Unit commitment by parallel simulated annealing

U. D. Anna kkag e T.Numnonda N.C. Pahalawaththa

Indexing terms: Parallel simulated annealing, Annealing, Unit commitment, Speculatiie computation, Serial subset

Abstract: The paper investigates the application of parallel simulated annealing for unit commitment problems. Two parallel simulated annealing concepts, specdative computation and serial subset, are applied to a unit commitment problem of ten thermal generators. The authors also propose a combined scheme where speculative computation is used in the initial phase and the serial subset is used in the final phase. The parallel simulated annealing schemes are tested with an example problem and the results show that the parallel schemes can considerably speed up the computation of simulated annealing.

List of symbols

i = index for generating unit j = index for time interval nu = number of generating units nt = number of time intervals P = power output of a generating unit D = total power demand R = spinning reserve UC(., .) = unit commitment state matrix C = cost of operation F = fuel cost of a generating unit E = total cost of operation (analogous system) Cs, = start-up cost of a generating unit T = temperature in simulated annealing

to energy of a

algorithm A4 = number of iterations at each temperature n = index for temperature loop N = total number of simulated temperature steps

1 Introduction

Unit commitment in power systems refers to the opti-

0 IEE, 1995 IEE Proceedings online no. 19952215 Paper received 3rd April 1995 U.D. Annakkage and N.C. Pahalawaththa are with the Department of Electrical and Electronic Engineering, tlnivenity of Auckland, Auckland, New Zealand T. Numnonda is with the Departmmt of Electrical and Electronic Engineering, Khon Kaen University, Khon Kaen, Thailand

IEE Proc.-Gener. Transm. Distrib., Vol. 14.2, No. 6, November 1995

misation problem of determining the switching pattern of generating units that minimises the operating cost for a given time horizon. A practical unit commitment problem is a combinatorial optimisation problem that involves a large number of constraints such as, demand, spinning reserve, minimumup and minimum- down times of units.

The unit commitment problem, characterised by a large number of constraints, is a challenging problem that has attracted the attention of many researchers. Optimisation techniques presently avaiiable range from highly heuristic techniques to more accurate dynamic programming techniques [I]. Due to the complexity of handling the constraints, many simplifications are used in presently available algorithms at the expense of the closeness to the optimal solution.

Simulated annealing (SA) is a comparatively new optimisation technique that has been proved to be very effective in solving large combinatorial optimisation problems. The simplicity of incorporating constraints into this algorithm makes the SA algorithm very attractive for the unit commitment problem.

In SA, the optimisation problem is simulated as an annealing process. The natural process of optimisation that takes place in a slowly cooling metal (annealing) guarantees that the structure of the metal reaches the crystal structure corresponding to the minimum energy [2,3]. In this natural process, a transition from a struc- ture corresponding to an energy level of E to that cor- responding to E+AE takes place, with a probability given by the Boltzman function e-AHkT. As the temper- ature decreases the probability of such transitions determined by the Boltzman function becomes lower. The above process allows a slowly cooling metal to escape from crystal structures corresponding to local minimum energy states in its search for the globally minimum energy state.

In SA, the objective function to be minimised is anal- ogous to the energy in the crystal structure and the temperature is analogous to a control parameter in the algorithm.

The application of SA for unit commitment was reported by Zhuang and Galiana in 1990 [4]. They have compared the SA algorithm with Lagrangian relaxation and iterative improvement algorithms and have shown that SA is easily adaptable to problems with complicated constraints. However, the conver- gence time of the SA algorithm has been highlighted as a limiting factor.

It is well known that the major disadvantage of SA is the large computational demand. This has motivated the development of algorithms that can reduce the computational burden [5-111. The most widely used

595

technique to speed up SA is to implement an algorithm on a parallel computer architecture.

There are two parallel SA algorithms available at present: speculative computation (SC) [7] and serial subset (SS) [12]. In this paper, a technique derived by combining the advantages of these two techniques is presented and the efficiencies of the three techniques are compared for a unit commitment problem of ten thermal units.

2 Unit commitment by SA

The object of unit commitment is to decide which of the available generators should start-up and shut-down over a given time horizon so that the overall operating cost is minimised subject to demand and spinning reserve constraints.

To solve this problem using SA, the switching-on and switching-off of the generating units over the given time horizon can be treated as a pattern. Suppose there are n, units and nt time intervals. The unit commitment state matrix UC (nu x nt)

is defined as uc,, = 1 when the unit is ON uc,, = 0 when the unit is OFF

where the suffixes i and j refer to unit number and the time interval, respectively.

Then the unit commitment problem is to determine the optimum pattern for UC. The SA algorithm to determine the optimum UC pattern is described below.

The UC matrix is assigned an arbitrary state and an initial temperature. Then a new state is generated from the current state by giving a random perturbation, and the change in cost function AE is calculated. If the change in cost is negative then the system transits to the new state. Otherwise the transition takes place with a probability given by the Boltzman function. This process is repeated for a large number of times (say M times). The temperature is then decreased according to a certain cooling equation and the above steps are car- ried out again for M times at the new temperature. The whole procedure is repeated until a given stopping cri- terion (usually a lower limit for the temperature) is sat- isfied. The selection of the initial temperature, the cooling equation and the number of perturbations required at a given temperature ( M ) are discussed in Section 5.

Step 1: Initialise UC, select initial temperature T and the number of iterations M and find the total operating cost E for the given UC. Step 2: Repeat steps 2.1 and 2.2 until the temperature is sufficiently low (a suitable value is determined exper- imentally) Step 2.1: Repeat step 2.1.1 to step 2.1.5 A4 times Step 2.1.1: Randomly disturb UC (described below) Step 2.1.2: Find the new value of total operating cost En,, (described below) Step 2.1.3: Compute AE: AE = E,,, - E Step 2.1.4: Accept or reject the new UC according to the Boltzman function (described below) Step 2.1.5: If new UC is accepted then update UC and E

The algorithm is as follows:

596

Step 2.2: Decrease the temperature Steps 2.1.1, 2.1.2 and 2.1.4 are described below. Step 2.1.1: Randomly disturb UC A random disturbance is given to UC such that mini- mum-up and minimum-down time constraints are not violated. This is done by choosing a generating unit randomly, choosing a time to switch off the unit ran- domly and choosing a duration of shut down ran- domly, but not to violate the minimum-up and minimum-down time constraints. The procedure is as follows: (a) select a generating unit i (b) set uq to unity for a l l j (c) randomly select a time interval jshul (d) randomly select a shutdown period tShut so that the minimum-up and minimum-down time constraints are not violated (e) if not equal to zero set UCllshut to UCgshut+tshut-l equal to zero. Step 2.1.2: Find the new value of total operating cost For a given UC, the Economic Load Dispatch (ELD) subproblem can be defined as

n 7L minimise : C, = F,UC~, for j = 1 to nt (I)

i=l subject to (i) power balance constraints

ne,

ucij pij = ~j for j = 1 to nt 2= 1

(ii) spinning reserve constraints nu

ucY P F ~ ~ 2 R, + D, for 1 = 1 to nt ( 3 )

(4)

%== 1 (iii) capacity limits of generating units

(iv) feasibility of generating the required demand Pzmzn i: Pz, 5 Ptmax for z = 1 to nu

n,,

( 5 ) a= 1

(v) minimum-up time and minimum-down time con- straints. Minimum-up time is the minimum number of hours a unit must be run before it can be shut off. Whereas, minimum-down time is the minimum number of hours a unit must be offline before it can be brought online again. (vi) crew constraint The number of units that can be brought online at any given time is limited due to constraint of crew.

is assigned a penalty p7 otherwise the minimum Cj is cal- culated subject to constraints (i) and (iii) by solving the economic load dispatch problem [13]. (The penalty p is chosen to be a value which is 10% higher than the cost of supplying the peak demand with the most expensive combination of generating units.)

If the constraint (ii), (iv), (v) or (vi) is violated,

The total cost E,,, is then computed by n+ n.

IEE Pvoc.-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995

The start-up cost, C,, is expressed as a function of the number of hours (d,) the unit has been down and it is given by [ 141

C S T ~ = &,,(I - e x ~ ( - ( d , - dmzn)d3,,)) + d2,% (7) where d,,, is the minimum-down time, and d l , z , d2,t and d3,, are real constants. Step 2.1.4: Accept or reject the new UC according to the Boltzman function In a simulated annealing algorithm the criterion of accepting or rejecting is based on the Boltzman func- tion and is well known as the Metrapolli’s criterion [2]. It can be described as follows: IF A23 is negative THEN accept the new configuration ELSE compute the probability of acceptance = e-AE/T, and generate a random number between 0 and 1 from a uniform distribution. If the probability of acceptance is greater than the ran- dom number then accept the new configuration, other- wise reject it.

3 P(aralle1 simulated annealing concepts

In the conventional SA algorithm, a large number of perturbations (say A4) are evaluated sequentially at each temperature value. Each evaluation involves the three basic steps (a) generate a random perturbation (b) evaluate the new cost and (c) accept or reject the perturbation. In a parallel SA algorithm, a block of perturbations (say m perturbations) are evaluated simultaneously and the required number of perturba- tions (M) is thus reached much faster than a conven- tional lSA algorithm. The speed up factor is given by

round-up(M/m) M speed up factor =

where the function round-up(x) is defined as the small- est integer greater than x.

Parallel SA algorithms exploit the fact that there are only two outcomes possible from each perturbation: accept it or reject it. Two concepts of parallel algo- rithms, speculative computation (SC) and Serial Subset (SS), axe discussed below.

3. I Speculative computation (SC) The concept of speculative computation is to perform work before it is known whether or not it is needed. One pirocessor, known as the root processor can be assigned to work on a given iteration and two other processors, known as children processors, can be assigned to perform Speculative Computation as illus- trated iln Fig. 1. One of the children processors specu- lates that the perturbation evaluated by the root processor will be rejected and it proceeds to the next iteration. Meanwhile the other child processor specu- lates acceptance and the next iteration is performed by using the new configuration results from the accepted perturbation. Therefore some work in the next iteration has already been performed before the decision is made and thius speeds up the computation. This approach can be extended to higher levels with additional chil- dren processors, for instance Fig. 2 illustrates specula- tive computation for three iterations.

The acceptance ratio (~(7)) at temperature T can be defined as

,?m = (JIM where a is the number of perturbations accepted out of the total of M perturbations evaluated at T.

In SA, the acceptance ratio is close to 1 at high val- ues of T and it has a value close to 0 at low values of T. Therefore the utilisation of the processors can be improved by allocating more processors as accepted children at high values of T, and allocating more proc- essors as rejected children at low values of T.

root processor

rejected child processor accepted child processor

Fig. 1 Speculative computation approach: two iterations

root processor

accepted child processor

rejected child processor

Fig. 2 Speculative computation approach: three iterations

3.2 Serial subset approach (SS) The concept used in the SS approach can be explained as follows. In conventional SA the solution evolves as a sequence of transitions which take place only when a perturbation has been accepted. To illustrate this, con- sider the state transition diagram shown in Fig. 3 .

Fig. 3 State tvansition diagram

The initial configuration is indicated by 0. A new configuration 1 is generated by perturbing the configu-

597 IEE Proc-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995

ration 0. This new configuration is rejected by the algo- rithm and the state returns to configuration 0. Again a new configuration 2 is generated from 0 and that too is rejected by the algorithm. Then another new configura- tion 3 is gcnerated from 0 and the algorithm accepts it. So, a transition takes place from configuration 0 to configuration 3. The next configuration is generated (not shown in the diagram) by perturbing the configu- ration 3.

It is clear that the configurations 1, 2 and 3 were gen- erated from the configuration 0. Therefore the above sequence of events can be performed simultaneously. In this example, three evaluations, out of the required number of A4 evaluations, can be carried out in paral- lel. In general, a sequence of rejected perturbations and an accepted perturbation can be evaluated independ- ently and thus in parallel.

3.3 Combined SC and SS approach A combined SC and SS approach is also investigated in this paper. In this, the algorithm begins by using SC at high temperatures. An unbalanced tree approach is chosen in this regime in which more processors are allocatcd as acccpted childrcn (Figs. 4-8). Thc shapc of

Fig.4 Tree structures fo r ~ ( c ) Is 0.8

A

Fig.6 Tree structures for 0.6 2 ~ ( c ) > 0.4

598

the tree structure is altered as the temperature decreases by allocating more processors as rejected children.

At the low temperatures, the speculative computa- tional approach is replaccd with the scrial subset approach.

Fig.7 Tree structures for 0.4 t x(c) 0.2

Fig.8 Tree structures for 0.2 2 ~ ( c )

4 Implementation of parallel SA

4. ’I Speculative computation To implement this method, the algorithm begins with an arbitrary UC matrix. A perturbation is applied to this UC matrix and the root processor evaluates whether to accept it or not. If an ‘accept’ child proces- sor is present in the tree, it will assume that the out- come of the root processor is to accept the perturbation and so the child processor is assigned a UC matrix generated by giving a random perturbation to the UC matrix accepted by the root processor. If a ‘reject’ child processor is present in the tree, it will reject the UC matrix evaluated by the root processor and thus it returns to the previous UC matrix. Then a new perturbation is given to it and the resulting UC matrix is evaluated. This procedure is applied to all olher processors in the tree by treating each child proc- essor as a root processor to the processors in the next level.

At the end of the evaluations, the tree is traced according to the outcome of the evaluation, until a ter- mination of the tree is reached. If the number of nodes traced is m then this parallel evaluation is equivalent to m number of evaluations in a conventional SA (CSA) algorithm. The final state after the parallel evaluation is determined by the state of the last node traced and

IEE Proc -Gener Transm Dlstrib , Val 142, No 6, November 1995

its outcome. The parallel evaluation is repeated until Zm is equal to or greater than the required number M .

4.2 Serial subset According to the Serial Subset concept explained ear- lier, a set of rejected perturbations together with an accepted perturbation is equivalent to a block of per- turba!tions evaluated in CSA.

Thlarefore, to implement this method, each processor is assigned a UC matrix randomly generated by per- turbing the current UC matrix. Then all the processors evaluate the assigned UC matrices simultaneously. At the end of the parallel evaluation, one of the accepted UC matrices is chosen randomly. Let K be the number of processors used and r be the number of processors that decided to reject the assigned UG matrix. Accord- ing to the SS approach, these K evaluations are equiva- lent to a sequence of m evaluations in a CSA algoriithm. m consists of one of the accepted perturba- tions plus some rejected perturbations. Since only one out of K - r accepted perturbations is selected the cor- rect ,fraction of the rejected perturbations must be taken as valid. Therefore the value of nz is calculated as m = I + r/(K - r) if at least one UC matrix is accepted, otherwise m = K.

5 Simulation results

Three parallel SA algorithms, the Serial Subset, the Speculative Computation and the combined approaches, were implemented for a ten generating unit test system [15]. These algorithms were actually imple- mented on a conventional computer, but it simulated a parallel computer with fifteen processors. In terms of the Speculative Computation approach, different shapes of tree structures are employed in different value regimes of the temperature. Those structures are illustrated in Figs. 4-8. The underlying idea for choos- ing ea.ch unbalanced tree shape is to ensure that the probability the algorithm will terminate after log2(K) iteratitons (the number of iterations which can be per- formed by the balanced tree at each parallel evaluation) is greater than 50% at those acceptance ratios. For the combined algorithm, the low value regime of the tem- perature is defined as those values which have an acceptance ratio less than 0.35.

4 8 12 16 20 24 periods

Fig.9 Loud curve

In this study, the initial temperature for annealing was selected to ensure that at least 95% of the trials are accepted at this temperature and the temperature T is decreased according to the exponential rule as Tn+, = T,, x 0.9. The lowest temperature is defined as the tem- perature corresponding to n = nmax.

IEE Proc -Genu Trunsm Dirtrib, Vol 142, No 6 November 199.5

Table 1: Characteristics of the generating units

Operating cost Start up cost Capacity , coefficient coefficient

a

1

2

3

4

5

6

7

8

9

10

15

25

40

32

29

72

105

100

49

82

b C

1.4 0.0051

1.5 0.00396

1.35 0.00393

1.4 0.00382

1.54 0.00212

1.35 0.00261

1.3954 0.00127

1.3285 0.00135

1.2643 0.00289

1.2136 0.00148

4 85

101

114

94

113

176

267

282

187

227

4 ___I

20.588

20.594

22.57

10.65

18.639

27.568

37.749

45.749

38.617

26.641

d3 - 0.2

0.2

0.2

0.18

0.18

0.15

0,09

0.09

0.130

0.11

pmax pmin

60 15

80 20

100 30

120 25

150 50

280 75

520 250

150 50

320 120

200 75

The example problem considered in this study has the load curve as shown in Fig. 9. The spinning reserve requirement in each hour is assumed to be 10% of the power demand during that hour. The problem consist of ten generating units, and their parameters are given in Table 1 [14]. It is assumed that only one generating unit can be brought online at any given time due to crew constraint. In addition, the minimum-up and min- imum-down times are assumed to be 5 and 2h, respec- tively.

The CSA algorithm and the following three parallel SA approaches were applied to this example problem: (i) using only the Serial Subset approach, (ii) using only the Speculative Computation approach, (iii) using the combination of both approaches. The simulated annealing is a stochastic optimisation technique. There- fore, the problem was executed 20 times to test the con- sistency of the algorithms. The final result (cost function) of each run was recorded and a comparison between the algorithms is shown in Table 2, The CSA algorithm was used with M = 150 and nmax = 120. The optimum unit commitment obtained with the CSA algorithm is illustrated in Fig. 10. The minimum oper- ating cost for this solution is 59 520 units. The same result was obtained from the parallel SA algorithms, but with less computational time.

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

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0

= 5 0 0 0 5 6 0 0 0

7 0 0 0 8 0 0 0 9 0 0 0

1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000000 00000000

Fig. 10 Best unit commitment 0 on 0 off

Table 2: Final results of the example problem

00 00 00 00 0. 00 00

Results Nuber of

best worst average hit the best runs ._I_ Algorithm

CSA 59520 59588.2 59538.4 10

Serial PSA 59520 59548.5 59532.8 11

Speculative PSA 59520 59548.5 59527.3 15

Combination PSA 59520 59548.5 59540 6

s99

It is not possible to compare the computational time by using actual executing time since the parallel SA algorithms can be implemented using a parallel compu- ter. The most time consuming part in the algorithm is the evaluation of the cost function. Therefore, the number of times the cost function is evaluated with each algorithm is considered as a good measure of the speed of the algorithm. These figures are illustrated in Table 3. In addition, the Table shows how parallel implementations could speed up the computation time. The average speed up of the three PSAs were obtained from the experimental study.

Table 3: Number of times that each test algorithm evalu- ates the cost function

Parallel implementation cost evaluatons Number of average Algorithm

processors speed up

- - CSA 16450

Serial PSA 16450 15 5.66

Speculative PSA 16450 15 5.93

Combined PSA 16450 15 6.03

The three parallel implementation schemes were eval- uated for 15 processors and the speed up factors obtained range from 5.66 to 6.03. These speed up fac- tors are well above 3.0, which corresponds to the speed up that can be achieved with 15 processors in a bal- anced tree of the Speculative Computation technique. This shows that there is a clear advantage of imple- menting one of the parallel schemes proposed in this paper. Furthermore, the combined scheme performs marginally better than the other two at no additional cost.

6 Conclusions

This paper has considered the application of simulated annealing for the unit commitment problem. Although SA has earlier been applied to this problem, it requires large computational effort and consequently its conver- gence is very slow. In order to reduce this computa- tional burden, this paper has investigated the use of parallel simulated annealing algorithms.

Two conventional parallel SA algorithms, speculative computation and serial subset, have been reviewed. Since these algorithms can perform efficiently in differ- ent value regimes of the temperature, it has been sug- gested that the combination of these two algorithms would be effective. The parallel SA algorithms have been tested using computer simulations. It has been found that the combined algorithm could speed up the serial SA for the particular unit commitment problem by a factor of six (using 15 processors).

In conclusion, this paper demonstrates that (a) the parallel SA algorithm can significantly improve the speed of solving the unit commitment problem and (b) it is effective to combine the SC and SS approaches in implementing the parallel SA algorithm.

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