Daily combined economic emission scheduling of hydrothermal systems with cascaded reservoirs using self organizing hierarchical particle swarm optimization technique

Expert Systems with Applications 39 (2012) 3438–3445

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Daily combined economic emission scheduling of hydrothermal systems with cascaded reservoirs using self organizing hierarchical particle swarm optimization technique K.K. Mandal ⇑, N. Chakraborty Jadavpur University, Department of Power Engineering, Kolkata 700098, India

a r t i c l e

i n f o

Keywords: Self-organizing particle swarm optimization with time-varying acceleration coefficients (SOHPSO_TVAC) Combined economic emission scheduling (CEES) Cascaded reservoirs Hydrothermal systems

a b s t r a c t Daily optimum economic emission scheduling of hydrothermal systems is an important task in the operation of power systems. Many heuristic techniques such as differential evolution, and particle swarm optimization have been applied to solve this problem and found to perform better in comparison with classical techniques. But a very common problem with these methods is that they often converge to sub-optimal solution prematurely. A reliable and efficient method termed as self-organizing hierarchical particle swarm optimization technique with time-varying acceleration coefficients (SOHPSO_TVAC) is presented in this paper to avoid premature convergence. A multi-chain cascaded hydrothermal system with non-linear relationship between water discharge rate, power generation and net head is considered in this paper. The water transport delay between connected reservoirs is also taken into consideration. The problem is formulated considering both cost and emission as competing objectives. The effect of valve point loading is also taken into account in the present problem formulation. The feasibility of the proposed method is demonstrated on a sample test system. The results of the proposed technique are compared with other heuristic techniques. It is found that the results obtained by the proposed technique are superior in terms of fuel cost, emission output etc. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Daily optimum scheduling of hydrothermal systems plays an important role in the economic operation of electric power systems. The generation scheduling problem consists of determining optimum operation strategy for allocation of generations to different units so as to minimize the total operational cost subjected to a variety of constraints. The operational cost of hydroelectric plants is insignificant as water resources for electric power production is assumed to be renewable. Thus, the problem of minimizing the operational cost of a hydrothermal system reduces to minimizing the fuel cost of thermal plants subjected to variety of constraints of hydraulic and power system network. Cascaded hydropower plants are related to each other both in terms of electric power and hydraulic properties. The thermal power plants based on fossil fuels releases harmful emission such as oxides of carbon, sulphur oxides, oxides of nitrogen, etc. These emissions not only affect human but also the entire plant and animal lives over the globe. Due to growing concern over environmental protection, many international agreements have ⇑ Corresponding author. Tel.: +91 33 23355813; fax: +91 33 23357254. E-mail address: [email protected] (K.K. Mandal). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.09.032

been made to reduce the emission below certain tolerable limits over the years. This has forced the electric utilities all over the world to reduce the plant emission level below certain specified limits. Several methods and strategies to reduce the emission from thermal power plants have been proposed and discussed. These include switching to low emission fuel, replacement of inefficient fuel burners, installation of pollutant arresting equipments and so on. These methods involve considerable amount of capital investment, time and can be termed as long term strategies. Economic emission scheduling in an attractive alternative and it addresses both economy and reduction of emission simultaneously. One of the major complications in the above considerations is that the cost and emission functions are of conflicting nature. In other words, minimizing pollution increases cost and vice versa. Emissions in power dispatch problems have been included either in objective function or treated as additional constraints by many research groups (El-Keib, Ma, & Hart, 1994; Talaq, El-Hawary, & El-Hawary, 1994). Several methods have been applied successfully to reduce the emissions. Some of the important methods are evolutionary algorithm based multi-objective technique (Abido, 2003), evolutionary programming technique (Wong & Yuryevich, 1998), improved back-propagation neural network methodology (Kulkarni, Kothari, & Kothari, 2000), improved genetic algorithm

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based multi-objective approach (Chiang, 2007), fuzzy satisfaction maximizing decision approach (Huang, Yang, & Huang, 1997), Hopfield neural network methodology (King, El-Hawary, & El-Hawary, 1995), evolutionary optimization algorithms (Deb, 2008) and algorithms based particle swarm optimization techniques (Agrawal, Panigrahi, & Tiwari, 2008; Wang & Singh, 2009). The generation scheduling problem of hydrothermal systems has been the subject of intensive research work for several decades. Scholars have used many methods to solve this difficult optimization problem. Some of these solution techniques are dynamic programming approach (Chang, Chen, Fong, & Luh, 1990), decomposition techniques (Pereira & Pinto, 1982) and Lagrangian relaxation technique (Guan & Peter, 1998). In recent times optimal hydrothermal scheduling problems have been solved by different heuristic techniques such as simulated annealing (Wong & Wong, 1994), genetic algorithm (Orero & Irving, 1998; Ramirez & Ontae, 2006), evolutionary programming (Yang, Yang, & Huang, 1996), cultural algorithm (Yuan & Yuan, 2006) and evolutionary strategy (Werner & Verstege, 1999). Lakshminarasimman and Subramanian (2008) applied modified hybrid differentia evolution technique to solve short-term hydrothermal generation scheduling problems with promising results. A comparison of particle swarm optimization and dynamic programming for large scale hydro unit load dispatch was made by Cheng, Liao, Tang, and Zhao (2009). Recently, mixed-integer quadratic programming method was applied to determine scheduling of head dependent cascaded hydro systems by Catalão, Pousinho, and Mendes (2010). Multi-objective shortterm hydrothermal scheduling problems have also been solved by many heuristic techniques such as fuzzy decision-making methodology (Dhillon, Parti, & Kothari, 2002), multi-objective differential technique (Qin, Zhou, Lu, Wang, & Zhang, 2010) and methods based on heuristic search technique (Dhillon, Dhillon, & Kothari, 2007) and so on. Particle swarm optimization (PSO) happens to be a comparatively new combinatorial metaheuristic technique which is based on the social metaphor of bird flocking or fish schooling (Kennedy & Eberhart, 1995). The PSO technique has been applied to various fields of power system optimization such as economic dispatch (Park, Lee, Shin, & Lee, 2005) hydrothermal scheduling (Wu, Zhu, Chen, & Zhang, 2008; Yu, Yuan, & Wang, 2007) etc. A novel parameter automation strategy called self-organizing hierarchical particle swarm optimization technique with timevarying acceleration coefficients (SOHPSO_TVAC) has been applied in this paper for daily combined economic emission scheduling of cascaded hydrothermal systems to address the problem of premature convergence. In this case, the particle velocities are reinitialized whenever the population stagnates at local optima during the search. A relatively high value of the cognitive component results in excessive wandering of particles while a higher value of the social component causes premature convergence of particles. Hence, time-varying acceleration coefficients (TVAC) (Ratnaweera, Halgamuge, & Watson, 2004) are employed to strike a proper balance between the cognitive and social component during the search. The effectiveness of the proposed SOHPSO_TVAC technique is tested on a sample test system comprising of four cascaded hydro units and three thermal units. The main constraints included are the cascaded nature of the hydraulic network, the time coupling effect of the hydro sub problem where the water inflow of an earlier time interval affects the discharge capability at a later period of time, the varying hourly reservoir inflows, the physical limitations on the reservoir storage and turbine flow rate, the varying system load demand and the loading limits of both thermal and hydro plants. The effect of valve point loading is also included in the problem formulation. A comparison with improved quantumbehaved particle swarm optimization technique (Sun & Lu, 2010) is presented here which shows SPHPSO_TVAC could provide quite

encouraging results. It is observed that the proposed technique performs effectively in comparison to other population based heuristic techniques. 2. Problem formulation Daily combined economic emission scheduling of hydrothermal systems is an optimization problem that considers both economy and emission as objectives. As the fuel cost of hydroelectric plants is insignificant in comparison with that of thermal power plants, the objective is to minimize the fuel cost and as well as the emission of thermal units, while making use of the availability of hydro-resources as much as possible. 2.1. Economic scheduling The pure economic load-scheduling (ELS) problem may be described as the minimization of the total fuel cost of the thermal units under several operating constraints. Thus, for a given hydrothermal system, the problem may be described as minimization of total fuel cost as defined by (1) under a set of operating constraints:

Minimize FðPsit Þ ¼

Ns T X X  v  fit ðPsit Þ t¼1

ð1Þ

i¼1

where F(Psit) is the total fuel cost, T is the total number of time interval for the scheduling horizon, Ns is the total number of thermal generating unit, Psit is the power generation of ith thermal generating unit at time t and fitv ðPsit Þ is the fuel cost function. The fuel cost function of each thermal unit considering valve point loading effects may be expressed as the sum of a quadratic and sinusoidal function. So fitv ðPsit Þ can be defined by (2) as:

fitv ðPsit Þ ¼ asi þ bsi P sit þ csi P2sit þ jesi  sinffsi  ðP min  Psit Þgj si

ð2Þ

where asi, bsi, csi, esi, fsi are the fuel cost coefficients of the ith thermal generating unit. 2.2. Emission scheduling The solution of pure economic load-scheduling (ELS) problem determines the amount of active power to be generated by different units at a minimum fuel cost for each time interval during the entire scheduling period. But the amount of emission or emission cost is not considered in the above pure ELS problem. The emission generated can be expressed as a sum of a quadratic and an exponential function. The economic emission-scheduling (EES) problem is thus described as the minimization of total amount of emission release defined by (3) as:

EðPsit Þ ¼

Ns h T X X t¼1

i

asi þ bsi Psit þ csi P 2sit þ gsi expðdsi Psit Þ

ð3Þ

i¼1

where E(Psit) is total amount of emission and asi, bsi, csi, gsi, dsi are the emission coefficients of the ith unit. 2.3. Combined economic and emission dispatch The economic load scheduling and emission scheduling are combined together that will consider both economy and emission simultaneously. The combined economic emission scheduling (CEES) problem can be expressed as follows by introducing a price penalty factor ht (Kulkarni et al., 2000):

Minimize TC ¼ FðPsit Þ þ ht  EðPsit Þ

ð4Þ

where TC is the total operational cost of the system and ht is the price penalty factor during time t. Now, for a trade off between fuel cost and emission cost, (4) can be revised as follows:

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Minimize TC ¼ w1  FðP sit Þ þ w2  ht  EðPsit Þ

ð5Þ

where w1 and w2 are the weight factors. For pure economic load scheduling w1 = 1 and w2 = 0; for pure economic emission scheduling w1 = 0 and w2 = 1 while w1 = w2 = 1 yields results for combined economic emission scheduling (CEES). The price penalty factor ht can be found out by a practical method as discussed in (Kulkarni et al., 2000). The following steps can be used to find out the price penalty factor for a particular load during each time interval over the entire scheduling period. Find out the average cost of each generator at maximum power output. Find out the average emission of each generator at its maximum output. Divide the average cost of each generator by its average emission and thus hit is given as: max FðPmax si Þ=ðP si Þ max max ¼ hit EðPsi Þ=ðPsi Þ

ð6Þ

$=lb

Table 1 Hourly load demand. Hour

PD (MW)

Hour

PD (MW)

Hour

PD (MW)

1 2 3 4 5 6 7 8

750 780 700 650 670 800 950 1010

9 10 11 12 13 14 15 16

1090 1080 1100 1150 1110 1030 1010 1060

17 18 19 20 21 22 23 24

1050 1120 1070 1050 910 860 850 800

(i) Active power balance The total power generated must balance the power demand plus losses, at each time interval over the entire scheduling period Ns X

Psit þ

i¼1

Arrange the values of price penalty factor in ascending order. Add the maximum capacity of each unit (Pmax ) one at a time P max si Psi P PDt is realized. starting from the smallest hit unit until At this stage, hit associated with last unit in the process is the price penalty factor ht for the given load during the time t. From the above description, it is clear that the value of the price penalty factor ht is dependent on the total power demand during each time interval and hence it will have different values for different power demand. It is also important to note that the value of the price penalty factor ht will be the same for ELS, EES and CEES as long as the power demand is the same. The above objective function described by (5) is to be minimized under a variety of constraints as follows:

Ι h1

Ι h2

-----Reservoir 1 - - - - - Qh1

------ - - - - - Reservoir 2 Qh 2

Ι h3 Reservoir 3

Nh X

Phjt  PDt  PLt ¼ 0

ð7Þ

j¼1

where Phjt is the power generation of jth hydro generating unit at time t, PDt is power demand at time t and PLt is total transmission loss at the corresponding time. In this work the power loss is not considered for simplicity. However, it may be calculated by using B-loss matrix directly. The hydropower generation is a function of water discharge rate and reservoir storage volume, which can be described by (9) as follow:

P hjt ¼ C 1j V 2hjt þ C 2j Q 2hjt þ C 3j V hjt Q hjt þ C 4j V hjt þ C 5j Q hjt þ C 6j

ð8Þ

where C1j, C2j, C3j, C4j, C5j, C6j are power generation coefficients of jth hydro generating unit, Vhjt is the storage volume of jth reservoir at time t and Qhjt is water discharge rate of jth reservoir at time t. (ii) Power generation limit

P min 6 Psit 6 Pmax si si

ð9Þ

max P min hj 6 P hjt 6 P hj

ð10Þ

where P min and P max are the minimum and maximum power si si and Pmax are generation by ith thermal generating unit, Pmin hj hj the minimum and maximum power generation by the jth hydro generating unit respectively. (iii) Water dynamic balance

------------Qh 3 Ι h4

------Reservoir 4 - - - - - - Qh 4

V hjt ¼ V hj;t1 þ Ihjt  Q hjt  Shjt þ

Ruj   X Q hm;tsmj þ Shm;tsmj

ð11Þ

m¼1

Where: Ι hj : natural inflow to j th reservoir Qhj : discharge of j th plant

Plant

1

2

3

4

Ru 0 0 2 1 td 2 3 4 0 Ru : no of upstream plants t d : time delay to immediate downstream plant Fig. 1. Hydraulic system network.

where Ihjt is natural inflow of jth hydro reservoir at time t, Shjt is spillage discharge rate of jth hydro generating unit at time t, smj is the water transport delay from reservoir m to j and Ruj is the number of upstream hydro generating plants immediately above the jth reservoir. (iv) Reservoir storage volume limit max V min hj 6 V hjt 6 V hj

ð12Þ

max V min hj ;V hj

where are the minimum and maximum storage volume of jth reservoir. (v) Water discharge rate limit max Q min hj 6 Q hjt 6 Q hj

ð13Þ

where Q min and Q max are the minimum and maximum water hj hj discharge rate of the jth reservoir respectively.

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K.K. Mandal, N. Chakraborty / Expert Systems with Applications 39 (2012) 3438–3445 Table 2 Hydrothermal generation (MW) schedule for ELS using SOHPSO_TVAC. Hour

Ph1

Ph2

Ph3

Ph4

Ps1

Ps2

Ps3

Total

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

95.1006 64.2479 85.7413 94.9959 72.8921 80.7418 78.6619 62.9463 96.7278 77.6102 70.3350 90.4967 64.5000 83.4600 69.8320 94.8863 95.2994 96.9788 64.6050 61.7848 90.8171 72.4979 77.4039 80.0452

61.4948 84.7611 73.3652 78.4565 78.0096 50.7131 53.7896 58.7682 53.1091 68.3063 57.4123 74.9660 44.5662 58.4851 56.7240 35.2052 46.9474 56.4173 30.0830 39.2927 9.8543 16.9546 13.2735 27.1106

38.7812 0.0000 39.5211 48.2748 0.0000 45.7951 54.2005 25.5332 58.9846 0.0000 50.4333 57.2741 52.9182 55.3748 58.1927 52.0714 33.2301 59.7761 32.6918 53.4615 55.3151 46.6291 22.9039 59.9171

262.5733 249.1560 222.4691 142.3747 164.3830 165.1706 210.9901 271.7310 300.4223 266.4650 231.9173 309.2425 241.7376 248.7730 278.8357 324.9290 318.5855 340.9950 309.4102 313.6466 302.6740 335.6640 278.2192 326.4660

99.8666 22.3280 20.0000 31.9264 152.3033 20.0000 21.3058 167.5068 139.4814 138.0747 175.0000 88.5101 175.0000 153.1742 22.9457 103.2880 102.1812 113.7402 102.8458 147.8618 105.8424 20.8696 20.0000 37.1139

139.6971 40.0000 208.9033 122.2370 48.4646 206.9135 300.0000 286.9186 210.7106 300.0000 285.5144 300.0000 211.4300 291.3060 293.5647 40.0000 129.9085 129.0628 115.9522 292.4014 207.6518 137.9750 211.0977 40.0000

52.4864 319.5070 50.0000 131.7347 153.9474 230.6659 231.0521 136.5954 230.5643 229.5440 229.3873 229.5105 319.8484 139.4272 229.9056 409.6205 323.8481 323.0302 414.4122 141.5513 137.8453 229.4098 227.1018 229.3472

750 780 700 650 670 800 950 1010 1090 1080 1100 1150 1110 1030 1010 1060 1050 1120 1070 1050 910 860 850 800

Table 3 Hourly plant discharge (104 m3) for ELS using SOHPSO_TVAC. Hour

Qh1

Qh2

Qh3

Qh4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

12.5917 6.5244 9.9531 12.9631 8.0754 9.7301 9.5407 6.8926 11.6285 9.4185 7.9239 12.2834 6.9104 9.9889 7.3942 12.6536 13.5063 7.9209 6.7962 6.3968 12.3862 8.3823 12.2123 10.3597

7.9148 13.2483 14.6989 10.7880 8.8223 7.6202 8.2352 9.5318 8.7851 12.8131 10.5401 13.3465 8.9831 13.0761 14.1591 8.9880 12.4783 8.4795 9.8668 14.6423 7.6825 8.8462 10.3551 12.5747

26.2050 28.5982 18.0211 18.4131 26.6203 17.3167 12.7747 22.9603 20.1461 28.2132 15.3666 23.6449 14.1426 14.9749 13.9749 18.3717 22.9557 23.8210 23.3699 18.6515 17.8262 20.4017 25.1097 15.5850

8.8661 19.2873 18.4224 8.0229 6.1726 10.8577 12.0997 18.9068 13.7290 16.3608 12.8428 14.7449 12.4622 11.9836 14.6102 19.2249 18.9448 9.1357 17.6320 18.0946 16.3102 19.7057 14.0417 17.7189

3. Overview of some PSO strategies 3.1. Classical PSO The particle swarm optimization (PSO) is one of the modern powerful heuristic optimization techniques. The method is inspired by the natural phenomenon of fish schooling or bird flocking. Kennedy and Eberhart (1995) originally developed the PSO concept based on the behavior of individuals (i.e. particles or agents) of a swarm or group. PSO is a population-based search procedure in which each individual is referred to as a particle and represents a solution for the problem under consideration. The particle flies around the multidimensional search space with a velocity that is dynamically adjusted according to its own experience and the experience of neighboring particle.

Let in a physical d-dimensional search space, the position and velocity of the ith particle (i.e. ith individual in the population of particles) be represented as the vectors Xi = (xi1, xi2, . . . , xid) and Vi = (vi1, vi2, . . . , vid) respectively. The previous best position of the ith particle is recorded and represented as pbesti = (pbesti1, pbesti2, . . . , pbestid). The index of the best particle among all the particles in the group is represented by the gbestd. The modified velocity and position of each particle can be calculated using the current velocity and the distance from pbestid to gbestd as shown in the following formulas:   k k k V kþ1 id ¼ w  V id þ C 1  randðÞ  ðpbest id  X id Þ þ C 2  randðÞ  gbest d  X id

i ¼ 1;2;...;Np ; d ¼ 1;2;...;N g ð14Þ where Np is the number of particles in a swarm or group, Ng is the number of members or elements in a particle, V kid is the velocity of individual i at iteration k; w is the weight parameter or swarm inertia, C1 and C2 are the acceleration constants, rand() is uniform random number in the range [0 1] and X kid is the position of individual i at iteration k. The updated velocity can be used to change the position of each particle in the swarm as depicted in (15) as:

X kþ1 ¼ X kid þ V kþ1 id id

ð15Þ

Suitable selection of inertia weight w provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. In general, the inertia weight w is set according to the following equation:

w ¼ wmax 

wmax  wmin  iter iter max

ð16Þ

where itermax is the maximum number of iterations and iter is the current number of iterations. 3.2. Concept of time-varying acceleration coefficients (TVAC) It is observed from (14) that the search toward the optimum solution is heavily dependent on the two stochastic acceleration components (i.e. the cognitive component and the social component). Thus, it is very important to control these two components

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Table 4 Hydrothermal generation (MW) schedule for EES using SOHPSO_TVAC. Hour

Ph1

Ph2

Ph3

Ph4

Ps1

Ps2

Ps3

Total (MW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

91.1514 88.5264 93.2535 88.2540 88.8990 87.5459 65.7025 83.1667 87.7204 67.4809 75.0176 88.5229 87.8218 85.2002 84.3555 80.1606 65.4609 83.9533 79.5672 74.1539 72.6252 68.0858 36.1298 42.9892

52.885 88.7729 84.6470 42.3498 47.0394 83.0984 52.4703 60.2389 42.8641 55.2180 72.2509 76.7996 46.2141 59.3329 55.3433 56.9074 16.3948 45.6067 32.2222 35.2038 43.1932 25.9159 11.0555 10.4060

0.0000 46.6983 8.4327 0.0000 49.0880 23.9828 27.3841 54.4679 49.3643 36.6128 43.6846 62.5259 56.4486 54.3696 59.8080 42.1091 61.5749 54.3164 61.8911 41.1925 54.9914 41.9691 56.9051 63.2909

132.4231 251.9537 160.5644 232.6974 152.0241 223.3453 279.4771 208.9616 215.2553 224.2156 299.0524 316.1477 230.7239 302.6108 261.7651 308.2668 340.9681 279.5634 311.4193 203.3454 341.0023 217.0015 349.2191 362.4329

175.0000 131.7875 175.0000 111.5847 159.2128 121.1851 175.0000 175.0000 175.0000 175.0000 170.9127 134.0707 157.9194 175.0000 175.0000 175.0000 175.0000 175.0000 175.0000 175.0000 147.3480 162.1343 135.6124 146.0670

158.7560 94.6983 125.2563 125.1141 123.7367 124.5639 209.8517 209.5912 290.4359 292.4999 300.0000 236.2201 256.3868 210.3141 209.8514 257.4106 209.6785 210.5424 270.0793 291.5880 139.1320 209.0722 121.2713 124.8140

139.7845 77.5629 52.8461 50.0000 50.0000 136.2786 140.1143 218.5738 229.3600 228.9728 139.0818 235.7127 274.4850 143.1724 163.8767 140.1455 180.9228 171.0178 139.8209 229.5164 111.7079 135.8212 139.8068 50.0000

750 780 700 650 670 800 950 1010 1090 1080 1100 1150 1110 1030 1010 1060 1050 1020 1070 1050 910 860 850 800

Table 5 Hourly plant discharge (104 m3) for EES using SOHPSO_TVAC. Hour

Qh1

Qh2

Qh3

Qh4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

11.2663 10.7003 12.3805 11.3434 12.3567 13.8632 8.1207 14.7299 5.7476 8.6252 10.0394 5.3434 13.7981 13.0380 12.9049 11.4612 8.0586 14.1292 13.9969 13.6629 6.1191 12.7082 5.5086 6.2075

6.5605 14.5365 14.9068 6.0472 6.4454 14.6360 8.0055 9.8161 6.9949 9.0607 13.8719 13.3238 8.9401 12.6073 12.3822 14.7779 6.0123 12.1223 10.0219 11.8429 7.4839 9.8549 6.9737 6.6789

29.5845 23.8220 24.0360 28.4669 11.4033 22.6721 22.0726 12.4494 22.5942 21.3242 19.4001 16.1447 11.5295 17.2598 11.1038 22.2004 11.1722 20.2289 17.6585 23.4223 24.6095 23.4032 19.7725 12.0001

6.0402 7.8672 8.9587 18.6173 7.9338 13.0857 18.8100 9.9862 8.0070 10.2429 16.5954 10.5239 6.5771 15.2077 11.3701 15.2242 19.1254 12.8834 15.9994 7.0533 17.2054 7.7844 18.5856 19.8412

properly in order to get optimum solution efficiently and accurately. It is reported (Kennedy & Eberhart, 1995) that a relatively higher value of the cognitive component, compared with the social component, results in excessive roaming of individuals through a larger search space. On the other hand, a relatively high value of the social component may lead particles to rush toward a local optimum prematurely. In general, for any population-based optimization method like PSO, it is always desired to encourage the individuals to wander through the entire search space, during the initial part of the search, without clustering around local optima. In contrast, during the latter stages, it is desirable to enhance convergence towards the global optima so that optimum solution can be achieved efficiently. For this, the concept of parameter automation strategy called time varying acceleration coefficients (TVAC) had been

introduced (Ratnaweera et al., 2004). The main purpose of this concept is to enhance the global search capability during the early part of the optimization process and to promote the particles to converge toward the global optimum at the end of the search. In TVAC, this can be achieved by changing the acceleration coefficients with time. With a large cognitive component and small social component at the beginning, the particles are encouraged to move around the search space, instead of moving towards the population best prematurely. On the other hand, during the latter stage of optimization, a small cognitive component and a large social component encourage the particles to converge towards the global optimum. The concept of time varying acceleration coefficients (TVAC) can be introduced mathematically as follows (Ratnaweera et al., 2004):

iter þ C 1i itermax iter C 2 ¼ ðC 2f  C 2i Þ þ C 2i itermax

C 1 ¼ ðC 1f  C 1i Þ

ð17Þ ð18Þ

where C1i, C1f, C2i, C2f are constants representing initial and final values of cognitive and social acceleration factors, respectively. 3.3. Self-Organizing hierarchical PSO with TVAC (SOHPSO_TVAC) The classical PSO is either based on a constant inertia weight factor or a linearly varying inertia weight factor. It is reported that the particles in classical PSO may converge to a local minimum prematurely due to lack of diversity in the population, particularly for complex problems (Ratnaweera et al., 2004). In SOHPSO_TVAC, the previous velocity term in (14) is kept at zero. It is observed that in the absence of previous velocity term the particles rapidly rush towards a local optimum solution and then stagnate due to the lack of momentum. To overcome this difficulty, the modulus of velocity vector of a particle is reinitialized with a random velocity (called reinitialization velocity) whenever it stagnates in the search space. Stagnation of particles highly influences the performance of PSO in searching global optimum. When a particle is stagnated, its pbest remains unchanged over a large number of iterations. When more particles are stagnated, the gbest also remains unchanged and the PSO algorithm converges to a local minimum prematurely. The necessary momentum is imparted to the particles by

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K.K. Mandal, N. Chakraborty / Expert Systems with Applications 39 (2012) 3438–3445 Table 6 Hydrothermal generation (MW) schedule for CEES using SOHPSO_TVAC. Hour

Ph1

Ph2

Ph3

Ph4

Ps1

Ps2

Ps3

Total (MW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

89.8021 83.2811 93.0474 94.7478 89.3973 86.3217 89.0701 79.5523 64.9276 78.3442 71.9447 66.9496 73.1153 67.8024 68.5881 65.6505 50.1060 88.0318 58.1968 82.8408 56.4355 83.9855 73.0347 61.4282

63.1679 78.0924 70.8291 48.3937 85.0062 85.8596 73.2954 44.1666 73.4768 50.8911 57.0108 46.7805 68.817 57.6724 57.2552 48.6599 60.0800 75.1307 40.0496 55.5995 51.5151 40.7437 31.2955 32.2283

12.8825 19.5357 0.0000 12.827 0.0000 38.3524 50.8735 0.0000 44.0218 50.0019 0.0000 51.5695 32.5694 52.0682 50.6051 14.1712 7.0816 47.0423 54.1086 52.6734 57.6442 54.2780 55.8303 58.0427

145.1053 148.0795 231.4541 222.5870 155.3191 227.5914 280.6561 270.0767 287.5499 278.4152 287.7578 308.5275 292.2036 237.9210 334.8471 284.8038 323.0420 289.6959 328.6209 300.6100 368.2088 313.0170 231.0194 312.3550

175.0000 175.0000 105.6272 102.4995 165.2470 102.8553 175.0000 175.0000 175.0000 175.0000 175.0000 175.0000 163.5730 175.0000 175.0000 175.0000 175.0000 170.1878 175.0000 158.7223 113.0572 107.7135 175.0000 160.9854

123.7537 136.4541 125.2703 118.9450 125.0304 120.8123 141.2046 211.3451 215.1901 218.4220 279.0252 272.2124 297.7858 210.1503 184.0223 242.1698 209.8231 137.2111 210.1329 263.1638 127.5146 122.3328 144.0003 124.9604

140.2885 139.5572 73.7719 50.0000 50.0000 138.2073 139.9003 229.8592 229.8337 228.9257 229.2615 228.9605 181.9357 229.3857 139.6821 229.5449 224.8673 312.7004 203.8912 136.3902 135.6246 137.9295 139.8198 50.0000

750 780 700 650 670 800 950 1010 1090 1080 1100 1150 1110 1030 1010 1060 1050 1120 1070 1050 910 860 850 800

to overcome the difficulties of selecting appropriate reinitialization velocities.

Table 7 Hourly plant discharge (104 m3) for CEES using SOHPSO_TVAC. Hour

Qh1

Qh2

Qh3

Qh4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

10.9006 9.5001 12.0433 13.8305 12.7436 13.2715 11.5169 12.9840 8.8418 14.3222 11.2012 9.5095 5.3194 8.8598 8.6688 7.8961 5.4554 13.6021 6.6902 7.3919 6.5021 13.4804 10.1980 7.8060

8.2000 11.3060 10.1208 6.3186 13.9140 12.1959 8.0125 6.7466 13.2500 8.5410 9.7089 7.8275 13.1721 11.0187 11.3976 9.7505 13.8901 6.7392 9.7586 7.3718 14.6755 12.2389 10.3536 11.3669

25.5326 23.6388 26.5206 23.3565 25.8577 25.7910 28.9998 26.8772 16.6969 10.1858 28.6655 14.8475 23.7838 15.6919 16.4750 24.8236 25.6643 18.0158 10.8139 17.7274 14.6045 17.2293 16.3734 16.6634

7.0012 7.5438 16.9449 18.9187 9.3668 9.8864 7.4017 16.4169 17.4982 15.4554 15.2700 16.5833 14.8444 10.3124 18.5103 13.5283 16.6590 7.5373 16.7475 8.2033 19.6273 14.2370 8.2655 14.0094

4. Development of the proposed algorithm Now, an algorithm based on SOHPSO_TVAC is discussed to obtain quality solutions for daily economic emission scheduling problems for hydrothermal systems with cascaded reservoirs. The representation of individuals and their elements is very important for any population based evolutionary algorithm like PSO. For the present problem, the position of each particle (i.e. each individual in the population of particles) is represented by the discharge rate of each hydro plant and the power generated by each thermal unit. The algorithm starts with the initialization process. Let ð0Þ

kþ1 V id ¼



 iter ðC 1f  C 1i Þ þ C 1i  rand1  ðpbestid  X kid Þ iter max     iter þ ðC 2f  C 2i Þ þ C 2i  rand2  gbest d  X kid iter max

ð19Þ

If Vid = 0 and rand3 < 0.5 then

V id ¼ rand4  V d max else V id ¼ rand5  V d max

ð20Þ

Thus a series of particle swarm optimizers are generated automatically inside the main PSO according to the behavior of the particles in the search space until the convergence criteria is satisfied. The variables rand3, rand4 and rand5 are numbers generated randomly between 0 and 1. A time varying reinitialization strategy is used

ð0Þ

ð0Þ

ber of particles. For a system with Nh number of hydro units and Ns number of thermal units, position of kth individual of a population is initialized randomly satisfying the constraints defined by (14) and (10) and can be represented by

h iT ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ X k ¼ Q h1 ; Q h2 ; . . . Q hj ; . . . ; Q hNh ; Ps1 ; Ps2 ; . . . Psi ; . . . ; PsNs with

reinitialization of velocity vector with a random velocity. The above method can be implemented as follows (Ratnaweera et al., 2004):

ð0Þ

Pð0Þ ¼ ½X 1 ; X 2 ; . . . X k ; . . . ; X NP  be the initial population of Np num-

Q hj ¼ ½Q hj1 ; Q hj2 ; . . . Q hjt ; . . . Q hjT T ð0Þ

ð0Þ

ð0Þ

ð0Þ

ð0Þ

and

ð0Þ

ð0Þ

ð21Þ ð0Þ

P si ¼ ½Psi1 ; P si2 ; . . . ;

Psit ; . . . ; P siT T . The elements Q hjt and Psit represents the discharge ð0Þ

ð0Þ

ð0Þ

ð0Þ

rate of the jth hydro plant and the power output of the ith thermal ð0Þ ð0Þ unit at time t. The range of the elements Q hjt and Psit must satisfy the water discharge rate and the thermal generating capacity constraints as depicted in Eqs. (13) and (9) respectively. Assuming the spillage in Eq. (11) to be zero for simplicity, the water discharge rate of the jth hydro plant in the dependent interval is then calculated using (11) to meet exactly the restrictions on the initial and final reservoir storage. The dependent water discharge rate must satisfy the constraints in Eq. (13). At the same time, to meet exactly the power balance constraints, the thermal generation of the dependent thermal generating unit is calculated using (7). Thus, the initial generation is checked against all the constraints. If the constraints are satisfied then movement towards the next step is undertaken. The above scheme always generates individuals satisfying the constraints. Now, the algorithm can be described as follows:

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K.K. Mandal, N. Chakraborty / Expert Systems with Applications 39 (2012) 3438–3445

Fig. 2. Convergence characteristics for minimum fuel cost.

Table 8 Comparison of cost and CPU time for ELS.

Fuel cost ($) Emission (lb)

ELS

EES

CEES

41983.00 24482.00

44432.00 16803.00

43045.00 17003.00

Step 1: Initialize randomly each particle according to the limit of each unit including individual dimensions, searching points and velocities according to (21). These initial particles must be feasible candidates for solutions that satisfy the practical operating constraints. Step 2: For each particle, calculate fitness value according to (5). Step 3: If the fitness value of individual is better than the best fitness value in history, set current value as the pbest. Step 4: Modify the member velocity of each particle according to (19) and reinitialize according to (20). Step 5: Choose the particle with the best fitness value of all the particles as the gbest. Step 6. If the number of iterations reaches the maximum, then go to Step 7 else go to Step 4. Step 7. The individual that generates the latest gbest is the solution of the problem. 5. Simulation results To demonstrate the effectiveness and performance of the proposed algorithm, it was applied to a test system that consisting of a multi-chain cascade of four hydro units and three thermal units (Sun & Lu, 2010). The scheduling period has been kept to 24 h with 1 h time interval. The hydro sub-system configuration and network matrix including water time delays are shown in Fig. 1. In this case, four hydro plants are cascaded and the power generation of plants at lower stream is affected by the delay as well as the discharge of the plants at higher stream as shown. Hydro unit power generation coefficients, reservoir inflows, reservoir

limits, generation limits, emission coefficients and cost coefficients of thermal units are the same as that of (Sun & Lu, 2010) and hence not reproduced here. However, the hourly load demand is shown in Table 1. The performance of PSO algorithm is quite sensitive to the various parameter settings. Tuning of parameters is essential in all PSO based methods. Based on empirical studies on a number of mathematical benchmark functions (Ratnaweera et al., 2004), it has been reported the best range of variation as 2.5–0.5 for C1 and 0.5–2.5 for C2. The idea is to use a high initial value of the cognitive coefficient to make use of full range of the search space and to avoid premature convergence with a low social coefficient. We experimented with the same range. Several tests were performed with various combination of initial values of C1 and C2 and the best results were obtained for the range of 2.5–1.3 for C1 and 0.7–2.5 for C2 out of 50 trial runs. The optimization is done with a randomly initialized population of 50 swarms. The iteration number was increased in step of 50 and beyond 1000 no improvement in results was obtained. Hence, the maximum iteration was set at 1000. The problem was solved by an in-house MATLAB program on 2 GB RAM, 3.00 MHz PC. The problem is initially solved as pure economic load scheduling (ELS) with w1 = 1 and w2 = 0. Then it was tested for a pure economic emission scheduling (EES) with w1 = 0 and w2 = 1. Finally it is solved again as a case of combined economic emission scheduling (CEES) incorporating w1 = 1 and w2 = 1. Tables 2 and 3 show the optimal hydrothermal generation schedule and optimal hourly water discharge rate obtained by the proposed algorithm respectively for ELS. Optimal fuel cost is found to be $41983.00 for this case while computation time is found to be 112.00 s and, while amount of emission is found to be 24482.00 lb. Optimal hydro-generation schedule and optimal hourly water discharge rate are shown in Tables 4 and 5 respectively for EES. The optimal amount of emission and computation time for EES is found to be 16803.00 lb and 112.56 s respectively, while fuel cost is found to be $44482.00. Optimal hydro-generation schedule and optimal hourly water discharge rate are shown in Tables 6 and 7 respectively for combined economic emission scheduling (CEES). The computation time is found to be 120.00 s, while suboptimal fuel cost and amount of emission is found to be $43045.00 and 17003.00 lb respectively for this case. Convergence characteristic of minimum fuel cost for ELS is shown in Fig. 2. Table 8 shows a comparison of fuel cost, amount of emission and computation time for ELS, EES and CEES cases. The conflicting nature of the two objectives (minimum fuel cost and minimum emission) is evident from the results. It is also seen from the Table 8 that pure ELS produces minimum fuel cost but the amount of emission is higher than pure EES and CEES. In the case of pure EES, the amount of emission is minimum, but at the cost of higher fuel cost. The CEES produces a better solution with a little increase in fuel cost and a large reduction of emission in comparison to pure ELS. This shows that huge reduction in emission release is possible with some compromise in fuel cost. The results of the proposed method are compared with the results obtained by other population based heuristic techniques. Table 9 compares the results with that of improved quantum -behaved particle swarm optimization (QPSO) (Sun & Lu, 2010).

Table 9 Comparison of results obtained by proposed SOHPSO_TVAC method with improved quantum-behaved particle swarm optimization (QPSO). ELS

Proposed SOHPSO_TVAC QPSO

EES

CEES

Fuel cost ($)

Emission (lb)

Fuel cost ($)

Emission (lb)

Fuel cost ($)

Emission (lb)

41983.00 42545.00

24482.00 31205.00

44432.00 46288.00

16803.00 17735.00

43045.00 44122.00

17003.00 18102.00

K.K. Mandal, N. Chakraborty / Expert Systems with Applications 39 (2012) 3438–3445

It is clearly seen that the proposed method yields comparable results in terms of fuel cost, amount of emission for all the cases i.e. ELS, EES and CEES. 6. Conclusion Environmental concern is one of the important issues in the operation of present day power systems. In this paper, an algorithm termed as SOHPSO_TVAC has been proposed and successfully applied to solve daily combined economic emission scheduling problem of cascaded hydrothermal systems. The proposed method has been applied on a sample test systems comprising of a multi-chain cascade of hydro units and three thermal units to evaluate the effectiveness and performance of algorithm and results are presented. The results obtained by the proposed algorithm have been compared with other population based technique like QPSO technique. It is found that the proposed method can produce comparable results in terms of fuel cost and amount of emission. Acknowledgments We would like to acknowledge and thank Jadavpur University, Kolkata, India for providing all the necessary help to carry out this work. Appendix A. List of symbols

Psit max P min si ;P si

asi, bsi, csi, esi, fsi PDt Phjt max P min hj ;P hj

Qhjt Vhjt max Q min hj ;Q hj max V min hj ;V hj

C1j, C2j, C3j, C4j, C5j, C6j Ihjt Ruj Shjt

smj Ns Nh

output power of ith thermal unit at time t lower and upper generation limits for ith thermal unit cost curve coefficients of ith thermal unit load demand at time t output power of jth hydro unit at time t lower and upper generation limits for jth hydro unit water discharge rate of jth reservoir at time t storage volume of jth reservoir at time t minimum and maximum water discharge rate of jth reservoir minimum and maximum storage volume of jth reservoir power generation coefficients of jth hydro unit inflow rate of jth reservoir at time t number of upstream units directly above jth hydro plant spillage of jth reservoir at time t water transport delay from reservoir m to j number of thermal generating units number of hydro generating units

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