Application of Genetic Algorithm to Optimal Reactive Power Dispatch including Voltage Stability Constraint

This paper presents a Genetic Algorithm (GA) based approach for solving optimal Reactive Power Dispatch (RPD) including voltage stability limit in power systems. The monitoring methodology for voltage stability is based on the L-index of load buses. Bus voltage magnitudes, transformer tap settings and reactive power generation of capacitor banks are the control variables. A binary-coded GA with tournament selection, two point crossover and bit-wise mutation is used to solve this complex optimization problem. The proposed algorithm has been applied to the IEEE 30-bus system to find the optimal reactive power control variables while keeping the system under safe voltage stability limit, and found to be more effective for this task. ______________________________________________________________________________________ Nomenclature P loss Network real power loss Pi, Qi Real and reactive powers injected into network at bus i Gij, Bij Mutual conductance and susceptance between bus i and bus j Gii, Bii Selfconductance and susceptance of bus i Qgi Reactive power generation at bus i QCi Reactive power generated by i capacitor bank t k Tap setting of transformer at branch k Vi Voltage magnitude at bus i Vj Voltage magnitude at bus j θij Voltage angle difference between bus i and bus j Sl Apparent power flow through the l branch g k Conductance of branch k NB Total number of buses NB-1 Total number of buses excluding slack bus NPQ Number of PQ buses Ng Number of generator buses Nc Number of capacitor banks NT Number of tap-setting transformer branches Nl Number of branches in the system i δ Voltage phase angle of i generator bus S. Durairaj et al./Journal of Energy & Environment 4 (2005) 63 – 73 64 Introduction The purpose of the reactive power dispatch (RPD) in power system is to identify the control variables that minimize a given objective function while satisfying the unit and system constraints. Scheduling of reactive power in an optimum manner reduces circulating VAR (volt ampere reactive), thereby promoting a uniform voltage profile which leads to appreciable power saving on account of reduced system losses. RPD is a complex non-linear optimization problem. Linear programming (LP), non-linear programming and gradient based techniques have been proposed in the literature [1-4] for solving RPD problems. However, due to the approximations introduced by linearized models, the LP results may not represent the optimal solution for inherently non-linear objective functions such as the one used in the reactive power dispatch problem. It is very difficult to calculate the gradient variables and a large volume of computations is involved in this approach. Also, these conventional techniques are known to converge to a local optimal solution rather than the global one. Lately, expert system approach [5] has been proposed for the reactive power control computations. This approach is based on “If-then” based production rules. The construction of such rules requires extensive help from skilled knowledge engineers. Evolutionary computational techniques like Genetic algorithm (GA) [6], Evolutionary programming (EP) [7] and Evolutionary strategy [8] have also been proposed to solve the optimal reactive power dispatch problems. Generator voltages, transformer tap positions and number of switchable shunt capacitor banks were used as the control variables in the work [6] and they are represented as integer vector in the genetic population. In addition to the crossover and mutation operations, AI-based rules were applied to improve the solutions. Evolutionary programming was applied in [7] to solve the reactive power dispatch problem with minimization of active power loss as the objective function. IEEE 30-bus system was employed to carryout the simulations and the results obtained using the EP-based approaches were found to be better than the results obtained using the conventional method. Although, these works have solved the RPD problem successfully, none of them has considered the line flow and voltage stability constraints, which are important for any practical implementation of RPD. If a contingency occurs in an already stressed system both angular and voltage stability may be lost. Many voltage instability i.e. voltage collapse events have been experienced by the utilities in the recent years. This was mainly due to reactive power shortage during the peak load. These events warrant inclusion of the voltage stability constraint in the RPD for maintaining the security of modern power systems. This paper is concerned with application of GA for optimal reactive power dispatch with line flow and voltage stability constraints. The L-index defined in the work [9] is used in this paper to compute the voltage stability level of the system. This index uses information from a normal power flow and is in the range of zero to one. In the present work some restrictions are applied on the maximum value of L-index in the normal operating condition so that even if a contingency occurs on the system the L-index value does not reach an alarming level. Thus voltage stability constrained reactive power dispatch problem is solved in this paper using the genetic algorithm. IEEE 30-bus test system has been used to carry out the simulation study. Problem Formulation The RPD problem aims at minimizing the real power loss in a power system while satisfying the unit and system constraints. This goal is achieved by proper adjustment [10-12] of reactive power variables like generator voltage magnitudes (Vgi), reactive power generation of capacitor banks (Qci) and transformer tap settings (tk). S. Durairaj et al./Journal of Energy & Environment 4 (2005) 63 – 73 65 This is mathematically stated as;