A Layered Matrix Cascade Genetic Algorithm and Particle Swarm Optimization Approach to Thermal Power Generation Scheduling Siew Chin Neoh, Norhashimah Morad, Chee Peng Lim*, and Zalina Abdul Aziz School of Electrical and Electronic Engineering University of Science Malaysia, Engineering Campus 14300 Nibong Tebal, Penang, MALAYSIA *Corresponding author:
[email protected]
Abstract. A layered matrix encoding cascade genetic algorithm and particle swarm optimization approach (GA-PSO) for unit commitment and economic load dispatch problem in a thermal power system is presented in this paper. The tasks of determining and allocating power generation to different thermal units in a way that the total power production cost is at the minimum subject to equality and inequality constraints makes unit commitment and economic load dispatch challenging. A case study, based on the thermal power generation problem presented in [1], is used to demonstrate the effectiveness of the proposed method in generating a cost-effective power generation schedule. The schedule obtained is compared with that of Linear Programming (LP) as reported in [1]. The results show that the proposed GA-PSO approach outperforms LP in solving the unit commitment and economic load dispatch problem for thermal power generation system in this case study. Keywords: Genetic Algorithms, Particle Swarm Optimization, Thermal Power Scheduling
1 Introduction Thermal power generator scheduling represents a critical activity in power system operations. It includes a number of production constraints such as types and number of generation units available, minimum and maximum generation level, hourly cost of running at different generation level, generator start up cost, and also expected reserved capacity. According to [2], the thermal generator scheduling is a combinatorial problem that consists of two sub problems; the first is to develop a daily generator schedule whereas the second is to evaluate the total production cost associated with the generated schedule. The power generation scheduling problem can also be viewed as the unit commitment (number of power generation units dedicated to serve the load demand) and economic dispatch problem (the allocation of power generation to different generator units). In [3], the commitment and dispatch problems are decoupled, while the first attempt to consider the coupled problem is presented in [4]. Various numerical optimization techniques have been employed to optimize power generation scheduling, e.g. priority list methods, dynamic programming, branch-and-bound methods, and Lagrangian relaxation. According to [3], the priority list method gives a solution that is far from the optimum, whereas dynamic programming is computationally prohibitive due to its increasing search space. A simulatedannealing approach that is able to find the global optimum schedule for unit commitment is developed in [5]; however, the approach does not always satisfy the constraints. Branch-and-bound method has a drawback on the exponential growth in the execution time with the size of unit commitment problem. Lagrangian relaxation may suffer from numerical convergence even though it could provide fast solutions [6]. Besides the methods mentioned above, Genetic Algorithms (GAs) have been identified to be able to find solutions for problems involving multiple constraints and objectives, especially in various scheduling problems [7-9]. Similar to GA, Particle Swarm Optimization (PSO) is a population based optimization method that has been shown to be very effective when applied to diverse set of optimization problems [6, 10]. In this research, we investigate the effectiveness of a GA-PSO hybrid approach, using a layered matrix encoding representation structure, in finding solutions to the thermal generator scheduling problem.
2 Problem Description Unit commitment and economic load dispatch are important in power generation scheduling. In this study, the problem and data are taken from [1]. The daily electricity load demand that need to be fulfilled according to five different time periods is shown in Table 1. Three types of thermal power generating units are available in this study: twelve of type 1, ten of type 2, and five of type 3. For each type of generator, the minimum and maximum operating levels, hourly costs of running for each generator at the minimum level or above the minimum level, and the start up cost need to be considered in generating the power generator schedule. The details of each type of thermal power generator are given in Table 2. Table 1. Daily Electricity load demand according to time period
Time Period
Electricity Load Demand (Megawatts)
12 p.m. to 6 a.m. 6 a.m. to 9 a.m. 9 a.m. to 3 p.m. 3 p.m. to 6 p.m. 6 p.m. to 12 p.m.
15,000 30,000 25,000 40,000 27,000
Table 2. Information of thermal power generator
Generator Type
Units Available
Minimum Level (MW)
Maximum Level (MW)
Cost per hour at minimum (£)
Type 1 Type 2
12 10
850 1250
2000 1750
1000 2600
Type 3
5
1500
4000
3000
Cost per hour per megawatt above minimum (£) 2 1.30 3
Startup Cost (£) 2000 1000 500
Besides meeting the electricity load demand as shown in Table 1, there must be sufficient generators to support an increase of load of up to 15%. In other words, the extra load requirement must be supported by the chosen generators, without starting up any more generators. Generators that are already operating should be adjusted within their permitted limits to accomplish the increase of 15% electricity load demand. By fulfilling the demand constraint and generator operation level constraint, the main objective of this case study is to decide the best combination of generators that should be working in each particular period over a day to generate the best power dispatch in order to minimize the total production cost. 2.1 Constraints and Mathematical Formulation The main variables in this problem are as follows: nik = number of generating units of type k working in period i (where i =1,2,3,4,5 are the five period of the day, and k =1,2,3 are the generator type) zik = number of generator of type k started up in period i. xik = output power from generator of type k in period i. In the proposed model, the power scheduling problem is represented in two layers: (1) randomly generate the unit commitment schedule, (2) randomly allocate power to each generator type based on the generated schedule for unit commitment. These two layers are decision outputs for the power generation scheduling and the overall representation structure is given in Fig.1. In layer 1, nik, represents the number of generator unit to be running for generator type k in period i; whereas in layer 2, xik refers to the amount of power to be allocated to generator type k in period i based on layer 1.
There are three important constraints in the proposed model: − Demand constraints − Generator’s operation constraint − Start-up generator constraint 2.1.1 Demand Constraint The electricity load demand, as shown in Table 1, must be met in each period. Therefore, the sum of power to be generated by each generator type k in period i should fulfill the demand for period i, as indicated in Equation (1), where Di is the demand in period i. k =3
∑x
ik ≥
Di
(1)
k =1
The extra guaranteed load requirement must be able to be met without starting up any more generators, as shown in Equation (2) for all period i. k =3
∑M n k
k =1
ik
≥
115 Di 100
(2)
where Mk is the maximum output levels for generators of type k and nik is the committed units of generator k at period i. 2.1.2 Generator’s Operation Constraint The output power of each generator must lie within the limits of the minimum and maximum operation levels. Equations (3) and (4) show the operation constraints of the power generator.
xik ≥ mk nik
(3)
xik ≤ Mk nik
(4)
where mk and Mk are the given minimum and maximum operating levels for generator type k, xik is the generated output power of generator k in period i, and nik is the committed units of generator k at period i. 2.1.3 Start-up Generator Constraint The number of generator started up in period i, zik, can be obtained by Equation (5).
n ik - n (i - 1) k if generator n ik ≥ n ( i − 1 ) k z ik = 0 if otherwise
(5)
where nik refers to the committed units of generator k at period i. Note that when i=1, period i-1 is taken as the 5th period 2.1.4 Evaluation Function The weighted-sum approach is applied to minimize the total power generation cost. Equation (6) shows the objective function used. Objective
Function
(6)
i=5 k =3 = minimize ∑ ∑ (C k (x ik - m k n ik ) + E k n ik + F k z ik ) i =1 k =1
where Ck is the cost per hour per megawatt above the minimum level of generator k multiplied by the number of hours in period i, Ek is the cost per hour for operating at the minimum level of generator k multiplied by the number of hours in period i, and Fk is the start up cost of generator k.
3 The GA-PSO Optimization 3.1 A Layered Matrix Structure Many real world problems involve multi-decision and multi-constraint. As a result, multi-dimensional encoding structure may be necessary in order to incorporate all required constraints and decisions into one single solution representation. However, it is clear that increase in dimension for a multi-dimensional encoding structure (e.g. 4 or 5 dimensions) makes the solution representation and evaluation process tedious. This paper proposed a layered matrix encoding structure to solve multi-dimensional encoding problems. The proposed layered matrix encoding structure is different from the existing multi-dimensional encoding approaches in which it separates different decision outputs into different layers so as to simplify the problem representation. With the layer structure, constraints and decision outputs can be analyzed more distinctively and effectively, and at the same time makes the evaluation process much easier. One of the main advantages of the layered encoding structure is that it could allow multi-stage cascade optimization where different optimizers can be used to optimize different decision based on the characteristic of the decision search space. In the proposed model for power generation scheduling, a two layers 2D matrix is used to represent the power generation schedule for each period throughout a day which is similar to the structure shown in Fig.1. Each layer represents different decision outputs, layer 1 is used to decide unit commitment (number of generator units to be running), and layer 2 is used to decide economic power dispatch (determination of the amount of power to be generated from the selected generator to fulfill demand). Layer 1: Unit Commitment Generator (Gk)
Period (Pi) P1 P2 P3 P4 P5
G1 n11 n21 n31 n41 n51
G3 n13 n23 n33 n43 n53
Layer 2: Power Dispatch Generator (Gk)
Period (Pi) P1 P2 P3 P4 P5
G2 n12 n22 n32 n42 n52
G1 x11 x21 x31 x41 x51
G2 x12 x22 x32 x42 x52
G3 x13 x23 x33 x43 x53
Electricity Demand (Di) D1 D2 D3 D4 D5
Electricity Demand (Di) D1 D2 D3 D4 D5
Fig. 1. The layered matrix encoding structure for power generation schedule in the proposed model
3.2 Cascade GA-PSO Optimization For every single schedule of unit commitment, there could be thousands of possible schedules of power dispatch that could fulfill the similar unit commitment. Therefore, it is easy to imagine how large the
searching space of stochastic search can be. This paper proposed a layered encoding GA-PSO cascade optimization to narrow down the searching space and fasten the process of finding both global and local optimums. The flow of cascade GA-PSO optimization model is depicted in Fig.2. Layers in the layered matrix structure can be viewed as the optimization stages in cascade optimization. In the proposed cascade optimization for power generation scheduling, a population of possible unit commitment schedules is generated based on the power generation constraints in the first stage. For each schedule of unit commitment, a population of possible power dispatch is generated in the second stage. The GA is used to find the optimal power dispatch for each schedule of unit commitment. The best power dispatch for each unit commitment schedule will be returned back to the first stage. Subsequently, based on all the best combinations of unit commitment and power dispatch in the first generation, PSO is used to fine-tune the possible unit commitment schedules and thereafter generate the second generation of unit commitments. The evolution loop of GA-PSO proceeds until the best solution converges for 20 iteration.
Fig. 2. A cascade GA-PSO optimization model
The GA is a stochastic search method that mimics the metaphor of natural biological evolution [11], whereas PSO is an optimization tool driven by the social behavior of organisms [12]. The GA is used to search for the best power dispatch to represent a particular schedule of unit commitment. After each
schedule of unit commitment has its best representative of economic dispatch, PSO will be used to learn from the behavior of the pool of representatives and adjust the search for better unit commitment. The combination of GA and PSO provides good balance between exploration and exploitation which will also helps to balance the individuality and sociality of the search. The cascade GA-PSO optimization is as follow: Step1: First Stage Initialization The schedules of unit commitment (positions of particles, nikq) and the initial velocity of each position of the particle, Vikq, are randomly initialized. Check the feasibility of each particle. Regenerate the particle if it does not satisfy the constraints. Step2: Second Stage Initialization Based on each schedule of unit commitment, generate a population of appropriate and feasible power dispatch. Integrate GA to find the best representative of power dispatch for each unit commitment schedule. Return the best representatives into first stage. Step 3: Particles Updating Adjust the position and velocity of each particle with Equation (7) and (8) to generate a new generation of unit commitment schedules. Step 4: Repeat Repeat step 2 to step 3 until the number of PSO iteration reaches T, a maximum PSO iteration which is specified before termination. 3.3 PSO in Unit Commitment PSO is a stochastic population-based approach in which each potential solution (also called as particle) searches through the problem space, refining its knowledge, adjust its velocity based on the social information that it gathered and update its position. The best found position for a particle is given as pbest whereas the best found position for all particles is denoted as gbest. Equation (7) and (8) calculate the velocity (Vikq) and position (nikq) of particle q in the dimensional search space of i×k respectively. Position nikq refers to the unit commitment, nik, of particle q in the population of particle swarm. V ikq = w × V ikq + c 1 × rand ( ) × ( pbest − n ikq ) + c 2 × rand ( ) × ( gbest − n ikq )
(7)
nikq = nikq +Vikq
(8)
where w is the inertia weight, c1 and c2 are two constant factors, and rand( ) is a randomly generated value between 0 and 1. A decreasing w from the range of 0.9 to 0.4 is applied in this case study as it has been shown to perform well in [13, 14]. The values of c1 and c2 are both set to 2 in this research as mentioned in [12] where the recommended choice for constant c1 and c2 is integer 2 since it on average makes the weight for “social” and “cognition” parts to be 1. The PSO loop is terminated after convergence for 20 iterations. 3.4 GA in Power Dispatch In the proposed model, layer 2 of the layered matrix structure is used to represent chromosomes in the GA population pool. A probabilistic selection is performed based upon the individual’s fitness such that the better individuals have an increased chance of being selected. Stochastic Universal Sampling which is a single phase sampling algorithm with minimum spread and zero bias is used as the selection function in the proposed model. The basic strategy underlying this approach is to keep the expected number of copies of each chromosome in the next generation [15]. 3.4.1 GA Operators For this case study, Randomly Selected Crossover is used as the crossover operator whereas Randomly Selected Mutation is used as the mutation operator. The principle of Randomly Selected Crossover for the power generation scheduling problem is depicted in Fig. 3. As the chromosomes are represented by 2D
matrices, Randomly Selected Crossover is applied to the selected matrix rows. The crossover probability is given as Pxovr, in which each individual has Pxovr chances to be selected for a crossover. For each crossover, the number of rows that are randomly selected from a 2D matrix will be inter-exchanged with another individual to produce offsprings (new individuals). The purpose of having Randomly Selected Crossover is to allow inter-exchange of power scheduling distribution among individuals in the population and, at the same time, to ensure that the load demand for a particular period, Di, is always met. Randomly Selected Mutation is applied to randomly regenerate output power for a selected period over a day. The probability for an individual to be selected to undergo mutation is given as Pm. In Randomly Selected Mutation, the randomly selected rows from the 2D matrix (selected period of power schedule) will be regenerated (Fig. 4). Period (Pi) P1 P2 P3
Period
Generator G1 … … …
G2 … … …
(Pi) P1 P2
G3 … … …
P3
Generator G1 … … …
G2 … … …
G3 … … …
Fig. 3. Randomly Selected Crossover
Period (Pi) P1 P2 P3
Generator G1 … … …
G2 … … …
G3 … … …
Regenerate power schedule for period i
Fig. 4. Randomly Selected Mutation 3.4.2 Genetic Parameter Selection The techniques developed in this model were implemented on a 4.0GHz PC using MATLAB language. According to experience, the GA population size and the maximum number of generation convergence for GA termination are both selected as 20; whereas the crossover probability, Pxovr, and mutation probability, Pm, are set as 0.7 and 0.3 respectively.
4 Results The results for power dispatch and unit commitment based on the layered matrix cascade GA-PSO approach and LP (reported in [1]) are depicted in Tables 3 and 4, respectively. From the tables, the layered matrix cascade GA-PSO approach gives lower daily power production cost (£ 954,260) compared to that of LP (£ 988,540). A total cost of £ 34,280 has been saved using the proposed approach. It is clearly showed that layered matrix GA-PSO cascade optimization outperforms LP in this case study. As a result, it can be used as an alternative way of solving unit commitment and power dispatch problem in thermal power generation scheduling. In the point of view of soft computing, the layered matrix encoding structure of GAPSO cascade optimization helps to narrow down the search space and fasten the process of finding both local and global minimum. Searching by layers in a GA-PSO cascade flow is easier then searching randomly in a big pool of possible solutions without a proper direction.
5 Conclusions A layered matrix cascade GA-PSO approach has been developed and applied to solve unit commitment and power dispatch problem in thermal power generation scheduling. Constraints of the power generation
scheduling have been formulated and explained in this paper and the advantages of the layered matrix structure and the cascade GA-PSO approach is mentioned. The detail procedure of cascade GA-PSO has been given and the performance between the layered matrix cascade GA-PSO model and LP in saving total daily power generation cost has been compared. The intrinsic nature of particles position and velocity update in PSO and the genetic evolution character in GA makes the cascade GA-PSO a superior approach for solving unit commitment and power dispatch problem. Similarly, the layered matrix structure that is able to reduce the complexity of multi-dimensional representation and to allow combination of different optimizers enhanced the construction of the cascade GA-PSO model. Table 3. Power dispatch and unit commitment using the layered matrix cascade GA-PSO model Period Period 1 Period 2 Period 3 Period 4 Period 5
Units commitment for generator type
Power dispatch (MW)
Total daily cost
12 of type 1 3 of type 3 5 of type 1 4 of type 2 5 of type 3 12 of type 1 4 of type 2 11 of type 1 3 of type 2 5 of type 3 12 of type 1 7 of type 2
10450 4550 5300 16800 7900 10300 14700 15850 16450 7700 10500 16500
£ 954,260
Table 4. Power dispatch and unit commitment using LP (as reported in [1]) Period
Period 1 Period 2 Period 3 Period 4
Period 5
Units commitment for generator type
Power dispatch (MW)
Total daily cost
12 of type 1 3 of type 2 12 of type 1 8 of type 2 12 of type 1 8 of type 2 12 of type 1 9 of type 2 2 of type 3 12 of type 1 9 of type 2
10200 4800 16000 14000 11000 14000 21250 15750 3000 11250 15750
£ 988,540
References 1. Williams, H. P.: Model Building in Mathematical Programming. 4th edn. John Wiley and Sons Inc (1999). 2. Wong, K. P. and Wong, Y. W.: Thermal generator scheduling using hybrid genetic / simulated -annealing approach. IEE Proceedings Part C –Generation Transmission, and Distribution, Vol. 142, No.4 (1995) 372-380. 3. Aldridge, C. J., McKee, S., McDonald J.R., Galloway, S. J., Dahal, K. P., Bradley, M. E., and Macqueen, J. F.: A knowledge-based genetic algorithm for unit commitment. IEE Proceedings on Generation, Transmission and Distribution, Vol. 148, No. 2 (2001) 146-152.
4. Garver, L. L.: Power scheduling by integer programming. IEEE Trans. Power Apparatus and Systems, Vol. 81 (1963) 730-735. 5. Zhuang, F. and Galiana, F. D.: Unit commitment by simulated annealing. IEEE Transactions on Power Systems, Vol. 5, No. 1 (1990) 311-317. 6. Ting, T.O., Rao, M.V.C., and Loo, C.K.: A novel approach for unit commitment problem via an effective hybrid particle swarm optimization. IEEE Transactions on Power Systems, Vol. 21, No. 1 (2006) 411-418. 7. Sarker, R. and Newton, C.: A genetic algorithm for solving economic lot size scheduling problem. Computers and Industrial Engineering, Vol. 42 (2002) 189-198. 8. Ip, W. H., Li, Y., Man, K. F., and Tang, K. S.: Multi-product planning and scheduling using genetic algorithm approach. Computer & Industrial Engineering, Vol. 38 (2000) 283-296. 9. Ono, I., Yamamura, M., and Kobayashi, S.: A genetic algorithm for job-shop scheduling problems using job-based order crossover. Proceedings of 1996 IEEE International Conference on Evolutionary Computation (1996) 547-552. 10. Eberhart, R.C. and Shi, Y.: Particle swarm optimization: developments, applications, and resources. Proceedings of the IEEE Congress on Evolutionary Computation, Seoul, Korea (2001) 81-86. 11. Holland, J.: Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor (1975). 12. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. Proceedings of IEEE International Conference on Neural Networks (1995) 1942-1948. 13. Naka, S., Genji, T., Yura, T., and Fukuyama, Y.: A hybrid particle swarm optimization for distribution state estimation. IEEE Transactions on Power Systems, Vol. 18, No. 1 (2003) 60-68. 14. Yoshida, H., Kawata, K., Fukuyama, Y., Takayama, S., and Nakanishi, Y.: A particle swarm optimization for reactive power and voltage control considering voltage security assessment. IEEE Transactions on Power Systems, Vol. 15, No. 4 (2000) 1232-1239. 15. Gen, M. and Cheng, R., Genetic Algorithms and Engineering Optimization. John Wiley & Sons Inc, New York (2000).
