Coordinated Tuning of a Group of Static Var Compensators Using Multi-objective Genetic Algorithm

The optimal coordinated tuning of a group of Static Var Compensators (SVC), in steady state, allows the Power Electric Systems (PES) to operate close to their overload limits, maintaining the voltage stability in several operating conditions. The mentioned tuning problem was considered as a Multi-objective Optimization Problem (MOP) with three objectives to optimize: the financial investment for acquiring the set of compensators, the maximum voltage deviation and total active power loss. The Genetic Algorithm (GA), which belongs to the group of Evolutionary Algorithms, was utilized and adapted for MOP, obtaining a Multi-Objective GA (MOGA). The parameters to be adjusted in each compensator are: the reference voltage and the minimum and maximum reactive power injected to the system. In this work, the number of compensators and their locations were calculated using the Q-V sensitivity curve, from the Load Flow algorithm, based on Newton–Raphson method. The proposed coordinated tuning method will be validated considering an example of PES, where was located and tuned a specific set of compensators. Time simulations were made for dynamic performing the steady state coordinated tuning.


Introduction
This paper is an extension of work presented in [1], where it was described the optimal robust tuning of the SVC parameters, considering a few operating conditions, in steady state.So, in this work, it was implemented an optimal coordinated tuning procedure for adjusting several compensators, simultaneously, considering different critical operating scenarios, in order to overcome the voltage stability, in steady and dynamic state.Then, the compensators adjusted optimally allow to any PES studied operates close to their overload limits, maintaining a good level voltage for any disturbance.
The SVC devices belong to the FACTS group (Flexible AC Transmission System), which combine the digital electronic and the AC (Alternative Current) electric circuits and power electronic, and offer high speed response and large operational reliability [2]; because of those attractive characteristics, the compensators are largely utilized in protection and voltage stability of PES and require minimum financial investment to evaluate and locate [3].This reactive compensation problem is solved, commonly, in two steps: a) Financial procedure, where the compensator parameters are adjusted and, b) operational procedure, where the feasibility of the tuned parameters is verified applying the Optimal Power Flow (OPF) method.If the parameter values, calculated in the financial step, do not satisfy the design requirements, in operational procedure step, the necessary reactive power is determined for injecting in the system, in order to satisfy the requirements.This reactive power value calculated, in the operational step in order to satisfy the voltage stability, is called as virtual reactive power.Then, considering the Bender Decomposition, new parameters are calculated in the financial step, taking into account the parameter values of the previous iteration, and the new parameters obtained are validated again.This iterative procedure is repeated until the virtual reactive power approaches to zero [4][5].Nowadays, the GA is going to be used in reactive compensation problems.In reference [6], is detailed a reactive location method based on GA, and in reference [7] was utilized the MOGA in order to locate and calculate capacitor banks in a determined PES, used as a test.In this work is proposed a coordinated tuning procedure for calculating the optimal parameter values of a group of compensators, based on the search technique of the GA, considering several operating conditions in steady state.It was used the GA because its recognized efficacy in global optimization of complex and large industrial problems [8].The parameters to be adjusted for each SVC are: a) the reference voltage of the Automatic Voltage Regulator (AVR) of each SVC, b) minimum reactive power, and c) maximum reactive power, to be injected to the system by each SVC devices.The coordinated tuning problem was considered as a MOP with three objectives to minimize: a) Financial Investment for acquiring the set of compensators, b) Maximum Voltage Deviation, and c) Maximum Power Loss.Then, the GA described in [8] was adapted in order to optimize several objective functions, simultaneously, obtaining a Multi-Objective GA (MOGA).The main methodology for adapting the GA for MOP, described in details in this paper, is the Pareto Dominance rules; where, several optimal solutions are classified and saved on a group of optimal solutions.The group of optimal solutions is classified in each iteration of the GA.This algorithm gets a family of optimal solutions [9], at the end of its execution.In order to compare the performance of the MOGA, based on Pareto Dominance rules, it was also implemented the Weighted Sum Method for adapting the GA for MOP, where the global evaluation function, or Fitness, is calculated by the weighted sum of several objective functions to optimize.This algorithm gets a unique optimal solution, at the end of its search procedure.In addition, this paper presents numerical results, which validate the proposed coordinated tuning procedure.The PES, used as a test, corresponds to an academic IEEE PES with fourteen buses (substations) [10].Time response simulations were made in order to evaluate the dynamic performance of the group of tuned compensators, using standard values for dynamic parameters of the AVR, of each SVC device.

Mathematical Model
In this section, it is described the steady state mathematical model of the SVC device, such as described in Fig. 1.According Fig. 1 (a), the compensator is composed by switching reactor L and capacitor C banks, controlled by thyristors [2], and they are connected in series.In Fig. 1 (b), a linearly susceptance represents, mathematically, the performance of the SVC device operating in the control region.However, the susceptance is a fixed value outside that region.The susceptance is associated to the reactive power injected to the system in order to maintain the voltage level between suitable limits, in the controlled substation 1 .Fig. 2 describes the mathematical expression regarding to the voltage level of the controlled substation with the reactive power injected to the system.Fig. 2 describes the relationship between the voltage value, in the controlled substation k, and the reactive power, Q SVC , injected to the system; where, r SVC is the slope of the characteristic curve.The b SVC varies linearly in the control region, but is a fixed value in the limit regions, because it has achieved the capacitive or inductive reactive power limit.These limits are associated to the capacity of the capacitor and reactor banks.
In order to represent each SVC device, operating in different regions, in the Load flow algorithm, based on the Newton-Raphson method, the corresponding Jacobian matrix is modified.Then, in the Jacobian matrix, the SVC entries as a control function where the variable parameter is ∆x i = Q i,SVC [11], and i ∈ {1, 2, …, p} | p is the number of compensators considered in the group, as indicated in equation (1).
The components of the Jacobian matrix, associated to the SVC group, set to different numerical values according to the operating condition, such as indicated in equation (1).All components, regarding to the SVC control function with the angle and the active power with ∆x i , are equal to zero. 1 Controlled substation is the substation where the SVC device is installed.Then, the compensator injects the necessary reactive power, throughout the referred substation, in order to maintain the voltage level in the whole system between suitable limits.
In the limit regions Control Region: The coordinated tuning of parameters of a SVC group was considered as a MOP with three objective functions to be minimized, such as indicated in equation ( 2): Minimize subject to the following restrictions: where P Gi , Q Gi , P Li , Q Li correspond to the active and reactive power generated and demanded in the substation i, such i ∈ {1, 2, …, nsubs} | nsubs is the number of substations of the PES.The index k identifies the controlled substation; so, V k0 corresponds to the reference voltage of the AVR associated to the compensators installed in substation k.In addition, there are restrictions which limit the active and reactive power generation, in those substations where are installed a group of generator machines: PG min,i ≤ PG i ≤ PG max,i , QG min,i ≤ QG i ≤ QG max,I ; and restrictions which limit the voltage level in substations regarding to load zones: V min,j ≤ V j ≤ V max,j | j ∈ {1, 2, …, nbL} | nbL is the number of load centers in the PES.

Decision Variables
The GA handles the vector of parameters (decision variables) such as shown in equation (3), and it was used the float point codification for representing each of them [12]: Such as described before, the parameters to be optimized for each SVC device are: a) the reference voltage, V REF (where, V REF = V k0 ), of the AVR associated, b) the minimum reactive power to be injected, Q min , and, c) the maximum reactive power to be injected by each SVC device, Q max .Each parameter value belongs to the following search space:

Objective Functions
In the MOP, for coordinated tuning of a group of SVC, there are three objective functions to be minimized: a) the Financial Investment, F 1 (x), for acquiring the set of compensators, b) the Maximum Voltage Deviation, F 2 (x), and, c) the Maximum Total Active Power Loss, F 3 (x), calculated by considering all selected critical operating scenarios.
In equation ( 4), F 1 (x) is directly proportional to the compensation capacity of each SVC, where B i is the monetary value for each MVAr of the i-th compensator; and, n SVC indicates the number of compensators to be adjusted in the PES.In this work, each B i = 1.0 monetary/MVAr.In addition, ng and nbL correspond to the number of substations with a group of installed generators and substations associated to the load zones, respectively.

GA Adapted for MOP
In this subsection will be described the two proposed algorithms, based on GA, adapted for MOP and applied for optimal coordinated tuning of parameters belonging to the group of compensators.

MOGA based on Pareto Rules
Considering the coordinated tuning problem of compensator parameters of a group, the Dominance Pareto Rules are described through the following mathematical expressions [9]: , where r and s ∈ (1, 2, …, N) | r ≠ s and N indicates the population size in the GA, and k ∈ {1, 2, …, f}, such f corresponds to the number of objective functions considered in the optimization procedure; ii) ∃ i, such that, at least one of the entries satisfies F i (x j ) < F i (x k ).
The Pareto Rules are applied on each solution of the GA population, in order to determine how many numbers of solutions are better than other one.This number defines the Dominance index for each feasible solution.The individual (solution), which Dominance index is null, is considered as an optimal solution.This classification method is made in each generation of the GA.Then, in each generation all optimal solutions obtained by the Pareto Rules are saved in a group, called the Pareto Front (PF); and, it is also actualized in each generation.In Fig. 3 is shown a pseudocode of the GA adapted for MOP by using the Dominance Pareto Rules.
These rules are applied on each solution, and then are compared with the rest of population, in order to determine how many solutions are better than the corresponding solution, defining its dominance index.

4.2.
Actualize the optimal solutions in PF(t)  All optimal solutions, which are obtained by applying the Dominance Pareto rules, are reproduced, in each generation, inside the PF(t) | t is an iteration counter.The PF(t) is actualized in each generation.In the mathematical expression, the Fitness calculation is directly associated to the Dominance index of the individual, such as indicated in Fig. 3 (b).The GA adapted for MOP using the Dominance Pareto Rules is called as GADP.

MOGA based on Weighted Sum Method
In this case, the GA such as described in [8] was adapted for MOP according to equation ( 5): The coefficients of equation ( 5) correspond to the normalization factors, where a q = c q / F max q and q ∈ {1, 2, 3}.F max q is the maximum value of the q-th objective function, and the coefficient c q is any value such that c q ≥ 1, this methodology avoids certain objective functions dominate over the rests [9], carrying to a local optimum [9].

Initial Population
In this work, N -D feasible individuals are generated randomly, where N = 50 individuals.The remaining D individuals are estimated through equation ( 6), varying the reference voltage, V REF,i , and then is calculated the necessary reactive power to be injected into the system, in the corresponding i-th controlled substation, which also defines the initial reactive compensation capacity of the i-th SVC device.
The equation ( 6) describes the Q-V sensitivity curve, obtained by the matrix equation of the active and reactive power deviation of the Load Flow algorithm [2].Each compensator is located in a substation, associated to the load zone, where is required a high reactive power value in order to maintain the associated voltage module in 1 p.u. So, this methodology also defines the i-th controlled substation.

GA Operators
A Stochastic Tournament Selection, with five individuals, was used in order to choose the probabilistic better solution for next generation.Then, it was utilized an Arithmetic Crossover [12], with probability p c = 0,7, and Mutation operator, with a constant probability p m = 0,01, for getting new individuals.

Characteristics of Power System Test
The implemented coordinated tuning algorithms are applied on IEEE14 system [10], show in Fig. 4.  In this paper, the coordinated tuning methodology adjusts two compensators.The Q-V sensitivity curve, which procedure was described in subsection 3.4, determined the load buses 13 and 14 for installing each SVC.

Computational Environment
The MatLab ® [13] was used for Load Flow implementation and GA adaptation for MOP.The Power System Analysis Toolbox (PSAT) [14] was used to evaluate the dynamic perform of each SVC.

Analysis of Numerical Results
The GA, based on Weighted Sum methodology (GAWS), was executed 5 times for each different Fitness.The different mathematical expressions for Fitness were obtained by modification of each coefficient, such as described in equation ( 5).In Table 2 are shown the different values considered for each coefficient, the best numerical result obtained by each GAWS execution, and the average computational time spent in each running.(p.u.)In Table 3 are shown the numerical results of GA, based on Pareto Dominance rules (GAPD).In Fig. 5, the variables: V REF , Q min and Q max correspond to the adjusted parameters, at steady state.The dynamic model, associated to each SVC, generates the necessary reactive power to be injected to the PES for correcting and regulating the voltage level of the whole system for any disturbance.The 5 th scenario is the most critical operating condition and is simulated dynamically using the software PSAT, and numerical results are shown in Fig. 6; where, TL 6 -13 and TL 9 -14 are disconnected at 1 and 2 s, respectively, after starting the time simulation.In Fig. 6 (a) the simulation was made without any SVC installed in the system test; but the dynamic results considering the compensators adjusted and installed in substations 13 th and 14 th are shown in Fig. 6     The disconnection of three transmission lines was simulated in order to validate the dynamic performance of the group of compensators, adjusted by applying the proposed multi-objective coordinated tuning algorithm.Then, the TL 6 -13, TL 9 -10 and TL 9 -14 are disconnected at 1, 2 and 3 s, respectively, after starting the time simulation, and the numerical results are shown in Fig. 7.
The simulation was made, firstly, without any compensator installed in the system, illustrating in Fig. 7 (a) the voltage drops, because insufficient reactive compensation.Then, in Fig. 7 (b), the response curve of each voltage module was simulated considering the compensator group adjusted by the proposed methodology.The numerical results, such as shown in Table 5 and dynamic simulations illustrated in Fig. 6 and Fig. 7, indicate the optimal performance of the SVC devices.The group of adjusted compensators maintains a good voltage level in the whole PES, in steady state, and, shows a good dynamic performance on single contingencies.

Conclusions
The two proposed coordinated tuning procedures are able to adjust several static compensator devices, considering several operating conditions, simultaneously.Both methodologies are based on the MOGA, the GAWS and the GAPD, and modify the Fitness calculation.In the GAWS algorithm, the Fitness is calculated with the weighted sum of the considered objective functions, and is obtained a unique optimal solution at the end of the execution.However, in the GAPD algorithm, the Pareto Dominance rules are applied in order to obtain a group of optimal solutions.The GAPD algorithm owns greater search capacity than the GAWS search procedure, according to the numerical results, despite of spending more computational time.The set of compensators, adjusted at steady state, also presents a good dynamic performance in single contingences, like transmission line disconnection.

Figure 1 .
Figure 1.Steady State model of the Static Var Compensator.(a) SVC structure.(b) SVC representations in different operating regions.In the control region

Figure 2 .
Figure 2. Voltage Characteristics versus Reactive Power to be injected.

Figure 3 .
Figure 3. Pseudocode of the MOGA based on Dominance Pareto Rules.

Figure 5 .
Figure 5. Dynamic model of the SVC voltage regulator.
(b).(a) Simulation without any SVC.(b) Simulation with the SVC group adjusted.

Figure 6 .
Figure 6.Time simulation of the 5 th Operating Condition.

Figure 7 .
Figure 7. Time simulation of Three Transmission Lines disconnection.

Table 1 .
Operating Conditions considered on IEEE14 system.

Table 3 .
Pareto Front of the GAPD algorithm, obtained in one execution.