Profit-Based Unit Commitment Problem Using PSO with Modified Dynamic Programming Anup Shukla, Vivek Nandan Lal, Student Members, IEEE and S. N. Singh, Senior Member, IEEE Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, India
[email protected],
[email protected],
[email protected] Abstract- This paper proposes a hybrid approach utilizing particle swarm optimization along with dynamic programming to solve unit commitment problem based on the profit under the deregulated power market. In the deregulated market, power and reserve prices are important factors in the decision process for unit commitment scheduling and offer freedom to utilities to schedule their generators to produce less than predicted load as well as reserve to maximize their profit. To solve the profit based unit commitment problem (PBUCP), the model is divided into exterior and interior dependent sub problems, which are discrete and continuous, respectively. The proposed model helps GENCOs to make a decision, how much power and reserve that must be put up for sale in the market, and how to schedule generators in order to receive the maximum profit. GENCOs with 3 and 10 generating units are used to demonstrate the effectiveness of the proposed approach. Index Terms - Competitive environment, profit-base unit commitment problem, dynamic programming, particle swarm optimization.
I. αi,βi,λi Dt FCost N PF Pit Pimax/ Pimin Rit RV SRt SCit tOFF,it/tON,it T TC TON,i/TOFF,i Uit ℑt
ℜt
ζ
NOMENCLATURE
Cost coefficients of the ith generating unit. System load demand at hour t. Fuel cost or production cost of units. Number of thermal units. Profit of GENCOs. Power generation of ith thermal unit at hour t. Max/Min generation of ith thermal unit. Reserve generation of ith thermal unit at hour t. Revenue of GENCOs. Spinning reserve requirements at hour t. Startup cost of ith thermal unit at hour t. Time period that ith thermal unit has been continuously down and up till period t. Number of time interval (hours). Total cost over the scheduling horizon. Minimum up and down time of ith thermal unit. Schedule state of ith thermal unit for hour t. Forecasted spot price at hour t. Forecasted reserve price at hour t. Probability of the reserve.
Authors acknowledge with thanks the partial financial support provided by the Department of Science and Technology (DST), New Delhi, India, under project no. DST/EE/20100127, to carry out the present research work. The authors are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India (e-mail:
[email protected] ;
[email protected];
[email protected]).
II.
INTRODUCTION
D
AILY generation scheduling is a critical task in modern energy management systems. The main task of unit commitment problem (UCP) is to find the ON/OFF status of the generating units to meet the forecasted demand, over a shortterm period, meeting all kinds of system and unit constraints to minimize total cost. Restructuring of power system has resulted in market-based competition to utilities by creating an open market environment and bringing changes in operation, planning and controlling of the traditional power system [1-2]. Hence, the objective function of minimum production cost is changed to profit maximization. Deregulation in power sector increases the efficiency of electricity production and distribution. It also plays a key role in lowering prices, with higher quality and reliability by allowing customers to choose their supplier. In deregulated market, power and reserve prices are important factors in decision making process for unit commitment scheduling, because the profit of GENCOs depend on revenue and production cost. Therefore, its offers freedom to utilities to schedule their generators to produce less than predicted load and reserve in the market which can produce maximum revenue with minimum production cost in order to maximize their overall profit. To solve PBUCP, several methods have been proposed, such as priority list (PL) [3], dynamic programming (DP) [4], Lagrangian relaxation (LR) approach [5], Muller method (MM) [6], etc. These methods are known as classical or numerical optimization techniques. Although, these methods are generally fast and simple, but most of them suffer from numerical convergence and solution quality problems. Some stochastic search methods such as genetic algorithm (GA) [7], particle swarm optimization (PSO) [8-9], artificial bee colony (ABC) [10], evolutionary programming (EP) [11], simulated annealing (SA) [12], etc. can successfully handle complex nonlinear constraints and provide high-quality solutions, but these methods suffer from the dimensionality problem. Later, efforts have been made to develop hybrid techniques, such as LR with GA (LRGA) [13], LR with EP (LR-EP) [14], LR with PSO (LR-PSO) [15], Tabu search based hybrid technique (TS-IRP) [16], etc., for better and faster optimum results to overcome these problems. Some other intelligent techniques, such as selective enumerative method with dynamic programming [17], multi-agent system (MAS) [18], new multi-agent approach [19-20] etc. also have been used to solve PBUCP effectively. Compared to other evolutionary techniques, the characteristic of the PSO is simple to implement to find a number of high 1
978-1-5090-0191-0/15/$31.00 ©2015 IEEE
quality solutions, and has stable convergence characteristics. Contrary to other evolutionary algorithms, it also has a flexible and well-balanced mechanism for improving and adjusting the global and local search capabilities. The objective of this work is to present the application of PSO algorithm in a profit based UC problem. This paper proposes a hybrid approach which consists of particle swarm optimization and modified dynamic programming to solve the unit commitment problem, based on the profit under the deregulated power market. Proposed model is divided into exterior and interior dependent sub problems, which are discrete and continuous, respectively. The exterior sub-problem (particle search space) is unit’s ON (1)/OFF (0) status. The interior sub problem (particle search) is economic load dispatch (ELD) of the thermal units. A search space consists of optimal combination of binary states of unit status for the movement of the particles, to maintain good exploration and exploitation search capabilities, and to select the best scheduling path, satisfying all system and units constraint for maximizing GENCOs profit. PROBLEM FORMULATION
III.
A profit based UC problem under competitive environment is an optimization problem. To solve this problem, the study period is divided into T time intervals. The short-term generation scheduling problem is formulated as follows: The objective function (1) Max.PF = RV − TC or (2) Min.T − R C
V
Subjected to the following constraints: A. Demand constraint: NT
∑U
it
⋅ Pit ≤ Dt
t =1,.........T
(3)
i =1
B. Reserve constraints: NT
∑U
it
⋅ Rit ≤ SRt
t =1,.........T
i =1
(4)
C. Power and Reserve limit: Pi m in ≤ Pit ≤ Pi m a x 0 ≤ R it ≤ Pi
m ax
R it + Pit ≤ Pi
- Pi
m in
m ax
i = 1 ,.......... N
(5)
i = 1 ,.......... N
(6)
i = 1,.......... N
(7)
D. Minimum Up and Downtime constraint: T O N ,i ≤ t O N ,it
i = 1 ,.... N a n d t = 1 ,..... T
T O F F ,i ≤ t O F F ,it
i = 1,.... N a nd t = 1 ,..... T
IV.
(9) (10)
MARKETING STRATEGY
In the deregulated power system, the amount of power and reserve sold in energy and reserve market depend on the way of payment made for the reserve. In this paper, two different methods of reserve payments are adopted, which are as follow [14]: A. Payment for Power Delivered (PPD) Revenue (RV) and total cost (TC) can be expressed as
T
N
T
N
RV = ∑∑( Pit ⋅ℑt ) ⋅ Uit + ∑∑ (ζ .ℜt ⋅ Rit ) ⋅ Uit t =1 i =1
t =1 i =1
T
N
T
N
TC = (1 − ζ )∑∑ F (Pit ) ⋅ Uit + ζ ∑∑ F ( Pit + Rit ) ⋅ Uit +SCit t =1 i =1
(11) (12)
t =1 i =1
In this method, reserve is paid when only reserve is actually used; here reserve price is higher than the power (spot) price. The fuel cost function for the unit i at any given time interval is assumed as quadratic function of the generator power output, Pit and given as 2 (13) Fi ( Pit ) = α i × ( Pit ) + β i × P it + λi B. Payment for Reserve Allocated (PRA) Revenue (RV) in this case can be expressed as T
N
T
N
RV = ∑∑( Pit ⋅ℑt ) ⋅ Uit + ∑∑((1 − ζ ).ℜt + ζ ⋅ℑt ) ⋅ Rit ⋅ Uit t =1 i =1
(14)
t =1 i =1
In this method, reserve price is much less than the spot price and total cost is same as of (12). V.
PROPOSED METHODOLOGY
Hybrid approach for PBUCP in a given scheduling time horizon is explained in this section. A. Dynamic Programming Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. Generally subset of possible decisions is associated with each sequential problem step and a single one must be selected, i.e., a single decision must be made in each problem step [4]. The UC is a complex decision-making process because of multiple constraints which may not be violated while finding the optimal commitment schedule. For solving PBUCP, there is a cost associated with each possible decision and this cost may be affected by the decision made in the preceding step. Here, the objective is to make a decision in each problem step which maximizes the total profit for all the decisions made [21]. The DP has many advantages over the enumeration scheme, the chief advantage being the optimality of the result. B. Overview of PSO Particle swarm optimizations, first introduce by Kennedy and Eberhart [22-23], is a population-based meta-heuristic search algorithm, inspired by the movement of a flock of birds searching for food. In PSO, swarm of particles are represented as potential solutions, and each particle i is associated with two vectors, i.e. velocity vector Vi=(vi1,vi2,…..vix) and the position vector Pi=(pi1,pi2,…..pix), where, x stand for dimension of the search space. The velocity and the position of each particle are initialized by random vectors within the corresponding ranges. During the evolutionary process, the velocity and position of ith particle on dimension x are updated as: (15) vixk+1 = w×vixk + c1 × rand1 ×(PBix − pixk ) + c2 × rand2 ×(GBgx − pixk ) pixk +1 = pixk + vixk +1
(16)
where, 2
Fig. 1 Particle search space (10 unit system)
c1, c2 are acceleration coefficients (cognitive and social); w is the inertia weight parameter, controlling the global and local exploration capabilities of the particle. Inertia weight (w) is varied according to the following equation itermax − iter (17) w = ( wmax − wmin ) × + wmin itermax where, iter is current iteration number and itermax is maximum iteration number (say 100). In (15), PBix is the position with the best fitness found so far for ith particle, (local best) and GBgx is the best position in the neighborhood (global best), and rand1 and rand2 are uniform random value between 0 and 1. C.
Proposed Approach
Hybrid approach is based on the particle search on the search space that consists of a combination of binary states of unit status for the movement of the particles, to maintain good exploration and exploitation search capabilities and to select the best scheduling path for maximization of the GENCOs profit. Proposed model is divided into two sub problems. a) exterior and b) interior. The exterior subproblem (particles search space) is unit’s ON (1)/OFF (0) status. Let us assume that there is N number of generating units. For particular hour, there are 2N-1 possible combinations of states and each hour states are grouped as N generating unit’s status. If the scheduling horizon is 24-hr, the particles search space can be considered as a matrix of (2N-1) * 24.Therefore, when the particle chooses a state from current hour to the next hour, it has to select a state from the large number of available states in the matrix column. Because of this disadvantage, a better method of determining the optimum combination of units in service is desirable for any given system and load condition. Due to freedom to utilities to schedule their generators to produce less than predicted load as well as reserve to maximize their profit. This search space can be reduced by eight times by assuming some initial status. This initial status depends on minimum per unit cost of thermal units. Per unit cost of thermal units is a function of fuel cost coefficients given in (13). For initial status units are turned ON according to minimum per unit cost to satisfy minimum load and reserve in 24-hr of scheduling interval. Also unit is kept OFF with maximum per unit cost for entire scheduling horizon. Therefore, the search space matrix will be reformed as (2(N-n)) *24, where, n equals to sum of number of units which are initially ON and OFF. For example, there will be 1023 states
and matrix space will be 1023*24, for N=10. After reforming, matrix space reduces to 128*24 (n=3). Particle search space for 24-hr of scheduling horizon is shown in Fig. 1. From the above particle search space, we can further reform a new search space matrix at each time period t based on load demand, which is given as fs*t. Where, fs are the feasible number of states satisfying the demand at given time period t. The interior sub problem (particle search) is ELD of the thermal units. In this case, the generator’s real and reactive power is allowed to vary within its limits to meet a particular load demand, including the reserves, such that the profit to the GENCOs over the scheduled horizon is maximized, subjected to the system and generator operational constraints. To perform optimization, PSO technique is used. Each binary state in a search space represents a group of generating units. Particles in each hour select an eligible state from its search space with the minimum transition cost reference to previous hourly state. The selected state for the (t+1)th hour should not violate the minimum up/down constraints from tth hour. This process is continued till the particles are able to find the best optimal commitment for 24-hr of scheduling interval. Flow chart for profit based unit commitment problem is given in Fig. 2, where, h and l are binary states from particle search space. VI.
RESULTS & DISCUSSIONS
The proposed method has been developed in MATLAB, executed on a computer with Intel Core of 3.40 GHz. To test the effectiveness of the proposed approach, simulations are carried out using two test systems with two payment methods (PPD and PRA), as discussed in sections IV (A) and IV (B) [14] and [19]. The first system consists of three generating units with 12-hr scheduling periods. The second system consists of ten generating units with 24-hr scheduling period. A simulation is done under the effect of ζ (probability of the reserve to be called and generated) and price. The parameters of PSO are as follows: • Population size = 20. • Maximum velocity = Pimax. • Minimum velocity = Pimin. • wmax=0.9 and wmin=0.4. • c1 =2 and c1=2. A. Three-unit test system Unit characteristics, cost coefficients, and other forecasted data are taken from [14]. The proposed method investigates the effect of ζ and reserve power. Simulation results obtained 3
PF (t,h ) = max[RV (t, h ) − Fcost (t,h ) − SC(t − 1, l : t, h )] {l }
PF (t,h ) = max[RV (t , h ) − Fcost (t,h ) − SC(t − 1, l : t, h ) + PF (t − 1, h ))] {l }
Fig. 2 Flow chart for PBUCP.
1 2 3 4 5 6 7 8 9 10 11 12
are compared with other existing methods. In this case, reserve price is fixed at 3 and 0.04 times of spot price for PPD and PRA, while ζ is fixed at 0.005. Scheduling plan of power and reserve for PRA method is shown in Table I. Initially unit 1 was OFF for 3-hr. As the minimum OFF time of unit 1 is 3-hr, so it can be committed. But, it continues to be in OFF state for next 4-hr, total OFF for 3+4=7-hr to maximize the profit. Similarly for unit 2, having minimum down time of 3-hr, and it is observed that unit 2 is OFF for last 3-hr of scheduling horizon and unit 3 is ON for whole scheduling interval. Thus, it can be observed that minimum ON/OFF time constraints (9) and (10) have been satisfied. Table II shows the profit comparison with other existing methods with execution time. Revenue and total cost are shown in Figs. 3 (a) and 3 (b). It is found that maximum profit is received at 7th hour of scheduling horizon. Figs. 4 (a) and 4 (b) shows the graph of revenue and total cost versus power at 7th hour, when three units (Convention UCP) and two units (Profit based UCP) are ON respectively. According to Fig. 4 (a) the maximum profit can be received when power is served between 850-950 MW. However, in the conventional UCP, forecasted load and reserve must be completely met. In the case of profit based UCP, utilities have given freedom to schedule their generator to produce less than predicted load and reserve in order to
maximize their profit. Fig. 4 (b) shows that, while running two units (unit 2 and unit 3) and power at 600 MW GENCOs can give maximum profit. TABLE II: PROFIT COMPARISON FOR 3 GENERATING UNITS Conventional UCP ($) Method-A Method-B Techniques Profit ($) Time (s) Profit ($) Time (s) 4,048.8 4,262.7 9,136.2 LR-EP[14] MM [6] Proposed 4,119.18 6.11 4,333.12 9.46 PBUCP($) Method-A Method-B Techniques Profit ($) Time (s) Profit ($) Time (s) 9,074.3 9,136.2 LR-EP[14] 9,030.5 MM [6] Proposed 9,074.3 5.14 9,136.2 7.69 R evenu e & Total Co st ($)
Hour
TABLE I. POWER AND RESERVE GENERATION OF PAYMENT FOR RESERVE ALLOCATED (ζ=0.005 AND RESERVE PRICE=0.04*SPOT PRICE) Conventional UCP Profit Based UCP Power (MW) Reserve (MW) Profit ($) Power (MW) Reserve (MW) Profit ($) U1 U2 U3 U1 U2 U3 U1 U2 U3 U1 U2 U3 0 100 70 0 0 20 132.77 0 0 170 0 0 20 537.67 0 100 150 0 0 25 360.64 0 0 200 0 0 0 570.00 0 200 200 0 40 0 114.31 0 0 200 0 0 0 300.00 0 320 200 0 55 0 318.61 0 0 200 0 0 0 390.00 100 400 200 70 0 0 -342.33 0 330 200 0 70 0 215.67 450 400 200 95 0 0 1049.68 0 400 200 0 0 0 1350.00 500 400 200 100 0 0 1074.52 0 400 200 0 0 0 1380.00 200 400 200 80 0 0 573.79 0 400 200 0 0 0 990.00 100 350 200 15 50 0 328.14 0 387 200 0 13 0 810.41 130 0 200 35 0 0 377.91 0 130 200 0 35 0 829.78 200 0 200 40 0 0 237.09 0 200 200 0 40 0 817.44 350 0 200 55 0 0 107.95 0 350 200 0 50 0 945.03 Total 4333.08 Total 9136.00
(a)
6000
10000
(b)
4000 5000 2000 0 2
4
6 8 Time (hr) Revenue
10
12
0
2 Total cost
4
6 8 Time (hr)
10
12
Profit
Fig. 3 Revenue, total cost, and profit curve (a) Convention UCP and (b) Profit based UCP.
4
TABLE IV. POWER AND RESERVE GENERATION OF PAYMENT FOR RESERVE ALLOCATED (ζ=0.005 AND RESERVE PRICE=0.01*SPOT PRICE) Profit Based UCP Power (MW) Reserve (MW) U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 Profit ($) 455 245 0 0 0 0 0 0 0 0 0 70 0 0 0 0 0 0 0 0 1838.95 455 295 0 0 0 0 0 0 0 0 0 75 0 0 0 0 0 0 0 0 1963.62 455 395 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 3348.57 455 455 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3258.20 455 455 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3804.20 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3094.04 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3186.04 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2822.04 455 455 130 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3020.24 455 455 130 130 162 68 0 0 0 0 0 0 0 0 0 12 0 0 0 0 11255.65 455 455 130 130 162 80 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13523.82 455 455 130 130 162 80 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15641.82 455 455 130 130 162 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5915.59 455 455 130 130 130 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 5674.36 455 455 0 130 160 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 3082.62 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2978.04 455 415 0 130 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 2747.03 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2718.04 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2874.04 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3342.04 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3810.04 455 455 0 130 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3654.04 455 445 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 3299.61 455 345 0 0 0 0 0 0 0 0 0 80 0 0 0 0 0 0 0 0 2632.55 109485.21 Total
Revenue & Total cost ($)
14000 12000
7000
(a)
6000
10000
5000
8000
4000
6000
3000
4000
2000
2000
1000
0 400
500
600
units (unit 1 to unit 6) and selling power at 1412 MW. Comparison of execution time with other methods available in literature is given in Table V. From simulation results it is found that proposed approach provides better solution with much lower execution times as compared to other approaches.
(b)
150
0 700 800 900 1000 1100 1200 150 200 250 300 350 400 450 500 550 Power (MW) Power (MW) Revenue
Total cost th
1200 Modified (f s )
N
Initial 2 -1
600
Profit
Fig. 4 Revenue, total cost, and profit curve at 7 hour, (a) Convention UCP and (b) Profit based UCP.
B. Ten-unit test system A ten unit system with 24 hour of scheduling time interval is used to demonstrate the advantage of proposed approach. Unit characteristics, cost coefficients, and other forecasted data are taken from [19]. Same methods of reserve payment are taken as discussed for three units system. Here, reserve price is fixed at 5 and 0.01 times of spot price for PPD and PRA method, while ζ is fixed at 0.05 for PPD and 0.005 for PRA method. The simulation results of proposed approach are shown in Table III. It is found that proposed approach provide more profit to GENCOs as compare to other existing methods such as improved PSO [9], improve SA [12], LR-PSO [15] and multi-agent approach [19], etc. Fig. 5 shows the trajectory of optimal states at which maximum profit is obtained at each hour for both original state matrix (2N-1) and modified search space matrix (fs). After reforming, search space matrix for each hour contains fs binary states (particle search space). Out of these states, particles will automatically select the optimum (eligible) state which provides maximum profit. This will continue up to 24-hr load period. Scheduling plan of power and reserve for PRA method is shown in Table IV. It is found that maximum profit to GENCOs is received at 12th hour of scheduling horizon, while running 6
100
1000
50
800
0
1
2
3
4
5
6
7
8
9
State number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
State number
Hour
600 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (hr)
Fig. 5 Eligible states trajectory for 24 hours.
TABLE III: PROFIT COMPARISON FOR 10 GENERATING UNITS Techniques Profit Based UCP ($) Payment Method-A Payment Method-B 1,13,122.81 LR [13] 1,13,122.81 GA [13] 1,13,134.12 LRGA[13] 1,12,818.93 1,07,838.5 LR-EP [14] 1,09,485.19 MAS [18] 1,13,018.7 Improved PSO [9] 1,13,134 Improved SA [12] 1,03,296 MM [6] 1,13,011 TS-IRP [16] 1,04,407 TS-RP [16] 1,07,838.57 LR-PSO [15] 1,13,256.3 1,09,332.6 Multi-agent [19] Proposed 1,13,922.08 1,09,485.21 TABLE V: EXECUTION TIME COMPARISON FOR 10 GENERATING UNITS Payment Method-A Techniques LR [13] GA [13] LR-GA [13] 111 752 1112 Time (s) Techniques TS-RP [16] TS-IRP [16] Proposed 109 63.84 Time (s) 79
5
VII.
CONCLUSION
This paper proposes a hybrid approach which consists of particle swarm optimization along with dynamic programming to solve profit based unit commitment problem under the deregulated power market. To solve the optimization problem (PBUCP), model is divided into exterior and interior dependent sub-problems, which are discrete and continuous, respectively. Exterior sub problem consists of reduced search space matrix and interior sub problem is solved with particle swarm optimization technique. The proposed model helps GENCOs to make a decision, for how much power and reserve that must be put up for sale in the markets, and how to schedule generators in order to receive the maximum profit. Two type of market strategies based on reserve payment method are simulated and compared with the existing literature methods. Simulation results confirm that proposed approach provide more profit to GENCOs in comparison to other existing methods. Future work will focus on the detailed evolution of proposed approach with some improvement to performance in large power system with reserve uncertainty. REFERENCES [1]
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