Available online at www.sciencedirect.com
1877–0509 © 2011 Published by Elsevier Ltd. doi:10.1016/j.procs.2011.08.081
Procedia Computer Science 6 (2011) 429–434Pro edia Computer cie ce 00 ( 010 0
Development of Hybrid-Coded EPSO for Optimal Allocation of
FACTS Devices in Uncertain Smart Grids
Hiroyuki Mori
*, Hajime Fujita
Meiji University,Tama-ku, Kawasaki,214-8571, Japan
This paper presents hybrid-coded EPSO (Evolutionary Particle Swarm Optimization) for optimal allocation of FACTS (Flexible AC Transmission System) devices in uncertain smart grids. The optimal allocation of FACTS devices is one of the important tacks that increase nodal loadability to maximizing the supply of active power at specified nodes in smart grids. However, it is not easy to determine the optimal location and the optimal variable output of FACTS devices due to the nonlinear mixed integer problem. Under such circumstance, it requires a lot of computational time in considering the uncertainties due to renewable energy. In this paper, a hybrid-coded scheme of EPSO is proposed to reduce computational time and maintain solution accuracy. The proposed method has advantage to deal with real-coded and integer-coded variables at the same time. The proposed method is successfully applied to a sample system.
Published by Elsevier B.V.
Keywords FACTS Devices; Loadability; Hybrid-Coded EPSO; Optimization; Meta-heuristics; Smart Grids
1. Introduction
This paper presents a hybrid-coded scheme of EPSO (Evolutionary Particle Swarm Optimization) for optimal allocation of FACTS (Flexible AC Transmission System) [1], [2] devices. The proposed method is applied to the maximization of loadability at certain nodes that corresponds to distribution companies. To maximize the nodal active power, FACTS devices are useful for controlling power flows, nodal voltage magnitude power quality, transmission capability, etc. As the devices, the followings are well-known: SVC, TCSC, STATCOM, UPFC, etc. UPFC is more attractive due to the flexibility that three variables of active and reactive power as well as voltage magnitude change the power flows. However, it is not easy to determine the optimal location and the optimal output of FACTS devices due to the nonlinear mixed integer problem. The former is expressed in discrete number while the latter is represented in continuous one. The conventional methods on the optimal allocation and the output optimal variables of FACTS may be classified as follows:
1) Sensitive matrix method [3] 2) Meta-heuristics [4], [5]
* Corresponding author to provide phone: +81-44-9347353; fax: +81-44-9347909.
E-mail address [email protected]
Complex Adaptive Systems, Volume 1
Cihan H. Dagli, Editor in Chief
Conference Organized by Missouri University of Science and Technology
2011- Chicago, IL
© 2011
Open access underCC BY-NC-ND license.
430 Hiroyuki Mori and Hajime Fujita / Procedia Computer Science 6 (2011) 429–434Mori Fuji a / Procedia Compu er Science 00 (201 ) 000–0 0
3) Hybrid meta-heuristics [6-8]
Method 1) has the limitation that it is based on local information. On the other hand, Method 2) is one of optimization methods that repeatedly make used of some rules or heuristics to obtain better solutions from a standpoint of global optimization. The key point is that meta-heuristics has a strategy to escape from a local minimum although the conventional methods easily get stuck in a local minimum. Method 3) consists of two phases that optimize the location and output of FACTS devices to evaluate the optimal solution. The process of Layers 1 and 2 is repeated to evaluate better solutions although it has a drawback to take the computational time.
In this paper, hybrid-coded EPSO is proposed to deal with the uncertainties of PV (Photovoltaic) systems. Unlike the conventional methods, this paper handles real and integer variables in hybrid code at the same time. To consider the uncertainties, Monte Carlo Simulation (MCS) is carried out to determine the optimal allocation of FACTS under uncertain smart grids. It is assumed that some nodes with PV systems bring about probabilistic variations of generation to smart grids. Hybrid-coded EPSO is a combination of discrete and continuous EPSO to reduce computational time and to improve the solution accuracy. The proposed method is successfully applied to a sample system.
2. EPSO
2.1 Outline of EPSO
This paragraph describes EPSO [12, 13] that is the improved version of PSO in a way that weights are adaptively tuned up to obtain better solutions. Miranda, et al. proposed EPSO to improve the solution quality of PSO through introducing the evolutionary strategy into PSO [14]. The conventional PSO has a drawback that it often gets stuck in a local minimum. To overcome it, EPSO improves the moving rule of PSO that makes use of replication, mutation, reproduction, and natural selection to modify the weights. The algorithm may be summarized as follows:
Step 1: Set up the initial conditions for parameters such as the weights, the maximum iteration counts, replication rate, initial agents, etc.
Step 2: Replicate each agent.
Step 3: Move the agents and the replicated ones with the moving rule of the velocity. Step 4: Evaluate all the agents and select them with the selection rule.
Step 5: Update the best solutions for each agent and the swarm.
Step 6: Stop if the algorithm reaches at the maximum iteration counts. Otherwise, return to Step 2. The moving rule of velocity may be expressed as
) * ( * ) ( * *0 1 2 1 t i i t i i i t i i t i w V w Pbest S w Gbest S V (1) ) 1 , 0 ( * w N w ik ik (2) ) 1 , 0 ( * 3 * Gbest w N Gbest i (3) ) 1 , 0 ( ' 3 * 3 w N wi i (4) 1 1 t i t i t i S V S (5) where
Vit: velocity of agent i at iteration t wi0-wi3: weights
Pbesti (Gbesti): best solution for agent i (swarm)
Sit: placement of agent i at iteration t
τ
: learning rates2.2 Binary EPSO
Binary EPSO [15] is explained in this paragraph. Basically PSO was developed to handle optimization problems with continuous variables. As a result, some ingenuity is required to deal with optimization problems with discrete
Hiroyuki Mori and Hajime Fujita / Procedia Computer Science 6 (2011) 429–434Mo Fu ita / Pro edia Co uter cience 00 (2011) 000–00 431 variables. Kennedy and Eberhart proposed binary PSO that modified PSO in updating the velocity and the placement as follows: ) ( ) ( 2 1 1 t t i t t i i i i V w Pbest x w Gbest x V (6) ) exp( 1 1 ) ( 1 t1 i t i V V s (7) 0 ;1 )) ( () ( 1 1 1 t i t i t i x else x then V s rand if (8) where
s(
·
): threshold value for judging whether the binary value is 0 or 1Binary PSO makes use of the sigmoid function to transform a continuous variable into binary one for the velocity through (8). The threshold value varies with the number of iteration counts and converges to a solution. This paper introduces the evolutionary strategy of EPSO into binary PSO to improve accuracy.
3. Proposed Method
3.1Outline of Proposed Method
In this paragraph, the proposed method is outlined. As mentioned before, mega solar systems are positively introduced to suppress the emission of CO2 in smart grids. To examine the influence of the mega solar system on the
loadability in smart grids, this paper evaluates the characteristics of loadability under uncertain smart grids through MCS. MCS and hybrid meta-heuristic method are combined to solve the nonlinear mixed integer problem of the optimal allocation of the FACTS devices so that loadability is maximized at several nodes. However, they have a drawback in terms of computational time. Therefore, the proposed method makes use of hybrid-coded EPSO that is a combination of discrete and continuous EPSO to reduce computational time and to improve the solution accuracy. Fig. 1 shows a concept of hybrid-coded EPSO in which the location and output of FACTS devices are given. The former is coded by the moving rule of discrete EPSO while the latter is coded by continuous EPSO. The key point is to transform two kinds of codes into one. Thus, this paper evaluates the influence of intermittent renewable energy on loadability at several nodes with hybrid-coded EPSO.
3.2 Mathematical Formulation
This paragraph describes the mathematical formulation of the optimal allocation of FACTS devices to maximize the loadability at several specified nodes. It is assumed that several specified nodes correspond to distribution companies that need active power as much as possible. The mathematical formulation may be written as
Cost Function: min f f f f f n m n l l m l n m m 3 2 2 1 1 1 1 1 2 1 1 1 0 ( ) (9) Constraints: 0 ) , (xu g (10) M i i m i V V V (11) M ij ij p p (12) M k n k1S1k S1 (13) M k n k1S2k S2 (14) M T T V V (15) M in in Q Q (16) where, m
432 Hiroyuki Mori and Hajime Fujita / Procedia Computer Science 6 (2011) 429–434
Fig. 1. Concept of Hybrid-Coded EPSO
0 k k P P ILR (17) 3 2 1 0, , , : parameters
Pk0: active power load at Node k for original power system conditions
Pk: active power load at Node k for power system conditions with controllers :
2
f active power loss :
(.)
g power flow equation :
x nodal voltage vector
:
u control variable vector corresponding to the optimal allocation of UPFC
: ) ( m i M i V
V upper (lower) bound of nodal voltage magnitude at Node i :
ij
P thermal limitation of the line that connects Node i with Node j :
) ( 1 1Mk Smk
S apparent power through the shunt (series) inverter at UPFC k :
M T
V upper bound of the applied voltage magnitude by UPFC
QinM: upper bound of the injected reactive power by UPFC
The first term is the adjustment parameter in (9). The second shows the maximization of ILR at the specified nodes in a way that coefficient
α
2is negative and the third one means equalizing ILR at each node. The last term indicates the minimization of active power network loss. Also, (10)-(16) show the constraints of this problem.4. Simulation
4.1 Simulation Conditions
1) The proposed method is applied to the IEEE 30-node system with 41 lines. It is assumed that the number of UPFC devices is two. As a result, the number of the UPFC location candidates results in 1640 if we consider the direction of the UPFC devices. The proposed method is repeated until the termination conditions are satisfied. 2) This paper makes assumption that the UPFC devices have the following constraints on voltage, current and angle:
Table 1. Simulation Conditions
Method A Method B Method C
1st Layer 2nd Layer 1st Layer 2nd Layer
No of Particles 10 10 10 10 10 No of Iterations 100 100 100 100 100 w1 0 01 0 01 0 01 0 01 0 01 w2 0 01 0 01 0 01 0 01 0 01 Wmax 1 1 1 1 1 Wmin 0 01 0 01 0 01 0 01 0 01 Replication Rate 2 2 2 τ 0 01 0 01 τ' 0 01 0 01 Output (Continuous Variables) Location (Integer Variables)
Hiroyuki Mori and Hajime Fujita / Procedia Computer Science 6 (2011) 429–434 Mori Fujita / Procedia om uter cience 0 (2011) 000–000 433 3) The parameters of the cost function are given as follows:
1 , 15 , 10 , 100 1 2 3 0 (21)
They are determined by the preliminarily simulation. For convenience, the following methods are defined: Method A: TLPSO (Two-layered PSO)
Method B: TLEPSO (Two-layered EPSO)
Method C: HCEPSO (Hybrid-Coded EPSO of Proposed Method)
Table 1 shows simulation parameters that are determined by the preliminarily simulation.
3) It is assumed that Nodes 2 and 9 have mega solar systems, where the capacity of mega solar at Nodes 2 and 9 correspond to 20% of the specified active power. Nodes 14 and 30 are selected as the target nodes in maximizing loadability. The uncertainty of PV systems is simulated in MCS according to the relationship between output and frequency. Three hundreds of scenarios are used to examine the performance of Methods A and B.
4.2 Simulation Results
Table 2 shows a comparison of each method, where the best, the worst and average cost functions are given. In addition, the standard deviation of the cost functions and computational time are shown. It can be observed that Method B is better than Method A in terms of the cost functions. That is because the problem to be solved has a lot of local minima and Method B has better strategies to escape from them. It is noteworthy that Method B succeeded in reducing 17.85% and 19.57% of the worst cost function and the standard deviation, respectively. On the other hand, Method B needs more computational time than Method A. Method C has almost the same performance as Method B in terms of the cost functions and standard deviation. However, it shows 45.86% improvement in terms of computational times. Fig. 2 gives the distribution of ILR. It can be seen that Methods B and C provide a set of solution sets that is far from the origin. That implies that the solutions more distant from the origin bring about more capacity of loadability. Also, the reason why the increase of loadability at Node 14 is larger than that at Node 30 is that Node 30 has the lines with the thermal limitations. To investigate the performance of the methods, let us define performance index ILR’as follows:
2 30 2
14
' ILRNode ILRNode
ILR (22)
Table 3 shows the frequency of ILR in Areas 1-4 for each method. It can be observed that Methods B and C give ' better solution sets than Method A. Table 4 shows the frequency and location of allocation of UPFCs.
Table 2. Comparison of Methods A, B and C
Cost Functions Methods
Best Worst Ave Standard
Deviations Computational Time[s]
A 11 26 53 33 33 19 11 96 5 867×105
B 11 01 43 81 28 61 9 620 8 269×105
C 11 09 44 13 28 82 9 980 3 791×105
Fig. 2. Distribution of ILR of Each Method ILR 14
434 Hiroyuki Mori and Hajime Fujita / Procedia Computer Science 6 (2011) 429–434 Mor , Fujita / Procedia Computer Science 00 (2011) 000–000
Table 3. Frequency of ILR’ of Each Method
Areas Method A Method B Method C
1 16 3 2 2 186 149 165 3 88 106 90 4 10 42 43
Table 4. Frequency and Location of Allocation of UPFCs
Patterns Frequency [%] UPFC 1 UPFC 2
1 89 33 14-15 27-30 2 5 000 12-14 27-30 3 3 667 14-15 29-30 4 2 000 12-14 29-30
5. Conclusion
This paper has proposed a hybrid-coded EPSO method for the optimal allocation of FACTS devices in uncertain smart grids. The proposed method is based on the combination of discrete and continuous EPSO to reduce the computational time and to improve the solution accuracy. As the hybrid-code, integer-codes are merged with continuous ones to express the optimal allocation and output of FACTS devices. It was compared with two-layered PSO and two-layered EPSO in the IEEE 30-node system with two mega solar systems. The simulation results have shown that the proposed method outperforms the conventional methods in terms of the cost functions and computational times. Therefore, the proposed method allows network planners to evaluate loadability in uncertain smart grids adequately.
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