 0.9827

-14.494

-20.53

0.872

1.0005

-10.545

-15.32

0.850

0.9669

-14.089 -19.37

0.764 0.9806

-13.138 -19.37

101.83 94.05

90.08 38.8

40.99 12.24

TABLE VI

ACTIVE POWER LOSS BEFORE AND AFTER OPTIMIZATION FOR STANDARD 6-BUS SYSTEM

Control variables Vmin Vmax Base

Optimized

Case Value

0.90 1.05 1.0500 1.1000

Active Power Loss PL(MW) Base Case 16.827

Optimized Value 9.05

0.90

1.10

1.1000

1.1000

0.90

1.10

1.1000

1.0265

0.90

1.10

1.0250

1.0037

0

5.5

1.000

5.5000

0

5.5

0.000

5.5000

0.90

1.10

1.1000

1.1000

0.90

1.10

1.1000

1.0265

0.90

1.10

1.0250

1.0037

0

5.5

1.000

5.5000

0

5.5

0.000

5.5000

The transmission real power losses are reduced from a base case value of 16.827 MW to 9.05 MW while satisfying

equality and inequality constraints. The reduction is about

7.777 MW and the loss reduction is significant. Table VI

gives the power losses in system at base case and after

optimization.

The control variables before and after optimization along with their lower and upper limits is shown the Table

IV. It is seen that the optimization procedure enforces these limit while minimizing the objective function. Some control variables are on the limits known as binding constraints.

The SQP based optimization procedure was tested on the 6-bus system and results provided in the Table IV and Table V. It is seen that the constraints on the control is enforced while the objective is minimized. The dependent

The single line diagram of 6-bus system is shown in Appendix A. SQP is applying in standard 6 bus system bus systems and the result showing in the table IV and table V with graph.

CONCLUSION

In the present work, the formulation of reactive power optimization to minimize the transmission real power loss was presented. The optimization problem was solved using a sequential quadratic programming method. The application of SQP to the test systems gives the optimized result satisfying all the constraints. The active power losses, in case 3-bus test system, were reduced by 34.36 MW i.e., by a significant 54.09%. It was found that losses are reduced by 46.18% in case 6-bus test system. In both test system losses are minimized while satisfying all the imposed constraints.

Fig 4 Voltage Magnitude Profile of 6 bus system

In Fig 4 shows the voltage profile of sample 6-bus system. It shows the variations of bus voltage magnitude before and after optimization along with minimum and maximum limit on the variation. From Fig 5 it is clear that the bus voltage magnitudes are maintained within the limit while reducing transmission loss.

Fig 5 Voltage Angle Profile of 6 bus system

The bus voltage angle profile of 6-bus system in base case and after optimization is shown in the Fig 5. The voltage angle at bus-1 is reduced to great extent and thereby reducing the angular displacement between bus-1 and bus- 2. The reduction in angle implies reduced real power flow and hence the power loss

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