Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy
Optimal Location of FACTS Devices to Enhance Power System Security S. Gerbex, R. Cherkaoui, and A. J. Germond, Member, IEEE
Abstract—This paper compares three heuristic methods (SA, TS and GA) applied to the optimal location of FACTS devices in a power system. The optimizations are made on three parameters: the location of the devices, their types and their sizes. The FACTS devices are located in order to enhance the system security. Five types of FACTS controllers are modeled for steady-state studies: TCSC, TCVR, TCPST, SVC and UPFC. Simulations are performed on an IEEE 118-bus power system for several numbers of devices. Results show that the three algorithms converge to similar optimal solutions. The security margin of the system may be increased with the use of FACTS devices, but some limitations are observed. The locations of the devices and their influence areas are analyzed. Index Terms—FACTS devices, Heuristic methods, Optimal location, Power system security.
devices, beyond which this loadability cannot be improved, has been observed [5]. It is important to choose the suitable type(s) of devices in order to reach the required goals. In this paper, we look for the optimal location of several different types of FACTS devices, with specific characteristics. They are modeled for steadystate analysis [6] and located in order to maximize the security margin of the system in terms of branch loading and voltage levels. The optimal location of a given number of FACTS devices nF is a problem of combinatorial analysis. To solve such kind of problem heuristic methods can be used. They permit to obtain acceptable solutions within a limited computation time. Three heuristic methods have been chosen for comparison: the Simulated Annealing (SA), the Tabu Search method (TS), and the Genetic Algorithms (GA).
I. INTRODUCTION
W
the electricity ymarket deregulation, the number of unplanned power exchanges increases, due to the competition among utilities and due to contracts concluded directly between producers and consumers. If these exchanges are not controlled, problems may appear with the power flows, which obey Kirchhoff’s laws and some lines located on particular paths may become overloaded. Before the opening of the market, the control of power flows was mostly realized by reallocating productions. In a deregulated environment, this kind of control is subject to an ancillary services market. Therefore it is in the interest of the Transmission System Operator (TSO) to have another way of controlling power flows in order to permit a more efficient and secure use of transmission lines. The FACTS devices (Flexible AC Transmission Systems) could be a means to carry out this function without the drawbacks of the electromechanical devices (slowness and wear). Studies and realizations have shown their capabilities in steady-state or dynamic stability [1]. With their ability to change the apparent impedance of a transmission line, FACTS devices may be used for active power control, as well as reactive power or voltage control. For a meshed network, an optimal location of FACTS devices allows to control its power flows [2] and thus to increase the system loadability [3] and the security margin [4]. However, a limited number of ITH
This work is supported by the Swiss foundations PSEL and CREE-RDP. S. Gerbex, R. Cherkaoui, and A. J. Germond are with École Polytechnique Fédérale de Lausanne, Laboratoire de Réseaux Électriques, 1015 Lausanne, Switzerland (e-mail:
[email protected]).
0-7803-7967-5/03/$17.00 ©2003 IEEE
II. FACTS DEVICES A. Generalities In a power system, the FACTS devices may be used to achieve several goals. In steady-state, for a meshed network, they can permit to operate transmission lines close to their thermal limits and to reduce the loop flows. In this respect, they act by supplying or absorbing reactive power, increasing or reducing voltage and controlling series impedance or phaseangle [1, 6]. The FACTS technology allows extending the transmission limits of a power system in a step-by-step manner as and when required. Furthermore, it offers the possibility to relocate such an installation when it is not useful anymore. Different types of devices have been developed. Three categories of FACTS controllers may be distinguished: • series controllers; • shunt controllers; • combined series-shunt controllers. Inside a category, several FACTS devices exist and each one has its own properties and may be used in specific contexts. The choice of the appropriate device is important since it depends on the goals to be reached. Five different types of devices have been chosen. Their models are developed to be suitable for steady-state. Each device may take a fixed number of discrete values. Only one FACTS device of a given type per branch may be allowed. On
the other hand, several devices of different types may be located in the same line. B. Steady-state models of FACTS devices The TCSC (Thyristor Controlled Series Capacitor) may have one of the two possible characteristics: capacitive or inductive, respectively to decrease or increase the impedance of the branch. It is modeled with variable series reactance. Its value is function of the reactance of the line XL where the device is located. It is in the range: −0.8 X L ≤ X TCSC ≤ 0.2 X L p.u. (1)
The TCVR (Thyristor-Controlled Voltage Regulator) operates by inserting an in-phase voltage to the main bus voltage so as to change its magnitude. It is modeled by an ideal tap changer transformer in series with the branch. Its value depends on the main bus voltage magnitude Vb of the line in which the device is located. The additional voltage is in the range: −0.15Vb ≤ VTCVR ≤ 0.15Vb p.u. (2) The TCPST (Thyristor-Controlled Phase Shifting Transformer) acts by adding a quadrature component to the prevailing bus voltage in order to increase or decrease its angle. It is modeled as an ideal phase shifter inserted in series with the branch. The scale of the angles is:
−15 ≤ δ TCPST ≤ 15 deg
(3)
The SVC (Static Var Compensator) may have two characters: inductive or capacitive, respectively to absorb or provide reactive power. The SVC is represented by a shunt variable susceptance inserted in the bus or at the mid point of the line. It may take values characterized by the reactive power QSVC injected or absorbed at the voltage of 1 p.u. The possible values are function of the considered power system. In this paper, we fixed:
−200 ≤ QSVC ≤ 200 MVar
(4)
The UPFC (Unified Power Flow Controller) is modeled by the simultaneous presence of several devices in the same line. A SVC at a bus and other devices in an adjacent branch may be also interpreted as an UPFC. The characteristics of the elements used to represent this device are the same as above for the TCSC, the TCVR, the TCPST, and the SVC.
III. OPTIMIZATION ALGORITHMS A. Heuristic methods Heuristic methods may be used to solve combinatorial optimization problems. These methods are called “intelligent”, because the move from one solution to another is done using rules close to the human reasoning. The heuristic algorithms search for a solution inside a subspace of the total search space. Thus they are able to give a good solution of a certain problem in a reasonable computation time, but they do not assure to reach the global optimum. The most important advantage of heuristic methods lies in the fact that they are not
limited by restrictive assumptions about the search space like continuity, existence of derivative of cost function [7]. In a very general manner, the principle of a heuristic method may be represented with the Fig. 1. The specificity of each method lies mainly in the way of moving from the current solution(s) to the new solution(s). define solution encoding generate initial solution(s)
evaluate objective function stop criterion reached ?
create new solution(s)
no
yes return best solution Fig. 1. General principle of a heuristic method
The three following heuristic implemented.
methods have been
1. The Simulated Annealing (SA) is inspired by the physical process of slowly cooling a metal [8]. A new configuration is constructed by imposing a random change to the current solution. Moves to worse solutions are occasionally accepted with a probability, the Metropolis criterion, controlled by a parameter called temperature. The probability of accepting worse solution decreases as the temperature decreases. 2. The Tabu Search method (TS) is based on selected concepts of artificial intelligence [9]. Solutions of the search space are visited using an operation called a move to define the neighborhood of any solution. The best solution in the neighborhood is selected to be the new solution, even if it is worse than the current one. In order to prevent cycling in the search process, the most recent moves are stored in a tabu list. These moves are forbidden during a specified number of iterations. However a tabu solution may be accepted according to an aspiration criterion. 3. The Genetic Algorithms (GA) are based on the mechanisms of natural selection [10]. The optimal solution is sought after from a population of solutions using random process. A new generation is created by applying to the current population the three following operators: selection, crossover and mutation. The common parts of the three algorithms, i.e. the solution encoding, the generation of the initial solution(s), the objective function, and the stop criterion are first described. The implementations of the three methods are then presented
separately. The parameters and the characteristics of each one are discussed.
The order of appearance of the devices in the string is indifferent. It may be important during the move from a solution to another one.
B. Solution encoding The goal of the optimization is to find the best location of a given number of FACTS devices. A configuration of nF FACTS devices is defined with three parameters: the location of the devices, their types and their sizes. Thus, a configuration is represented with three strings; each one is related to one of the parameters described below. 1. The first corresponds to the location of the devices and it contains the numbers of the elements (nodes and branches) where the FACTS are located. The possible values are identified following the Table 1. NUMBERING OF THE POWER SYSTEM ELEMENTS
Elements
1
bus 1
nn nn + 1
bus nn branch 1
nn + nb
branch nb
2. The second string is related to the types of the devices. A value is assigned to each type of modeled FACTS: 1 for TCSC, 2 for TCVR, 3 for TCPST, and 4 for SVC. 3. The last one indicates the size of the FACTS devices. It may take nv discrete values normalized to be in the range 0 to 1. The minimum value that the device can take corresponds to 0 and the maximum to 1. The Fig. 2 shows an example of configuration of five FACTS devices on a 7-bus, 11-branch network and the corresponding three coded strings. For instance, the third device is a TCSC located on branch 10 (element 17). Its value is a capacitance of -0.7 XL10, where XL10 is the reactance of the line 10. The two devices located on the element number 10 (branch 3) are a TCPST and a TCVR. Together, they represent an UPFC. branch 1 bus 1
branch 3 bus 3
bus 2 branch 4 branch 5
branch 2
TCSC
branch 6
TCVR
branch 7 bus 4
bus 5
TCPST SVC
branch 8
branch 9
UPFC branch 10 bus 6
D. Objective function The aim of the optimization is to enhance the security level of the system. The FACTS devices are located in order to remove and prevent overloads and over- or under-voltages. The objective function is based on indexes quantifying the severity of the contingencies in terms of branch loading and voltage levels [11]. Thus, for a given system load, we look for the best configuration of FACTS devices minimizing the following objective function:
TABLE 1
Values
C. Initial solution(s) For the three optimization methods, the initial solution is randomly generated. The locations of the devices, their types and their sizes are drawn using the encoding aforementioned. The constraint to not have more than one device of a specified type in given element of the network is taken into account. In the case of GA, ni solutions are generated to constitute the initial population.
bus 7 branch 11
a)
J = ¦ wi Vi − Vref i
n
i
§ S + ¦ wk ¨¨ k k © S max k
· ¸ ¸ ¹
n
(5)
where Vi and Vref i are the voltage magnitude and the nominal voltage at bus i. Sk is the apparent power in line k and Smax k is the apparent power rate of the line k. The weights wi and wk are calculated in order to have the same index value for voltage difference of 10% and for a branch loading of 100%. They could also be used to give more or less importance to specific elements of the system. The exponent n is equal to 4 in order to give more importance to high levels of voltage variations and overloads. E. Stop criterion In order to help the comparison between optimization methods, the same stop criterion has been chosen for the three algorithms. The simulations are stopped when a maximum number of solutions have been visited. This value is adapted in function of the size of the search space. In particular, it depends on: • nn the number of buses of the power system; • nb the number of branches of the power system; • nF the number of FACTS devices to locate; • nv the number of possible discrete setting value for a device. F. Simulated Annealing
b)
0.8 0.3 0.1 0.6 0.4 3 4 1 2 3 10 5 17 10 9
size type location
Fig. 2. Representation of a 5 FACTS devices configuration: a) power system, b) encoded solution
Starting from an initial temperature (its determination will be discussed below), a new solution is generated by applying random changes to the current solution. As shown on Fig. 3, the neighborhood of a solution is chosen in a way that the parameters of only one FACTS (location, type, size) are modified at time. If the objective function of the new solution
is lower than the objective function of the current solution, then the new solution is accepted and becomes the current solution (Fig. 3.a). On the other hand, if the new solution has a higher objective function, the Metropolis criterion is applied. A random number is generated in the range 0 to 1 and compared with the probability of acceptance:
pMet = e − ∆J T
(6)
where ∆J = Jnew_sol - Jcurrent_sol, and T is the current temperature. If the random number is smaller than pMet the new solution is accepted as current solution (Fig. 3.b). Otherwise it is rejected (Fig. 3.c). At a given temperature, the equilibrium is reached when the parameters of each device have been changed a certain number of times. The amount of new solutions generated increases when temperature decreases. a)
0.5 0.5 0.4 0.1 0.4 3 1 4 2 4 12 8 2 15 10
accepted
0.5 0.5 0.4 0.1 0.4 2 1 4 2 4 12 8 2 15 10
accepted
0.5 0.5 0.4 0.1 0.4 2 1 4 2 4 12 8 2 13 10
not accepted
b)
0.5 0.5 0.4 0.1 0.4 1 4 2 4 12 8 2 15 10 3
c)
0.5 0.5 0.4 0.1 0.4 1 4 2 4 12 8 2 15 10 d)
objective function
2
where nn is the number of buses of the system, and nb the number of branches. A tabu solution may become the new current solution if the aspiration criterion is satisfied. Such move is possible only if the objective function of the tabu solution is lower than the best objective function so far achieved. The Fig. 4 illustrates the transition to a new solution. The neighboring solutions are created by imposing a random change on the second device of the current solution (Fig. 4.a). The best neighboring solution not tabu is selected to become the new solution. The modification leading to the second solution is present in the tabu list and this solution does not satisfy the aspiration criterion. Thus the fourth solution becomes the new current solution (Fig. 4.b and c). b)
a)
a)
c)
search space
Fig. 3. Moves with the SA: a) solution directly accepted, b) solution accepted with the Metropolis criterion, c) solution not accepted, and d) objective function
Then the temperature is updated. It is decreased in the following way: Tnew = α Tcurrent (7) where α is a positive constant less than 1, usually close to unity. Its value is fixed to 0.95. The SA algorithm needs to start from high temperature. However, if this temperature is too high, the computation time will increase unreasonably. Initial temperature value T0 is computed so that almost all transitions to a neighboring solution are initially accepted with the Metropolis criterion. The initial acceptance ratio is equal to:
pin =
(9)
0.5 0.1 0.4 0.1 0.4 3 1 4 2 4 12 8 2 15 10
c) b)
sTL = nn + nb
number of moves accepted at T0 > 0.95 total number of moves attempted at T0
(8)
Temperature is increased in a step by step manner, until this initial acceptance is reached.
G. Tabu Search method The move to a new solution is done by selecting the best solution not tabu in the neighborhood of the current one. Such
objective function
0.5 0.1 0.4 0.1 0.4 1 4 2 4 12 8 2 15 10 3
as in the SA, a neighboring solution is obtained by applying an elementary modification on one device to the current solution. Only a part of the neighboring solutions are visited. They are randomly drawn. The tabu list contains the location and the type of the devices that have been removed from a branch to be located on another one. Its size sTL is fixed to:
current solution
0.5 0.1 0.4 0.1 0.4 3 1 4 2 4 12 15 2 15 10
not tabu
0.5 0.1 0.4 0.1 0.4 3 4 4 2 4 12 8 2 15 10
tabu
0.5 0.5 0.4 0.1 0.4 3 1 4 2 4 12 8 2 15 10
not tabu
0.5 0.1 0.4 0.1 0.4 3 1 4 2 4 12 10 2 15 10
not tabu
new current solution
search space
Fig. 4. Move with TS: a) current solution, b) neighboring solutions, and c) objective function
H. Genetic Algorithms
The objective function is computed for every individual and mapped to a fitness function. The value of the fitness function fit of a solution is given by: 1 fit = (10) J − J max 0 .1 + + 0.9 J max − J min where J is the value of the objective function of the solution, and Jmin, Jmax are respectively the minimum and the maximum values of the objective function of the generation. The obtained values are employed to create a biased roulette wheel. It is used for the move to a new generation (Fig. 5). After that, the operators of selection, crossover and mutation are applied successively to generate the offsprings. In turn, two individuals are randomly drawn from the population and reproduced (Fig. 6). The probability of drawing an individual is proportional to its part on the biased roulette wheel.
10 16 4 9
b) 0.8 0.3 0.1 0.6 0.4 3 4 1 2 3 0.9 5 17 0.6 10 0.2 1.0 9 0.4 1 4 3 2 4 0.7 3 11 0.2 10 0.5 0.7 6 1.0 4 2 4 4 1 10 0.6 18 0.2 0.0 1 0.6 9 0.7 2 3 4 4 2 11 4 5 14
a)
objective function
a)
b)
0.8 0.2 0.5 0.6 0.4 3 4 4 2 3 10 1 18 10 9 search space
0.4 0.2 0.5 0.6 0.4 4 1 2 3 10 5 18 10 9 3
Fig. 8. Mutations on an individual: a) before, b) after
Operations of selection, crossover and mutation are repeated until the number of desired offsprings is created to constitute the new generation.
c)
Fig. 5. Computation of the biased roulette wheel: a) population, b) objective function, and c) roulette wheel
IV. RESULTS A. Test system
2
1 2 1
0.7 0.2 0.5 0.7 1.0 4 2 4 4 1 4 10 18 1 9 0.8 0.3 0.1 0.6 0.4 3 4 1 2 3 10 5 17 10 9
An IEEE 118-bus, 187-branch power system is used for the simulations. The Fig 9 represents the result of a power flow computation for the base load. Two lines are overloaded (in dark red) and some voltages magnitudes are critical (less than 0.95 p.u.).
Fig. 6. Selection of 2 individuals
The crossover may occur with a probability pc; generally close to 1. A double crossover is applied as shown in Fig. 7. Two crossing sites are picked up uniformly at random along the individuals. Elements outside these two points are kept to be part of the offsprings. Then, from the first position of crossover to the second one, elements of the three strings of both parents are exchanged. As previously mentioned, only one FACTS device of a given type is authorized on an element of the power system. Therefore, if the crossover leads to place a second device of the same type on a bus or a branch, a correction has to be applied. An algorithm similar to the partially matched crossover (PMX) is used to remove the excess device [5].
0.975 p.u. ≤ V ≤ 1.025 p.u. 0.950 p.u. ≤ V < 0.975 p.u.
0.925 p.u. ≤ V < 0.950 p.u. 0.900 p.u. ≤ V < 0.925 p.u.
Fig. 9. Base load case of the IEEE 118-bus test system
a) 0.7 0.2 0.5 0.7 1.0 4 2 4 4 1 4 10 18 1 9 0.8 0.3 0.1 0.6 0.4 3 4 1 2 3 10 5 17 10 9
b) 0.7 0.3 0.1 0.7 1.0 4 4 1 4 1 4 5 17 1 9 0.8 0.2 0.5 0.6 0.4 3 2 4 2 3 10 10 18 10 9
c) 0.7 0.3 0.1 0.7 1.0 4 4 1 4 1 4 5 17 1 9 0.8 0.7 0.5 0.6 0.4 3 4 4 2 3 10 1 18 10 9
Fig. 7. Double crossover of 2 individuals: a) cross-points, b) before, and c) after correction
Mutations are possible independently on all elements of the three strings of an individual. A specific probability is applied for each string: pmL for the first string, pmT for the second and pmV for the last. These probabilities change with the generations. When a mutation occurs on a configuration, a new value among the set of the possible ones is randomly drawn (Fig. 8).
B. Optimization strategy
As explained previously, the FACTS devices are located to enhance the security of the system in terms of branch loading and voltage level according to (5). For several numbers of devices, we look for the best configuration of FACTS, i.e. the best location with the best values of the most appropriate controllers. When the number of devices is increased, the results obtained previously are not taken into account. In others words, FACTS devices may disappear from specific positions to reappear on others when their number is increased. C. System security
Simulations are performed five times consecutively with each of the three optimization algorithms. The Fig. 10 illustrates the evolution of the average of the best objective function obtained during the five optimizations. It shows that the three heuristic methods lead to similar solutions, but TS and GA converge faster than SA to a good solution.
22 Simulated Annealing Tabu Search method Genetic Algorithms
Best objective function J
20 18 16 14 12
TCSC TCVR
10
TCPST
0 0
2000
4000
6000
8000
10000
SVC
0.975 p.u. ≤ V ≤ 1.025 p.u.
UPFC
0.950 p.u. ≤ V < 0.975 p.u.
Number of visited solutions Fig. 10. Evolution of the best objective function with the number of solutions visited (average of 5 simulations and optimization for 5 FACTS devices)
Fig. 12. Optimal location of 5 FACTS devices
The Fig. 11 represents the dependence of the objective function with the number of FACTS devices. It shows that the security of the system increases (the objective function decreases) when the number of devices increases. As in [5], an asymptotical value of the objective function may be observed. According to the used optimization criterion and for the considered power system, the results show that the limit is about 30 devices. 15 ∆Pk ∆PFACTS ≥ 0.01 p.u.
Best objective function J
14
∆Pk ∆PFACTS 0
13
1
0.010 p.u. < ∆V ≤ 0.035 p.u.
12
0.035 p.u. < ∆V ≤ 0.075 p.u.
11 10
Fig. 13. Areas of influence on the power flows and voltage levels.
9 8
V. CONCLUSION
7 0 0
5
10
15
20
25
30
Number of FACTS devices
Fig. 11. Dependence of the objective function with the number of devices
D. Location of the devices
The Fig. 12 gives a configuration obtained for the optimal location of 5 FACTS devices. Compared to the case without FACTS device, under-voltages and overloads have been removed. In other words, the security of the system is increased. In a general way, TCSC and TCPST are used to control active power while SVC and TCVR are employed for reactive power and voltage control. The influence areas on the power flows and on the voltage levels are represented in the Fig. 13. It points out that the control area on the power flows is limited to the loops containing incident lines to the one with the device. The voltage variations are also limited to the neighboring buses.
Three heuristic methods (SA, TS and GA) have been implemented to the optimal location of FACTS devices in a power system. Five types of devices (TCSC, TCVR, TCPST, SVC and UPFC) have been modeled for steady-state studies. Optimizations were performed simultaneously on three parameters (location, type and size). The security of the system was employed as a measure of power system performance. The simulations show that FACTS devices can be used to enhance the security margin of the system. The security increases with the number of devices. However a limited value may be observed. The control area of a device is local for the power flows as well as for the voltages. The three methods lead to similar results, but generally TS and GA converge faster than SA to an optimal solution.
VI. REFERENCES [1] N. G. Hingorani, L. Gyugyi, Understanding FACTS: Concepts and Technology of Flexible AC Transmission Systems, IEEE Press, NewYork, 2000. [2] D. J. Gotham, G. T. Heydt “Power Flow Control and Power Flow Studies for Systems with FACTS Devices”, IEEE Trans. Power Systems, vol. 13, no. 1, pp. 60-65, Feb. 1998. [3] F. D. Galiana, K. Almeida, M. Toussaint, J. Griffin, D. Atanackovic, “Assessment and Control of the Impact of FACTS Devices on Power System Performance”, IEEE Trans. Power Systems, vol. 11, no. 4, pp. 1931-1936, Nov. 1996. [4] S.-H. Kim, J.-U. Lim, S.-I. Moon, “Enhancement of Power System Security Level through the Power Flow Control of UPFC”. Proceeding of the 2000 IEEE/PES summer meeting, pp. 30-43. [5] S. Gerbex, R. Cherkaoui, A.J. Germond, “Optimal Location of Multitype FACTS Devices in a Power System by Means of Genetic Algorithms”, IEEE Trans. Power Systems, vol. 16, no 3, pp. 537-544, Aug. 2001. [6] D. Povh and al, Load Flow Control in High Voltage Power Systems Using FACTS Controllers, CIGRÉ Task Force 38.01.06, Jan. 1996. [7] S.M. Sait, H. Youssef, Iterative computer algorithms with application in engineering: solving combinatorial optimization problems, IEEE Computer Society, 1999. [8] S. Kirkpatrick, C.D. Gellat, M.P. Vecchi, “Optimization by Simulated Annealing”, Science, vol. 220, pp. 671-680, 1983. [9] F. Glover, “Tabu Search”, CAAI Report 88-3, University of Colorado, Boulder, 1988. [10] D. E. Goldberg, Genetic Algorithms in Search Optimization & Machine Learning, Addison-Wesley Publishing Company, Inc., 1989. [11] A. S. Debs, Modern Power Systems Control and Operation, Kluwer Academic Publishers, 1988, pp. 119-122.
VII. BIOGRAPHIES Stéphane Gerbex received the M.Sc. degree in electrical engineering in 1997 from EPFL. He is now Ph.D. student at EPFL, LRE. His research interest is in optimization of FACTS devices location using heuristic methods.
Rachid Cherkaoui received the M.Sc. and the Ph.D. degrees in electrical engineering in 1983 and 1992, respectively, from EPFL. He is now a Senior Engineer at EPFL, LRE. His research interest is in power system simulation and optimization, and distribution system automation.
Alain Germond received the M.Sc. and the Ph.D. degrees in electrical engineering in 1966 and 1975, respectively, from EPFL. Since 1985 he has been Full Professor at EPFL and head of LRE. His research interest is in advanced techniques for power system analysis and control.
