Abstract: This study presents an algorithm, for solving Security Constrained Economic Dispatch (SCED) problem with Flexible AC Transmission Systems (FACTS) through the application of Evolutionary Programming (EP). The problem is decomposed into the optimal setting of FACTS parameters subproblem and the OPF with fixed FACTS parameters subproblem. These two subproblems are solved by Exponential Evolutionary Programming (EEP). Two types of FACTS devices are used: Thyristor-Controlled Series Capacitor (TCSC) and Thyristor-Controlled Phase Shifting (TCPS). The proposed approaches have been implemented on an adapted IEEE 30 bus system. The simulation results indicates are compared and discussed to show the performance of the EP technique.
INTRODUCTION
The Flexible AC Transmission Systems (FACTS) devices are integrated in power systems to control power flow, increase transmission line capability to its thermal limit and improve the security of transmission systems (Hingorani and Gyugyi, 1999). In addition to controlling the power flow in specific lines, FACTS devices could be used to minimize the total generator fuel cost in Optimal Power Flow (OPF) problem. For example, the linear programming based security constrained OPF method (Ge and Chung, 1999) has been proposed to solve OPF with FACTS devices. Load equivalent method (Chung et al., 2000) has been proposed to solve OPF with FACTS devices. Meanwhile, several heuristic methods including local search and Genetic Algorithms (GA) were proposed to determine the optimal parameters of FACTS devices when the power flow control in specific lines is not required (Chung and Li, 2000). Ongsakul and Bhasaputra (2001) have used TS/SA approach to optimal power flow with FACTS devices. Gerbex (2001) have used GA to set the optimal value of multi type FACTS devices in a power system. Nevertheless, the obtained results were far from the optimal solutions. Lai and Ma (1997) have used evolutionary program to solve the power flow problem in FACTS. Yang et al. (1996) have used the EP based algorithm for solving Economic Dispatch (ED) problem with non smooth fuel cost functions. Wong and Yuryevich (1998) have developed a hybrid EP and sequential quadratic programming, to solve the ED problem with non smooth fuel cost function. However, the works mentioned above, each parent generates an offspring with Gaussian mutation and better individuals among parents and offspring are selected as a population of the next generation. Gaussian probability distribution has finite variance therefore it has shortest flat tails comparing with other distribution. Due to the characteristics of probability distribution, global optimum solutions is not guaranteed.
In this study, Thyristor-Controlled Series Capacitor (TCSC) and Thyristor-Controlled Phase Shifting (TCPS) are integrated in OPF by using the reactance model and the injected power model, respectively. For OPF control, TCSC and TCPS are used to minimize the total generator fuel cost subject to power balance constraint, real and reactive power generation limits, voltage limits, transmission line limits and FACTS parameters limits. The proposed method solves the optimal settings of FACTS parameters in the first subproblem and conventional OPF subproblem. It is tested and compared to the GA and Hybrid TS/SA on the modified IEEE 30 bus system with TCSC and TCPS at the fixed locations.
MATERIALS AND METHODS
OPF With Facts Devices
A Static model of TCSC and TCPS are used in this study. TCSC can be
seen as a series reactance with control parameter Xs. Figure
1 shows the model of TCSC. It is integrated in the OPF problem by
modifying the line data. A new line reactance (Xnew) is given
as follows:
(1) |
The power flow equations of the line with a new line reactance can be derived as follows:
(2) |
(3) |
(4) |
(5) |
(6) |
δij is the voltage angle difference between bus i and j.
TCPS can be modeled by a phase shifting transformer with control parameter αp. Figure 2 shows the model of TCPS. The power flow equations of the line can be derived as follows.
(7) |
| Fig. 1: | Model of TCSC |
| Fig. 2: | Model of TCPSC |
| Fig. 3: | Injected power model of TCPS |
(8) |
(9) |
(10) |
| k | = | cos(αp) |
| δ | = | δij+αp |
The injected power is used to model TCPS as shown in Fig. 3. The injected real and reactive power flow TCPS at bus and bus are as follows:
(11) |
(12) |
(13) |
(14) |
| t = tan (αp) |
Problem formulation
The SCED problem with FACTS devices can be formulated as:
(15) |
where, ai, bi and ci are cost coefficients of generator i and Pi is the power generated by the ith unit, Fi (Pi) is the generation cost function for Pi generation at bus i, N is number of bus, subject to
| • | The power balance constraints |
(16) |
where, PD is the system load demand, Pi (αi) is the total injected power demand at bus i (MW), Vi is the voltage magnitude at bus i, Vj is the voltage magnitude at bus j, Yij (Xs) is the magnitude of the ijth element in Ybus with TCSC included, θij (Xs) is the angle of the ijth element in Ybus with TCSC included, αi is the phase shift angle of TCPS number i, NP is the set of TCPS indices.
| • | The inequality constraint on real power generation at bus i |
(17) |
where, Pimin and Pimax are, respectively minimum and maximum values of real power generation allowed at generator bus i.
| • | The power flow equation of the power network is given by |
(18) |
Where:
| Pi and Qi | = | Calculated real and reactive power for PQ bus i |
| Pinet and Qinet | = | Specified real and reactive power for PQ bus i |
| Pm and Pmnet | = | Calculated and specified real power for PV bus m |
| V and φ | = | Voltage magnitude and phase angles at different buses |
| • | The inequality constraint on reactive power generation Qi at each PV bus |
(19) |
where, Qimin and Qimax are, respectively minimum and maximum value of reactive power at PV bus.
| • | The inequality constraint on voltage magnitude Vof each PQ bus |
(20) |
where, Vimin and Vimax are, respectively minimum and maximum voltage at bus i
| • | The inequality constraint on phase angle of voltage at all the buses i |
(21) |
where, φimin and φimax
are, respectively minimum and maximum voltage angles allowed at bus i
| • | MVA flow limit on transmission line |
(22) |
| • | TCSC reactance limit |
(23) |
where, Xsi is the reactance of TCSC number i and
| • | TCPS phase shift angle limit |
(24) |
New OPF Formulation
The FACTS devices parameters in Eq. 15 are additional
control variables that cannot be solved by the conventional OPF because
these parameters will change the admittance matrix. Therefore, the OPF
with FACTS devices problem is decomposed in two subproblems. The first
subproblem is optimal setting of FACTS parameters and the second subproblem
is OPF with fixed FACTS parameters.
Optimal Setting of FACTS Parameters Subproblem
The proposed EEP method is used to determine the optimal setting of
FACTS parameters, minimizing the generator fuel cost within power flow
security limits. FACTS devices control variable in Eq.
16 will be fixed in the conventional OPF subproblem, which is solved
by EEP. The results from EEP OPF are used to evaluate the quality of FACTS
parameters.
Opf with Fixed FACTS Parameters Subproblem
The OPF with fixed FACTS parameters subproblem is expressed as:
(25) |
Subject to:
(26) |
(27) |
(28) |
(29) |
(30) |
EXPONENTIAL EVOLUTIONARY PROGRAMMING
Overview
The conventional EP employing the Gaussian mutation operator is called
as Classical Evolutionary Programming (CEP). The EP using Cauchy mutation
operator is called as Fast Evolutionary Programming (FEP) as it converges
faster than CEP.
Cauchy mutation is more likely to generate an offspring further away from its parent than Gaussian mutation due to its long flat tails. It is expected to have a higher probability of escaping from a local optimum, especially when the basin of attraction of the local optimum or the plateau is large relative to the mean step size.
The EP that uses the double exponential mutation operators is called as Exponential EP (EEP) as it has higher convergence rate compared to CEP and FEP (Narihisa and Kohmoto, 2006). Considering the shape of the three fundamental one-dimensional distributions shown in Fig. 4, Cauchy distribution has the longest flat tails and Gaussian distribution has the shortest flat tails. In other words, the shape of double exponential distribution has the middle long flat tails among three distributions. This fact can be expected that double exponential distribution may have the both merits of Gaussian and Cauchy distribution. Due to the characteristics of probability distributions, global optimum solutions with less time is guaranteed.
Double Exponential Probability Distribution
The one-dimensional probability density function of double exponential
probability distribution for parameter λ is given as:
| Fig. 4: | Distribution of N(0,1), C(0,1) and E(0,1) |
(31) |
The mean value of the probability density function
(32) |
The double exponential probability distribution based random number is
E (0, λ), which has the mean value of
Therefore,
IMPLEMENTATION OF EXPONENTIAL EVOLUTIONARY
PROGRAMMING TO SCED
Evolutionary programming is a probabilistic, global search technique
that starts with a population of randomly generated candidate solutions
and evolves towards better solutions over a number of generations or iterations.
The main stages of this technique include initialization, mutation and
competition and selection. The major steps involved in the evolutionary
programming approach are explained as:
Initialization
The initial population comprises combinations of only the candidate
dispatch solutions which satisfy all the constraints. It consists of [Xsj,
αj], j = 1, 2,.....I. Where, I is number of trial parent
individuals. The elements of a parent are reactance of TCSC and phase
shift angle of TCPS randomly chosen by a random number ranging over [0,
Xsimax] and [0, αimax].
Creation of Offspring (Mutation)
Using double exponential mutation, an offspring is created by:
(33) |
(34) |
where, Ei (0, λ) is a double exponential random number with parameter λ and is generated anew for each value of i. NS is the set of TCSC indices.
The standard deviation is given by the expression
(35) |
(36) |
where, β is the scaling factor which has to be tuned during the process of search for the optimum around the initial points, fj the fitness value of the jth individual and fmax is the maximum fitness among the I parents. Mutation results in creation of I offspring individuals. The parent individuals are candidate dispatch solutions which satisfy all constraints. However, after mutation, the elements of offspring Pi` may violate constraint Eq. 16. This violation is corrected as follows:
(37) |
However, after mutation, the elements of offspring P`i may
violate constraint Eq. 16. This violation is corrected
as follows:
In the case of SCED problems, the objective function given by Eq.
1 is augmented by a term for the violation of line limits as:
(38) |
where, k1 is a penalty coefficient.
The second term in Eq. 38 are equal to zero during
initialization and they get non-zero value after mutation only if
Competition and Selection
The 2I individuals compete with each other for selection. Fitness
function value is calculated for all the 2I individuals. The fitness function
values are arranged in ascending order. First I fitness functions and
the corresponding I individuals are selected as parents for next generation.
Steps 2 and 3 are repeated until there is no appreciable improvement. The same procedure is repeated for the second subproblem.
Parameter Selection
The final printed size of an au The total number of function evaluations
is fixed at 50 and population size is kept 50. The scaling factor β
is taken as 0.02 for 30 bus system and 0.04 for 10 bus system. The distribution
control parameter λ is discretely increased from 0.01 to 2.5 for
10 bus system and 0.1 to 2.5 for 30 bus system, respectively. Selection
of β and λ are problem dependent.
RESULTS AND DISCUSSION
The algorithm discussed earlier has been tested on an adapted IEEE 30-bus systems (Somasundaram and Kuppusamy, 2005) to assess the performance of the proposed algorithm. The algorithms for solving the examples were implemented on Matlab 6.5 platform. Their solutions are compared in the tables and graphs are plotted to show their relative convergence characteristics. The parameters of EEP approach are set as scaling factor β is self adaptive population size is 50 and maximum number of iterations is 50. The results are also compared with solutions of earlier methods such as genetic algorithm, TS/SA algorithm that were previously reported (Ongsakul and Bhasaputra, 2001).
For the example considered in this study, line security constraint violations can be taken into account by including additional terms with a penalty coefficient in Eq. 31.
Example IEEE 30-Bus System
The algorithm discussed earlier has been tested on adapted IEEE 30-bus
(Somasundaram and Kuppusamy, 2005) systems to assess the performance of
the proposed algorithm. The objective function is the total fuel cost
and the fuel cost curve of the units are represented by quadratic cost
functions. The adapted IEEE 30-bus system consists of 6 generators, 41
lines and a total demand of 283.4 MW. The fuel cost coefficient and the
generator data, load data, line data, transformer data and shunt capacitor
data for the system can be found in (Somasundaram and Kuppusamy, 2005).
Near optimal placements of TCSC and TCPS on the IEEE 30 bus system are
guided by the loss sensitivity index (Preedavichat and Srivastava, 1997).
In the experiments, the reactance limit of TCSC in pu is 0≤Xsi≤0.02
and phase shifting angle of TCPS in radian is 0≤αi≤0.1.
There are four case studies. Case 1 is OPF with TCSC at line 3-4; Case 2 is OPF with TCSC and TCPS at line 3-4; Case 3 is OPF with two TCSC at line 3-4 and line 19-20 and TCPS at line 3-4; Case 4 is OPF with two TCSC at line 3-4 and line 19-20 and two TCPS at line 3-4 and line 5-7.
Table 1 and 2 shows the solutions
and the times for convergence obtained by EEP technique. It is shown from
the Table 1 and 2 that, in all the
cases the proposed method gives better solutions compare with the solutions
obtained by TS/SA Ongsakul and Bhasaputra (2002). Their convergence characteristics
are shown in Fig. 5.
| Table 1: | Simulation results of cases 1-2 best solutions (demand 283.4 MW) |
| Table 2: | Simulation results of cases 3-4 best solutions (demand 283.4 MW) |
| Fig. 5: | Convergence characteristics of EEP for SCED for 30 bus system |
| Table 3: | Comparison of various methods from 20 runs for case 3 |
CONCLUSIONS
In this study, the EEP approach is effectively and successfully implemented to minimize the generator fuel cost in OPF control with TCSC and TCPS devices. The proposed EEP approach achieves better solutions than TS/SA on the modified IEEE 30 bus system with TCSC and TCPS fixed at given locations. Accordingly, the proposed EEP is potentially viable to OPF control due to generator fuel cost savings. Almost evolutionary programming that have been proposed till now commonly use Gaussian random number or Cauchy random number as the mutation of strategy parameter. The role of strategy parameter of evolutionary programming influences the search step size in solution search algorithm. Therefore, it must be small value within neighborhood of optimal solution. However, self-adaptive EP should obtain information concerning the convergence contribution of objective function for the sake of good solution. At this point of view, though almost EP algorithms use Gaussian random number in self-adaptation, it is expected that self-adaptation which uses double exponential random number can absorb the wide information concerning convergence contribution comparing with that of Gaussian random number. From the above mentioned reason, the performance of EEP is significantly better than TS/SA in terms of convergence rate and slightly better solutions. The experimental results show that EEP outperforms GA and TS/SA on applying to the function optimization problems. In the future, this EEP on applied to various types of optimization problems.