Interactive best-compromise approach for operation dispatch of cogeneration systems M.-T.Tsay, W.-M.Lin and J.-L.Lee Abstract: An interactive best-compromise approach is presented, based on evolutionary programming to solve the economical operation of cogeneration systems under emission constraints. A biobjective function including both the minimisation of cost and emission is formulated. The cost model includes fuel cost and tie-line energy cost. Emissions with SO, and NO, are derived as a function of fuel enthalpy. All constraints including fuel mix ratios in a boiler, operational constraints, and emission constraints must be met in the search process. The steam and fuel mix is found by considering the time-of-use dispatch between cogeneration systems and utility companies. Data from a real cogeneration system are given to illustrate the effectiveness of the proposed method. 1 Introduction Cogeneration systems offer a reliable, efficient, and economic means to supply both thermal and electrical energy. They can be constructed in urban areas and used as distributed electrical energy sources. Applications of cogen- eration systems are still growing, and more experience will be needed regarding efficient operation for greater energy saving. For more effective operation, efficient strategies have been developed in [l-61. Those strategies mainly operate in such a way that the operating cost is minimised regardless of the emissions produced. The passage of the 1990 US Clean Air Act Amendments [7] has forced utilities to modify their operating strategies to meet environmental standards set by legislation. In recent years some operating strategies including emissions dispatch and fuel switching have been developed [8-121. Emissions dispatch adds a second objective to the operating problem, which can obtain both emission reduction and minimise power production cost. Fuel switching uses the fuel cofiring tech- nique to reduce emission. These techniques not only intend to reduce emission into the atmosphere but also aim to minimise the operating cost. Due to the conflicting and noncommeasurable nature of fuel cost and emission control, a single objective function seems inappropriate for this problem. Considering emis- sion, a trade-off between economy and the environment needs to be considered in the optimisation process. With increased requirements for environmental protection, alter- native strategies are required. This is a complicated prob- lem, which includes two objectives and it requires an efficient and reliable solution. Most previous studies [7-121 0 IEE, 2001 ZEE Proceedhgs online no. 20010 163 DO1 10.1049/ipgtd20010163 Paper first received 16th February and in revised form 17th October 2000 M.-T. Tsay is with the Department of Electrical Engineering, Cheng Sku Insti- tute of Technology, Kaohsiung,Taiwan, Republic of China W.-M. Lin and J.-L. Lee are with the Department of Electrical Engineering, National Sun-Yat Sen University, Kaohsiung, Taiwan, Republic of China formulated this problem with only a single objective and emissions are treated as binding constraints. Since emis- sions are important to both the power utilities and custom- ers, it is beneficial to tackle the emissions as another objective function instead of just constraints. Thus, an interactive best-compromise (IBC) approach [ 131 with evolutionary programming (EP) [ 141 is presented to solve the problem. In the interactive procedure, it enables the decision maker (DM) to find a satisfactory strategy. In this paper fuel consumption and steam generation are first measured. A curve-fitting method [15] is then used to get the input-output (WO) curve for the heat input and steam generation output. A multi-fuelled unit model is for- mulated to get the U0 curve for a unit burning mixed fuels. Two important objectives are included, one represents the economic condition and the other is the pollution emission. To help the system operator to determine dispatch strategy, the IBC was used to conduct an exploration over the region of feasible alternatives for an optimal, or satisfactory near-optimal solution. Any improvement of one objective can be reached only by reduction of the other. The IBC approach will provide a flexible best compromise for opera- tion dispatch by following the intention of the DMs. 2 Problem formulation Fig. 1 shows the diagram of a cogeneration system with common high- and medium-pressure steam headers. The system has k back-pressure turbines and n extraction con- denser steam turbines for power generation. Fuels includ- ing FO, LNG, and coal were used in each boiler for producing high-pressure steam. Various models are used as follows: 2.1 YO cost curve of boilers It is assumed that the I/O curve of a boiler is a third-order polynomial Fbi(Mbi(t)) = A, + Ai x Mbi(t) +A2 x M:i(t) + A3 x M;i(t) (1) where Fb,(kfbit)) is the consumed enthalpy of the ith boiler at tth interval (MBTUh), Mbl{t) is the steam output of the ith boiler at tth interval (Th), Ao, A,, A2, A, are coefficients IEE Proc.-Gener. Transm. Distrib., Vol. 148, No. 4, July 2001 326
Transcript
Interactive best-compromise approach for operation dispatch of cogeneration systems
M.-T.Tsay, W.-M.Lin and J.-L.Lee
Abstract: An interactive best-compromise approach is presented, based on evolutionary programming to solve the economical operation of cogeneration systems under emission constraints. A biobjective function including both the minimisation of cost and emission is formulated. The cost model includes fuel cost and tie-line energy cost. Emissions with SO, and NO, are derived as a function of fuel enthalpy. All constraints including fuel mix ratios in a boiler, operational constraints, and emission constraints must be met in the search process. The steam and fuel mix is found by considering the time-of-use dispatch between cogeneration systems and utility companies. Data from a real cogeneration system are given to illustrate the effectiveness of the proposed method.
1 Introduction
Cogeneration systems offer a reliable, efficient, and economic means to supply both thermal and electrical energy. They can be constructed in urban areas and used as distributed electrical energy sources. Applications of cogen- eration systems are still growing, and more experience will be needed regarding efficient operation for greater energy saving.
For more effective operation, efficient strategies have been developed in [l-61. Those strategies mainly operate in such a way that the operating cost is minimised regardless of the emissions produced. The passage of the 1990 US Clean Air Act Amendments [7] has forced utilities to modify their operating strategies to meet environmental standards set by legislation. In recent years some operating strategies including emissions dispatch and fuel switching have been developed [8-121. Emissions dispatch adds a second objective to the operating problem, which can obtain both emission reduction and minimise power production cost. Fuel switching uses the fuel cofiring tech- nique to reduce emission. These techniques not only intend to reduce emission into the atmosphere but also aim to minimise the operating cost.
Due to the conflicting and noncommeasurable nature of fuel cost and emission control, a single objective function seems inappropriate for this problem. Considering emis- sion, a trade-off between economy and the environment needs to be considered in the optimisation process. With increased requirements for environmental protection, alter- native strategies are required. This is a complicated prob- lem, which includes two objectives and it requires an efficient and reliable solution. Most previous studies [7-121
0 IEE, 2001 ZEE Proceedhgs online no. 200 10 163 DO1 10.1049/ipgtd20010163 Paper first received 16th February and in revised form 17th October 2000 M.-T. Tsay is with the Department of Electrical Engineering, Cheng Sku Insti- tute of Technology, Kaohsiung, Taiwan, Republic of China W.-M. Lin and J.-L. Lee are with the Department of Electrical Engineering, National Sun-Yat Sen University, Kaohsiung, Taiwan, Republic of China
formulated this problem with only a single objective and emissions are treated as binding constraints. Since emis- sions are important to both the power utilities and custom- ers, it is beneficial to tackle the emissions as another objective function instead of just constraints. Thus, an interactive best-compromise (IBC) approach [ 131 with evolutionary programming (EP) [ 141 is presented to solve the problem. In the interactive procedure, it enables the decision maker (DM) to find a satisfactory strategy.
In this paper fuel consumption and steam generation are first measured. A curve-fitting method [15] is then used to get the input-output (WO) curve for the heat input and steam generation output. A multi-fuelled unit model is for- mulated to get the U 0 curve for a unit burning mixed fuels. Two important objectives are included, one represents the economic condition and the other is the pollution emission. To help the system operator to determine dispatch strategy, the IBC was used to conduct an exploration over the region of feasible alternatives for an optimal, or satisfactory near-optimal solution. Any improvement of one objective can be reached only by reduction of the other. The IBC approach will provide a flexible best compromise for opera- tion dispatch by following the intention of the DMs.
2 Problem formulation
Fig. 1 shows the diagram of a cogeneration system with common high- and medium-pressure steam headers. The system has k back-pressure turbines and n extraction con- denser steam turbines for power generation. Fuels includ- ing FO, LNG, and coal were used in each boiler for producing high-pressure steam. Various models are used as follows:
2.1 YO cost curve of boilers It is assumed that the I/O curve of a boiler is a third-order polynomial
Fbi(Mbi(t)) = A, + Ai x Mbi(t) +A2 x M:i(t) + A3 x M;i(t) (1)
where Fb , (k fb i t ) ) is the consumed enthalpy of the ith boiler at tth interval (MBTUh), Mbl{t) is the steam output of the ith boiler at tth interval (Th), Ao, A, , A2, A , are coefficients
Mh,l Mh. k 'h, k f l Mh, k+n I / Pg, 1 / pg, 1 / pg.1 / Pg,1 process
I I
1 1 I process steam D, i processsteam D, i i process steam D,
....... turbine @..; .......
demand PLD
PRVl turyne -@ .j
to plants , .......................................................................................... A Ptie
k+l
........ .- ........ . ~~
jelectricity j j utility i .....................
Mm,1 j M,, j M,,,, k+l
Fig. 1 Diugram of cogeneration system PRV pressure reducing valve
steam power
_ _ _ _ ............
&! &; condenser f M , , ~ + ~ condenser j
v
of the U 0 operation curve, and Mbtu is mega British ther- mal unit. With the mixed fuel used a proper modification should be adopted to represent the 110 operation curve of boilers. The dual-fuelled unit modelling had been formu- lated by [16]. Eqn. 2 is proposed for the U 0 curve simulta- neously burning three fuels
FbT(Mb(t)) = Fbl(Mb(t)) x (Al(t) +%/a A 2 ( t ) + %/3 A3(t)) (2)
where FbAMb(t)) is the total consumed enthalpy at tth interval (MBTUih), Fbl(Mb(t)) is the consumed enthalpy of fuel 1 at tth interval (MBTUih), qIl2 is the efficiency ratio of fuel l/fuel 2, ~ 7 , ~ ~ is the efficiency ratio of fuel 1 /fuel 3, and h,(t), &(t), &(t) are the mixed ratios of fuel 1, 2, and 3 at tth interval, With dl(t) + &(t) + &(t) = 1. The I/O cost curve of boilers can be described by
FBCT(t) = FbT(Mb(t)) x BCT(t) (3)
B C T ( ~ ) = BC1 x Al(t) + B C 2 x A 2 ( t ) +BC3 x A s ( t )
where FBCdt) is the total operation cost of boilers at tth interval (NT$), BCAt) is the total cost of fuel mixture at tth interval (NT$/MBTU), BC,, BC2, BC3 is the cost of fuels 1, 2, and 3 (NT$/MBTU), and NT$ is new Taiwanese dollar.
(4)
2.2 YO operation curve of steam turbine For a back-pressure turbine generator, the power equation for turbine i can be formulated by
P&) = KO, + Kl,w"t) + K 2 X & ( t )
2 = 1 , 2 , .... k (5)
For an extraction condenser turbine generator, the power equation for turbine i can be written by
Pg,(t) = KO% + ~ l % K r u ( t ) + K 2 % M W J % ( t )
i = 1 , 2 , . . . . n (6)
where M,,(t) is the medium-pressure extraction flow of
IEE Proc -Genu Transm Distrib , Vol 148, No 4, July 2001
turbine i at t-interval and Mwi{t) is the exhausted flow of turbine i, Pgi is the generated electric power from turbine i, and Koi, ICli, are the coefficients of turbine i which can be found by the curve-fitting technique.
2.3 Emission model Two primary power plant emissions are sulphur dioxide (SO,) and nitrogen oxide (NO,). Emission models may be defined as the amount of fuel consumed or as a function of boiler steam. The emissions are modelled as a function of fuel enthalpy dependent on the emission factor [I 11
ES(Mb) = YS x Fb(Mb)
EN(Mb) = 'YN x Fb(Mb)
(7)
(8) And the emission model with mixed fuels can be formu- lated by
si(.) = (YSliAldt) + YsniAai(t) + YS3i(t)XSi(t))
ENi(.) = (YNl iAl i ( t ) + Y N 2 i A 2 i ( t ) + YNSi(t)A3i(t)) xFbTi(Mbi(t), AIz(t), A 2 i ( t ) i A 3 i ( t ) ) (9)
xFbTi(Mbi(t), xli(t), A 2 i ( t ) , A S i ( t ) ) (10) where, E,(.) is the amount of pollutant SO, for the ith boiler at the tth interval (Tih), &I(.) is the amount of pollutant NO, for the ith boiler at the tth interval (T/h), FbTi(.) is total enthalpy for the ith boiler at the tth interval (Th), ysli, yNli are emission factors of SO, and NO, with oil for the ith boiler (T/Mbtu), ymi, ymi are emission factors of SO, and NO, with LNG for the ith boiler (T/Mbtu), ys3i, ymi are emission factors of SO, and NO, with coal for the ith boiler (T/Mbtu), and hli{t), hi{& d3[{t) are the mixed ratios of fuel 1,2, and 3 for the ith boiler at tth inter- val.
2.4 Objective function and constraints The optimisation needs to meet the demand of in-plant process steam and electric demand. The biobjective func- tion including cost model C(.) and emission model E(.) can
327
be formulated by
Min . [C(* ) , E(*)] (11) T
C(*) = t=l
T
E ( * ) = t=l
IC+ n
Esz(Mbt(t), A2,(t), A3t(t>)
1 k+n {c
+ ENt(Mbz(t), AZt(t)) 2= 1
C, and CN are the charged pollution emission fees for SO, and NO,. EC(t) is the TOU rate as shown in Table 1 [17]; Pl,,(t) is the electricity purchased from or sold to the utility and z is the time interval. The constraints considered are (i) Equality constraints including power balance, steam balance for boilers, turbine, and industrial process
k + n
i= 1
k + n k+n
Mbi(t) - Dh(t) - Mhi(t) = 0 (13) i=l i=l
k+n k+n n
k+n
i=1
also
Ali(t) + h ( t ) + ASi(t) = 1.0 i = 1,. . . , k + n (16)
PLD(t) is the load demand for utility at the tth interval (MW), Dh(t) is the high-pressure steam demand for indus- trial process at the tth interval (T/H), Dm(t) is the medium- pressure steam demand for industrial process at the tth interval (T/H) and Mht{t) is the high-pressure injection flows of turbine i at the tth interval (T/H)
Table 1: Time-of-use rates
Electricity sale price (NT$/kWh) Utility
Level-1 Level-2 NT$/kWh buyback price,
peak period 3.04 2.748 3.04 semipeak period 1.83 1.5767 1.83
off-peak period 0.69 0.4729 0.69
Level-I: power exported under 20% rated capacity Level-2: power exported over 20% rated capacity
(ii) Inequality constraints for boilers, steam turbine, power generation, and emission control
328
M $ ~ 5 ~ , ; ( t ) 5 ~ 2 , pzZn 5 ~ , i ( t ) 5 p z u x
vi = 1,. . . ,n (20)
i = 1,. . . , I C + n (21) Mbimin, Mbimax are lower and upper limits of flows for boiler i, Mhimin, Mhimax are lower and upper limits of high-pressure injection flows for turbine i, Mmimin, M m y x are lower and upper limits of medium-pressure extraction flows for turbine i, AIwi"", M,vimax are lower and upper limits of high-pressure exhausted flows for turbine i, and Pgimin, Pgimax are lower and upper limits of the generated electric power for turbine i.
3 EP solution methodology and implementation
The objective function shown in eqn. 11 is a biobjective function. The improvement of one objective can be only reached by retarding the other. The IBC approach is devel- oped to deal with the dilemma by using EP.
3.1 Initial ideal and non-ideal solution In eqn. I1 we first solve the single goal problem by using the EP procedure. The detailed EP procedure is shown in the Appendix Section 7. The optimisation can provide the best solution and then the worst solution of E(.) and C(.) The best solutions of C(.) and E(.) are defined as cost-ideal and SN0,-ideal, and the worst solutions of C(.) and E(.) are defined as cost-nonideul and SNOx-nonideul.
Best solution Min. cost = cost-ideal Min. S N O , = SNO,-ideal Worst solution Max. cost = costnonideal Max. SNO, = SN0,-nonideal
3.2 A minimum least square error approach The minimum least square approach is defined by
C(%i) - costideal cost-nonideal - cost-ideal Min T ( & ) =
E(!I&) - SNO,-ideal + (SN0,-nonideal - SN0,ideal
eqns. 12-21
Subject to cost-ideal I C ( Ea) <cost-nonideal S N O , - i d e a l ~ E ( E i ) ~ S N O , -nonideal
(22) {
!Rj an individual as defincd in the Appendix. If the minimi- sation of T(&) occurs in !Rimin, !Rimin will be an IBC solu- tion with
cost = C(9 i min)
(23) { S N O , = E(Xi min)
IEE Proc.-Gener. Transm. Distrib., Vol. 148, No. 4, July 2001
3.3 Satisfaction factor An IBC solution Rimin may not fit company policy. To choose a desirable solution, Rimin should be judged by the DMs. A satisfaction factor was defined for DMs in eqn. 24.
x 100% Icost-nonideal - cost1
Icost-nonideal - costideal I ISNO,-nonideal- SNO,I
ISNO,-nonideal- SNO,-ideall
SR-C =
x 100%
(24)
SR-E =
3.4 Alternation of decision region For cases where C(&) needs further reduction, E(Ri) will be chosen as the compromised term and the parameters cost-nonideal and SNOx-ideal will be adjusted by
(25) cost-nonideal = cost SNO,-ideal = SNO,
Conversely
(26) cost-ideal = cost SN0,-nonideal = S N O ,
3.5 Definition of goal index Following these steps the decision region will become smaller and smaller with the DMS an important decision factor. A goal index defined in eqn. 27 could provide the information of the maximal improvement which the next search can attain. The DMS will then decide if further
start
input data
find ideal solution by EP cost- min=cost- ideal 1 SNO, min=SNOX ideal
find nonideal solution by EP cost- max=cost- nonideal
The proposed algorithm was tested on the system of a Petroleum company shown in Fig. 1 which contains five back-pressure steam turbine, . one extraction condenser steam turbine and six steam boilers. The maximal power produced are 10 and 50MW for the back-pressure and extraction condenser steam turbine, respectively. The fuels used are fuel oil, LNG and coal. The fuel consumption and steam generation were measured in the field. By using the measurement data, curve-fitting was used to get the I/O operation curves as in eqn. 1. Table 2 shows the coefficients of the I/O operation curve for boilers. The associated coef- ficients for the steam turbine are listed in Table 3.
Table 2: Coefficients of I/O operation curve for boilers
unit AOi AI i A?; A3i
boiler 1 -20.27984 3.47598 -0.00664 2.152e-5
boiler 2 523.56446 -17.82798 0.25662 -9.771e-4
boiler 3 -716.0406 25.69203 -0.23667 7.667e-4
boiler 4 27.23617 1.69342 0.00956 -3.189e-5
boiler 5 -14.45154 2.78372 -0.00288 6.2e-6
boiler 6 48.32322 1.58331 0.01253 -0.00004
Table 3: Coefficients of steam turbines
unit KO; Kl i K2i
gen 1 1.67419 0.02042 0.000216
gen 2 -1.46151 0.08497 0.00008
gen 3 -1.393504 0.093989 0.000037
gen 4 -0.23472 0.054951 0.000299
gen5 -3.2978 0.138129 -0.00029
gen6 10.79 0.10639 0.28915
Table 4 Efficiency of boilers with various fuels (%I
boiler Fuel oil LNG Coal
boiler 1 88 90 86
boiler2 87 89 86
boiler3 89 91 87
boiler4 86 89 85
boiler5 87 89 85
boiler6 91 93 88
According to eqn. 2 the fuel mixture ratio is required to get the U 0 operation curve for a boiler that is simultane- ously burning three fuels. Table 4 shows the efficiency of boilers with various fuels. All facilities including generator, boiler, and steam turbine have their capacity limitation. The rated limits are expressed in Tables 5 and 6.
The proposed algorithm is used to solve the optimal operation problem of the cogeneration system under TOU rates. In the study system the plant’s load is 120MW and the steam demands for the high- and medium-pressure process are 60 and 610T/h, respectively. The population size and the number of generation are all set to 100.
Table 7 shows the results of single-objective programming during the various periods that minimises operation cost and emission, respectively. In Table 7, if the operation cost achieves its minimum, the emission required is large. In the TOU rate, if the emissions are to be minimal, all generators
must produce the lower output and the imported power is about 60MW. Conversely, if the operation cost is minimal, the imported power is about 28MW since the cost of imported power is larger than the electric production cost.
Table 8 shows the IBC during the peak period. Compar- ing result 1 and result 2 it is shown that S R E has degraded from 100 to 55.14% when SR-C has improved from 0 to 50.56%. If the DMs find the result not to be suit- able for the utility regulators, further compromise can be made according to the direction dictated by DMs. The parameters of cost-nonideal and SN0,ideal are then mod- ified for the next search. The parameters of cost-ideul and SN0,rzonideal will be a new goal in the next step. The boundary of dis-cost and dis-SNOx represent the decision region that the DMs can estimate the maximal improve- ment for a further step. If the DMs find that the maximal improvement in the desired term is too small to continue searching, they can stop the process. Result 3 will supply another compromise result, which promotes the SR-C from 55.56 to 76.18%. Similarly, if the DMs want to reduce emission, the operation cost will be selected as a compro- mise term. Results 4-6 in C(.)-interactive are other solu- tions for the DMs.
Table 9 shows the operational strategy of result 2. The ratio of fuel in each boiler is summed to 1 and each fuel also used in each boiler. Note that all the constraints are satisfied in operation dispatch.
Fig. 3 shows the relationship between operation cost and satisfied factor during peak period. It provides utility plan- ners a wider range of alternatives, showing the various
Table 7: Results of single-objective programming during various periods ~~
Single-objective programming
Peak period Semipeak period Off-peak period
Min C(.) Min E(- ) Min C(*) Min E(.) Min CW Min E(*)
Operating CPU time Operating CPU time Operating CPU time cost (NT$) (seconds) cost (NT$) (seconds) cost (NT$) (seconds)
EP method 203730 127 170760 115 138530 102
GAmethod 203730 215 170760 198 138530 188
SQP method 229860 18.38 191670 15.75 156240 10.27
feasible regions. Instead of using maximal allowable limits for emissions as constraints, an appropriate operating strat- egy can be chosen to meet the desired level of emission or cost. Two curves intersected on 56.45 and 57.65% for SR-C and SR-E respectively. The intersection of the two curves may be a suitable dispatch strategy for DMs.
?j? ’ O 0 ~ 80 \- SR-C
4 60 60
a 401 20 2 SR-E \ 140m 20
1 0 ’is 30 35 i o i s 50 55‘ 6b 0
cast NT$xlOOOO Fig. 3 period
Relationship between operation co,sts and satisfied factor during peak
EP was also compared with genetic algorithm (GA) [18] and sequential quadratic programming (SQP) [ 191. Table 10 is the summary of operating cost and CPU time during the various periods. GA utilities the coded informa- tion of artificial strings and SQP is a conventional pro- gramming. The population size and the generation number for EP and GA are set to 100. The crossover rate and mutation rate parameters for GA are 0.6 and 0.1. Tests were run on an IBM-compatible Pentium-I1233 computer. From Table 10 the EP method converges faster than CA and has better performance. Although SQP has better per- formance, the operating cost is larger than for the EP method. It can be proved that SQP only leads to a local minimal.
5 Conclusions and dissussion
A biobjective function including operation and emissions was formulated for the economical operation of a cogen- eration system. An IBC approach with EP was used to solve the biobjective problem while meeting the require- ments of generated steam capacity and electrical power. This approach was tested on the system of a petroleum company. Results provide a practical and flexible frame- work for evaluating emissions. The optimisation generates a trade-off process between cost and emission based on multifuel dispatch. The simulations also proved that the
IEE Proc-Gener. Transm. Distrib.. Vol. 148, No. 4, July 2001
TOU rate has significant influence on the overall economy of a cogeneration system. This approach can support DMs in knowing where to look for improved solutions and how to recognise a final solution using an interactive procedure.
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References
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33 1
7 Appendix
EP searches from a population of points, not a single point. The population can move over hills and across valleys. It can search a complicated and uncertain area to find the solution. Therefore EP can discover a globally or near glo- bally optimal point. EP simulates an evolutionary process to find the fitness after the reproduction, mutation, compe- tition, and selection procedure. The EP procedure for the problem studied is delineated as follows.
7. I Initialisation and reproduction Let %i = {Mj’, @’, Pj’} be an individual, i = 1, 2 ,..., k. k is the population size and is set to 100 in this paper. All indi- viduals are set between the lower and upper limits with a uniform distribution as shown in eqns. 28-30.
Aj+m = X j min + U(O,1)’ * ( X j max - X j min) (29)
Pj+m = Pj min U(O, * (Pj malt - Pj min) (30)
7.2 Statistics The objective function C(.) is used as the fitness function by adding constraints as
ne
m=l nm
n=l
h(&) and g(Wi) are the equality and inequality constraints such as eqns. 12-21, ne and nm are the number of the equality and inequality constraints, jleq,m and Aineq,n are the penalty factors that can be adjusted in the optimisation procedure, and glim is defined by
If one or more variables violate their limits, the penalty factors will increase and the corresponding individual will be rejected to avoid generating the infeasible solution. The fitness values are arranged in descending order from the maximum to mini” F,min.
7.3 Mutation The mutation operation is carried out to double the popu- lation size from k to 2k. Each ‘Bi is mutated and created to
by using eqns. 33-35. The new individual is formed from the old one by adding to a Gaussian random variable N e ) as
Mt+k 3 = M j + N(0, 02) (33)
X;.+k = A;. + N ( 0 ) 0 2 )
P,”+h = Pji + N ( O , 2 )
(34)
(35 ) N(0, d) represents a Gaussian random variable with mean 0 and variance 02 shown by
(37)
(38)
4 g~ = P g * (Xmaz - Amin) * -
Fi,maz
F? OF = P g * (Pmax - Pmin) * -
Fi,max
pg is a mutation factor at the gth generation set within (0,1>.
7.4 Competition In this process, 2k individuals compete with other ran- domly selected individuals for a ‘win’ based on the fitness value. np individuals are selected. np is set to 20 in this paper. A Large np impedes performance and a small np could have a solution easily trapped to a local optimum. If the fitness Faj of a random individual i is smaller than of the FRr selected individual, a weight W, is assigned as 1 as in eqn. 3. The summation Wi was obtained after the com- petition process.
np
w i = w t t=l
0, otherwise where, r = [2ku2 + 11, [xJ denotes a greater integer less than x, and ul , y are random variables between 0 and 1.
7.5 Selection and reproduction After competition, the individuals !Iti, i = 1, 2, ... , k, are ranked in descending order according to Wi. The first k individuals are selected with F R ~ used for the next genera- tion. 100 generations are set in this paper as the stopping criteria. If the convergent condition is not met, the muta- tion, competition and selection process will be rerun.
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