Coordinated Tuning of a Group of Static Var Compensators Using Multi-Objective Genetic Algorithm
Enrique Ramón Chaparro Viveros
Facultad Politécnica de la Universidad Nacional del Este, Centro de Investigación
Itaipu Binacional, División de Estudios Eléctricos y Normas
Ciudad del Este, Paraguay
enriquerchv@fpune.edu.py, cver@itaipu.gov.py
and
]]> Manuel Leonardo Sosa RíosFacultad Politécnica de la Universidad Nacional del Este, Centro de Investigación
Ciudad del Este, Paraguay
Abstract
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.
Spanish abstract
El óptimo ajuste coordinado de un grupo de Compensadores Estáticos de Reactivos, en régimen permanente, permite a los Sistemas Eléctricos de Potencia (SEP) operar bien cercanos de sus límites operativos, manteniendo estable el nivel de voltaje en diversas condiciones de operación. El referido problema de ajuste coordinado fue considerado como un Problema de optimización Multi-Objetivo (PMO) con tres funciones objetivo a ser optimizados: la inversión financiera, para la adquisición del grupo de compensadores, el máximo desvío de tensión y la pérdida total de potencia activa. El Algoritmo Genético (AG), el cual pertenece a la familia de los Algoritmos Evolucionarios, fue utilizado y adaptado para PMO, obteniendo un AG Multi-Objetivo (AGMO). Los parámetros que serán ajustados en cada uno de los compensadores son: el voltaje de referencia, y el valor mínimo y máximo de la potencia reactiva inyectada al sistema. En el presente trabajo, el número de compensadores y sus localizaciones fueron determinadas usando la curva de sensibilidad Q–V, proveniente del algoritmo de Flujo de Potencia basado en el método de Newton–Raphson. El método de ajuste coordinado propuesto será validado considerando un ejemplo de SEP, en donde fueron localizados y ajustados específicos compensadores. Simulación en el tiempo fue realizada para evaluar el desempeño dinámico del ajuste coordinado en régimen permanente.
Keywords: Static Var Compensator, Coordinated Tuning, Multi-Objective Optimization Problem, Multi-Objective Genetic Algorithm.
Palabras–clave: Compensador Estático de Reactivos, Ajuste Coordinado, Problema de Optimización Multi-Objetivo, Algoritmo Genético Multi-Objetivo.
1 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.
2 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 substation1. Fig. 2 describes the mathematical expression regarding to the voltage level of the controlled substation with the reactive power injected to the system.
Figure 2. Voltage Characteristics versus Reactive Power to be injected.
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 Δxi = Qi,SVC [11], and i in {1, 2, …, p} | p is the number of compensators considered in the group, as indicated in equation (1).
(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 Δxi, are equal to zero.
3 Proposed Tuning Procedure
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 
(2)
where PGi, QGi, PLi, QLi 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, Vk0 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: PGmin,i ≤ PGi ≤ PGmax,i , QGmin,i ≤ QGi ≤ QGmax,I; and restrictions which limit the voltage level in substations regarding to load zones: Vmin,j ≤ Vj ≤ Vmax,j| j ≤ {1, 2, …, nbL} | nbL is the number of load centers in the PES.
3.1 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]:
(3)
Such as described before, the parameters to be optimized for each SVC device are: a) the reference voltage, VREF (where, VREF = Vk0), of the AVR associated, b) the minimum reactive power to be injected, Qmin, and, c) the maximum reactive power to be injected by each SVC device, Qmax. Each parameter value belongs to the following search space:
(4)
3.2 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, F1(x), for acquiring the set of compensators, b) the Maximum Voltage Deviation, F2(x), and, c) the Maximum Total Active Power Loss, F3(x), calculated by considering all selected critical operating scenarios.
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(4) In equation (4), F1(x) is directly proportional to the compensation capacity of each SVC, where Bi is the monetary value for each MVAr of the i-th compensator; and, nSVC indicates the number of compensators to be adjusted in the PES. In this work, each Bi = 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.
3.3 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.
3.3.1 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]:
Fk(xr) ≤ Fk(xs), 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;
∃i, such that, at least one of the entries satisfies Fi(xj) < Fi(xk).
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.
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.
3.3.2 MOGA based on Weighted Sum Method
In this case, the GA such as described in [8] was adapted for MOP according to equation (5):
(5)
3.4 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, VREF,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.
, ⇒
(6)
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.
3.5 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 pc = 0,7, and Mutation operator, with a constant probability pm = 0,01, for getting new individuals.
4 Experimental Results
4.1 Characteristics of Power System Test
The implemented coordinated tuning algorithms are applied on IEEE14 system [10], show in Fig. 4.
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Table 1. Operating Conditions considered on IEEE14 system.
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.
]]> 4.2 Computational EnvironmentThe 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.
4.3 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.
Table 2. Numerical results obtained by GAWS.
In Table 3 are shown the numerical results of GA, based on Pareto Dominance rules (GAPD).
Table 3. Pareto Front of the GAPD algorithm, obtained in one execution.
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In Table 2 and Table 3, the first SVC parameters correspond to the compensator installed in the 13th substation, and the second parameters belong to the compensator located in the 14th substation. Such as shown in Table 2, the set of SVC parameters, located on the 8th line, corresponds to the best solution obtained by the GAWS algorithm. The main computational time, spent by each GAWS running, is approximately equals to 198 s, but, it produces a unique solution at the end of its execution.
In contrast, the GAPD got a family of optimal solutions in a single run of the algorithm, according to the results shown in Table 3, and spent approximately 241 s for obtaining the solution set. The referred computational time, spent by the GAPD algorithm, is greater than the execution time of the GAWS, mainly due to the classification process of the optimal solution set, applying the Pareto rules, made in each generation and saved on the Pareto Front group. The 7th individual regards to the best solution in the Pareto Front of the GAPD. This solution is associated to the less financial investment, the identical voltage deviation and the same order of the active power loss value, comparing with the best numerical solution obtained by the GAWS algorithm.
The voltage values in the substations of the IEEE14, without any SVC installed and obtained by the Newton–Raphson Load Flow algorithm for each operating conditions described in subsection 4.1, are shown in Table 4.
Table 4. Voltage values in the IEEE14 system, without any SVC installed.
Table 5. Voltage values in the IEEE14 system, with SVC installed in substations 13th and 14th.
Such as illustrated on Table 5, the most critical operating condition is the 5th Scenario, where the voltage modules of the whole system are successfully corrected by installing the group of compensators, which parameters were adjusted by the GADP algorithm.
An optimal coordinated tuning of the compensators, at steady state, also determines a good dynamic performance on single contingencies. The most critical operating conditions, such as the 5th and 8th scenarios, were dynamically simulated in order to validate the referred hypothesis. For that reason, a standard dynamic model of the AVR, associated to each SVC [2], was used and it is described in Fig. 5.
In Fig. 5, the variables: VREF, Qmin and Qmax 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 5th 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 13th and 14th are shown in Fig. 6 (b).
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(a) Simulation without any SVC. |
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Figure 6. Time simulation of the 5th Operating Condition.
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Figure 7. Time simulation of Three Transmission Lines disconnection.
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.
5 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.
References
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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.