An Improved Tabu Search Algorithm and PSO for Unit Commitment Problem Solving Ahmad Ali Khatibzadeh*, Ghafur Ahmad Khanbeigi **, Mohamad Mehdi Bamdadian***, Heresh Naderi****, Mohamad Kazem Sheikh-el-Eslami***** * IEEE Student Member , Power system Lab, Tarbiat Modares University,
[email protected] ** IEEE Student Member, Amirkabir University of Technology,
[email protected] *** IEEE Student Member , Power system Lab, Tarbiat Modares University,
[email protected] **** IEEE Student Member , Power system Lab, Tarbiat Modares University,
[email protected] ***** Power system Lab, Tarbiat Modares University,
[email protected]
Abstract: One of the important tasks of system operators in electricity network is Unit commitment. Considering development of electricity market restructuring in generation side, the Generating and Transition costs will be decreased in a high scale by right committing of this operation. This paper proposes to represent compromise improved TS and enhanced PSO in the sense of accuracy and speed of solution. The priority of this method is determined more speed and less accuracy in the sense of PSO method by derived numerical results checking..
Keywords: Optimization; PSO; TS; UC. NUMINICALATURE:
Fit (Pit ) : Production cost of unit i at time t (s) Pit :Output power from unit i at time t Sit :Start-up cost of unit i at hour t Ft : Total Operating Cost PDt Rt Pi
: System Peak Demand at hour i
: System Reserve at hour t
min
:Unit i minimum generation limit
Pi max : Unit i maximum generation limit MDTi : Minimum down time MUTi : Minimum down time
Uit
: Unit i status at hour t
C1, C2: Acceleration Constants Rand ( ): Variable number in the period of [0 1] 1.
Introduction
In human systems, decision variables drove the optimization of these systems to a complicated problem. This kind of problems couldn't be solved easily by classic methods due to span searching area. The Local
optimization of problems are clearly revealed to be another difficulty of these problems. Electricity network is the biggest human handmade system. Generating units are component of the network for which the daily production allocation is vital. In another words, it should be determined which one of the Generation units and in what capacity is producing every day. This is called Unit Commitment (UC) in technical terms. Off course, it is become important in area of structural renewal to cause benefits and loss in each Generating units. So there are different methods for solving the problem. They can be divided in 3 general categories [1]: Classical Optimization: All methods such as Dynamic Programming (DP), Integer Programming (IP) and Lagrangian Relaxation (LR) etc. which are introduced clearly in books and articles. For example, the LR is reached to a suitable solution even in the span networks. Heuristic Method: the Priority list is obtained by this method due to expert decision. These methods can be reached to optimal solutions by decreasing the searching space. But we can't be exactly sure about the accuracy of this optimization. Artificial Intelligence Methods: These methods contain Neural Network (NN), Genetic Algorithms (GA) and Simulated Annealing (SA), Ants Algorithms, Taboo Search (TS), Particle Swarm Optimization (PSO) and etc. which are inspiring and growing methods. The GA, NN, TS methods are inspired from biological process. But the SA method is based on mathematical science. The NN method is widely used in applications of which operation can't be modeled. But it isn't normally used in optimization.
PSO algorithm is a method for solving non-linear optimization problems which wasn't possible to be solved neither by numerical methods nor enough time due to its complexity. Firstly, this method was presented by R.Eberhart and J.kennedy in 1995 [2].
Every Generating units has a maximum and minimum generation:
PSO as well as other non-linear optimization methods start with solutions by establishing initial population.
4) Minimum Up/Down Time When a Generating unit starts up (or shuts down), we couldn't shut down (or starts up) it for a period of time. This restriction is modeled by few ways:
Meanwhile, The Taboo Search (TS) is a powerful method for solving the complicated problems as well. Because exiting from local optimization and owning another features such as: non dependency of solutions to the start point or to the mathematical model for solving problem is a basic reason by which they can be widely used in every fields such as communication, VLSI planning, financial analyses, energy distribution, biological research etc. … This paper proposes to represent new creating restriction in TS and compromises it with PSO method. You will find the UC explanation in the first segment and TS explanation in the second segment. Moreover, in the third part has been seen PSO explanation. Finally, a case study is declared in the end. 2.
UC Implementation
One of the important tasks of system operators in electricity network is UC commitment. It is represent generation of each unit in different hours of day. For doing the production allocation, we have to prepare an objective function for optimizing. The function which is used for this purpose is:
Ft = ∑∑ (Fit (Pit ) + S it )
(1)
Following equation is sample of this function:
Fit ( p it ) = Ai p it + Bi
$/h
(2)
This problem contains some constraints as below: 1) Equality of generation and demand The Generation power should be equal to the demand in order to gain the stability of the network.
∑P
it
= PDt
(3)
i
2) Spinning Reserve A predetermined amount of power is always available in order not to encounter with a problem in the view of reliability ability. This amount of power can be obtained from subtracting potential generating power of Generating Unit from current consumption power.
∑P
max it
i
> (PDt + Rt )
3) Generation Limits
(4)
(5)
Pi min ≤ Pit ≤ Pi max
(U
t + MDT
it
(U
− U i ( t +1) )
∑U
it Pit < 0
t
t + MUT
i ( t +1)
(6)
− U it )
∑ U it Pit < 0
(7)
t
The first equation is for a minimum down time and the second is for minimum on time. 5) Initial Condition For implementing of UC, we need know the Generating Unit conditions before any new scheduling period time. It can be use as to turn on or turn off power plant based on minimum up time and minimum down time. Also its cost should add to the objective function. 6) Ramp up/down rate The up/down rate of power for each Generating Units is obtained due to the technology which is used in it. Anyhow, it isn't possible to decrease and increase the power in a Generating Unit simultaneously. 7) Transmission Limits In the real world, any lines which are laid along the different districts to transit power between them, have no unlimited capacity. Each line has its own capacity and it must be considered in Unit in order not to block the power in one district. 3.
TS method and PSO
3.1 TS The meaning of TABU is all things which are forbidden by God. But Tabu search doesn't relate to The God. In other words, Tabu search tries to reach to a suitable solution by creating restriction. This restriction is made in several methods, such as direct prohibition which is known as forbidden, or by evaluation and probability. The philosophy of this method is based on derivation of intelligent restrictions for solving problems. The origin of this method is based on variable memory. For saving the productions and derived structures, we need the variable memory. The main feature of owning such a memory is due to prevent us to be remained in local optimization. There are some definitions in the TS optimization which are explained below in brief [3,4,5,6]: 1) Memory Memory structure in the TS method is divided in 4 dimensions (Fig1). These methods are; Recency-base, Frequency-base, Quality-base, Influence-base.
Since the PSO uses each particle's data and the movement of the best particle for itself, it proposes below equation in order to find the next position if each particle in the future iteration will be:
Influence
Quality
Memory structure
Frequenc
Recency Figure 1. Memory Structure
Figure 2. Intensification and Diversification
2) Intensification and Diversification Two characteristics of tabu are intensification and diversification. Intensification is related to good solution neighborhoods indeed. Another issue in TS which is considered so much is Diversification. In this section, we look through somewhere that we haven't researched them before. As a matter of fact, we stand behind from the obtained good solution completely to get rid of falling in a tangle of local optimization. You can recognize these two concepts and their differences clearly in figure2. 3) Aspiration Criteria Another important component in TS is Aspiration Criteria, when an arrangement is considered as a restriction. In this stage, the solution is better than before. For this reason, we might ignore the forbidden and assume that solution acceptable. In technical term of view, it is said that we overcome the forbidden. 3.2 Particle Swarm Optimization (PSO) PSO starts with solutions by establishing initial population. Each member of this population is called Particle. Every particles move in D dimension space of the problem with a specific speed, so that speed and position of each particles depends on its own movement and position as well as another members do [7]. Position of member ith is shown as below: Xi= (Xi1, Xi2…Xid) The best position of each member ith (which is the best position due to objective function) is saved and shown as below which is called Pbest: Pi= (Pi1 …. Pid) Index g is used for the particle which owns the best position among others which is called gbest and reveals position and situation of the best particle. The speed of each particle ith is shown by the vector Vi= (Vi1….Vid).
Vidk +1 = w ⋅ Vidk + C1rand() ⋅ (Pid − Xid ) +
(8)
X idk +1 = X idk + Vidk +1
(9)
C2 rand() ⋅ (Pgd − Xid )
Part 1 of equation (8) indicates particle inertia in preposition state. The part 2 is related to self-discovery state of each particle which shows the best effect of each one. The part 3 is related to particles community which reflects the effect and cooperation of other particles with one another. Vi will be normally limited to a maximum value which causes the particles to remain in a district in last stage in order to reach to a convergent optimization solution. 3.2.1 Binary PSO PSO is working with real variables. Hence, for solving the problem, we have to decrease the number of variables. In Binary PSO [8], the position of each particle is shown with Binary PSO 0 and 1. The particle speed depends on a positive value without uncertainty which is selected in period (0-1). If the particle speed of its single dimension is big, the particle tends to select number near 1. This conclusion usually uses below function for a converge [9].
Sig ( Vi ) =
(10)
1 1 + e − vi
In other words, speed is calculated the same as standard method by equation (8). Then it is become binary by equation (10). Finally, the position of each particle is adjusted by below relation. (11) K
Xid =
{
1 0
if rand 〈 sig ( Vi K + 1 ) otherwise
4.
Improved TS
Usually in the TS method, for creating new variables operate single bite. In the other word, wrong forbidden path one by one and in case of mistake caused by the whole sample will be rejected. It is made the program be slow. New method of TS unlike previous methods study whole random sample in face and instead of deleting wrong sample correct it. In this method list of taboo save in the matrix. After creating random sample and using of mask algorithm, new sample will be created.
For easier understanding of method, should carefully see this example. Suppose that TL like Figure 3. If the last optimum response is like Figure 4 and new random sample is like Figure 5. So, in this stage we use matrix in Figure 3 as mask. Number one in matrix means that this field in new random should not differ from last optimum response. Figure 6 shows a new matrix for use in the solution. As stated, this method increase speed of creating sampling matrix. So, program achieves best response rapidly.
0 1 1
1 0 0
0 0 1
Figure 3. Tabu list
1 0 1
0 0 1
0 0 1
Figure 4. Last optimum response
1 1 0
1 0 0
1 1 0
Figure 5. Random sample
1 0 1
0 0 0
1 1 1
Case Study
In this section, we define a UC problem and solve it by using the TS method and PSO method including the results. There are 3 Generators in network with specifications as in the table II. It is supposed to do a UC production allocation for the next 12 hours. For this purpose, all load data are placed in table III and all data belong to transition lines between 3 districts are placed in table I. Solving this problem of UC for these 10 generators need a span space. In order to solve the problem faster, it should be changed to a linear problem. For solving of being nonlinear in objective function, in considered as a linear one. Meanwhile, for making the restrictions linear, they must be multiplied` by Uits. The operating time period for TS was 30 minutes and for PSO was 50
TABLE I. TRANSMISSION CAPACITY BETWEEN THE ZONES Transmission Receive Power Zone 1 Zone 2 Zone3 Zone 1 NA 550 100 Send
Zone 2 Zone 3
1000 200
NA 600
In table IV will be the solutions of UC and TS by implementing the program. On the other hand, in the amount of Economical Dispatching is determined by the operating program itself. Then the final cost of generation is 526000$. Also, the results represent that the second Generator is placed on their maximum production by lower production cost, and never be off in any hours. In this way, the high cost of operating start will not undertake on the next hours. On the other hand, the Generator 9 and 10 were on and off easily due to their minimum zero on/off. In table V will be the solutions of UC and PSO by implementing the program. Also, the final cost of generation is 525000$. In this Programming, the Generator 2 was off during the last hours in full time. On the other hand, the Generator 1 was off in the first and last time of operating as well due to expensive cost. It's off time period is selected so that the minimum off time is gained during this 12 hours in order not to be participated in the next period of programming. Refer to the results; there were close answers in these two optimizations. So the accuracy of improved TS was signified. The results represented that PSO had showed a better optimization but with a more time period than TS.
6.
Figure 6. New sample
5.
minutes. Also, computer specification was CPU 1.6MHZ & RAM 1GIG.
1000 NA
Conclusion
Solving the unit commitment problem in the power system is one of the most important tasks of system operator. The development of electricity market restructuring in generation side increase the importance of the problem and the accurate conduct of this operation lead to the decrease of the production's and transmission's costs. This paper proposes improved method of Tabu Search (TS) for solving the UC. Based on this method, since the short memory of computer is used, it is possible to escape from the local optimum. Numerical results indicate that it is faster than PSO, but the PSO method reach better response. It is meaning that it has less cost for generations.
TABLE II. GENERATION UNITS SPECIFICATION Zone 3 3 U10 U9 Generation Unit 0 0 Start Up Cost ($) 0 0 Minimum Up Time (hr) 0 0 Minimum Down Time (hr) 10 10 Ramp Up Rate (MW/min) 10 10 Ramp Down Rate (MW\min) 200 200 Maximum Power Supply (MW) 20.752 37.620 Maximum Power Supply (MW) 2 2 Initial Time (hr) 9500 7500 Base Cost ($)
TABLE III. HOURLY LOAD SPECIFICATION Hour Load (MW) Zone 1 Minimum Reserve Requirement (MW) Load (MW) Zone 2 Minimum Reserve Requirement (MW) Load (MW) Zone 3 Minimum Reserve Requirement (MW)
1 500 100 200 40 100 20
2 U8 1000 8 8 15 15 1000 35.536 4 16000
2 600 120 500 100 500 100
2 U7 500 2 2 5 5 600 97.120 -1 10000
3 800 160 500 100 700 140
4 1100 220 800 160 700 140
2 U6 500 2 2 5 5 600 93.432 -1 10000
5 1400 280 1400 280 1000 200
2 U5 500 2 2 5 5 600 8.598 2 10000
6 1500 300 1400 280 1000 200
7 1200 240 1400 280 1000 200
2 U4 0 0 0 10 10 200 22.287 2 7500
8 1000 200 1400 280 1000 200
1 U3 700 5 5 6 6 500 19.028 5 12000
9 1000 200 800 160 800 160
1 U2 350 3 3 8 8 200 0.839 -3 7500
10 1000 200 800 160 800 160
11 800 160 500 100 500 100
1 U1 500 2 2 5 5 600 21.793 5 14000
12 800 160 500 100 500 100
TABLE IV. OPTIMUM POWER DISTRIBUTION OF UNITS IN TS Real Power Output of every Generation Unit
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P10 (MW)
Time (hr) 1 21.79 0.83 19.02 22.28 8.59 0 0 689.88 37.61 0
2 21.79 0.83 19.02 0 330.21 93.43 97.11 1000 37.61 0
3 21.79 0.83 19.02 22.28 600 93.43 205.04 1000 37.61 0
4 21.79 0.83 19.02 22.28 600 315.51 600 1000 0 20.57
5 600 0.83 318.71 22.28 600 600 600 1000 37.61 20.57
6 600 0.83 418.71 22.28 600 600 600 1000 37.61 20.57
7 600 0.83 118.71 22.28 600 600 600 1000 37.61 20.57
8 21.79 0.83 19.02 0 600 237.79 600 1000 0 20.57
9 21.79 0.83 19.02 22.28 600 298.47 600 1000 37.61 0
10 21.79 0.83 19.02 22.28 600 315.51 600 1000 0 20.57
11 21.79 0.83 19.02 22.28 487.36 93.43 97.11 1000 37.61 20.57
12 21.79 0.83 19.02 22.28 524.97 93.43 97.11 1000 0 20.57
TABLE V. OPTIMUM POWER DISTRIBUTION OF UNITS IN PSO Real Power Output of every Generation Unit
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P7 (MW) P8 (MW) P9 (MW) P10 (MW)
Time (hr) 1 0 0.84 19.03 22.28 8.59 0 0 690.87 37.62 20.75
2 0 0.84 19.03 22.28 308.59 93.43 155.81 1000 0 0
3 21.792 0.84 19.03 22.28 600 93.43 184.24 1000 37.62 20.75
4 300 0.84 19.03 0 600 300 321.75 1000 37.62 20.75
5 600 0.84 318.5 22.28 600 600 600 1000 37.62 20.75
6 600 0.84 418.5 22.28 600 600 600 1000 37.62 20.75 [4]
7. References [1] [2] [3]
A.H.Mantawy, Y.L.AbdeI-Magid, S.Z.Selim, Unit commitment by tabu search, IEE, 1998. R. Eberhart, J. Kennedy,” A New Optimizer Using Particle Swarm Theory”, Sixth International Symposium on Micro Machine and Human Science, 1995. Fred Glover, Tabu Search Part 1, ORSA Journal of computing, Vol.1, No.3, Summer 1989.
[5] [6] [7]
7 600 0 119.34 22.28 600 600 600 1000 37.62 20.75
8 500.31 0 19.03 22.28 600 600 600 1000 37.62 20.75
9 200.31 0 19.03 22.28 600 300 400 1000 37.62 20.75
10 21.792 0 19.03 22.28 600 278.51 600 1000 37.62 20.75
11 0 0 19.03 22.28 365.25 93.43 300 1000 0 0
12 0 0 19.03 22.28 509.75 93.43 97.12 1000 37.62 20.75
Fred Glover, Tabu Search Part 2, ORSA Journal of computing, Vol.2, No.1, Winter 1990. Glover, F. and M. Laguna (1997) Tabu Search, Kluwer Academic Publishers, Boston. Manuel Laguna, A Guide implementing Tabu Search, Investigation Operative, Vol.4, No.1, April 1994. Xiao-Feng Xie, Wen-Jun Zhang, Zhi-Lian Yang,” Adaptive Particle Swarm Optimization on Individual Level”, International Conference on Signal Processing (ICSP), 2002.
[8]
T. O. Ting, M. V. C. Rao, C. K. Loo, "A novel approach for unit commitment problem via an effective hybrid particle swarm optimization" IEEE Trans. Power Syst, Feb. 2006. [9] T.Senjyu, S.Chakraborty*, A.Yousuf Saber, H.Toyama, A.Yona T.Funabashi,” Thermal Generation Scheduling Strategy Using Binary Clustered Particle Swarm Optimization Algorithm”, 2008 IEEE.
