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Electric power generation expansion: Planning with multiple objectives
Applied Energy 19 (1985) 301 319
Electric Power Generation Expansion: Planning with Multiple Objectives
M. A. Q u a d d u s and T. N. G o h Industrial and Systems Engineering Department, National University of Singapore, Kent Ridge, Singapore 0511 (Singapore)
SUMMARY A linear multi-objective modeljor power generation expansion is studied in this paper. The model is applied to the power system planning of Singapore and solved by a newly developed interactive method. The generation alternatives considered are oil-fired, coal-fired, natural-gas fired and nuclear power plants. The multi-objective model does not recommend nuclear plant jbr Singapore because of the high risks and environmental impact, though the single objective version considers it attractivejor economic reasons. Results obtained are analyzed and some prejerred expansion plans for Singapore are recommended.
INTRODUCTION Since the oil embargo of 1973, energy planning has been the most emphasized area of national activities for non-oil producing countries. One of the worst hit energy sources in these countries is electric power, owing to its heavy dependence on oil. Generally, careful study and research are necessary for the generation and expansion of electric power.~ Besides oil embargoes, planning for electric power generation 301 Applied Energy 0306-2619/85/$03.30 (© Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain
302
M. A. Quaddus, T. N. Goh
stems from other reasons: (i) high capital investment, (ii) risks associated with power generating plants, and (iii) environmental considerations for various plants. Planning for electric power generation expansion has multiple dimensions, and, as such, a traditional single-objective (or measure of performance) based approach will be inadequate to model the planning process in a realistic way. A comprehensive planning model would have multiple objectives, be highly non-linear and stochastic, 2 but such a model would also be very complex to understand and apply. Thus a workable model for power generation planning has to consider all the dimensions realistically, yet be easy enough for the Decision Maker (DM) to understand and apply, and to interpret the solutions. This paper studies a multi-objective model for power generation expansion planning which has been applied previously to the Turkish electrical system. 2 The model has been adapted for application to power-system planning in Singapore. A new interactive approach has been adopted to solve the model in a man-machine interactive environment. 3 This approach allows a DM to communicate with a computer and thus solve the model on-line. A brief description of the solution approach is included in the Appendix. In the following sections, the multi-objective model for powergeneration planning is described. The actual application, results and discussions are presented in detail.
MULTI-OBJECTIVE MODEL FOR POWER GENERATION Optimization-based methodologies are widely used for power-systems planning. 1,4 Anderson 5 provided a comprehensive state-of-the-art survey of mathematical modelling in electricity supply. However, very few applications of multi-objective methods to energy-policy planning are available in the literature. These include Kavrakoglu and Kiziltan, 2 some papers in Kavrakoglu, 1 and Bayraktar e t al. 4 The model due to Kavrakoglu and Kiziltan 2 has been selected in this study because of its application in the Turkish electrical system. The model is linear in nature and determines, for each period in the planning horizon, the types and sizes of generating plants to be installed and the operating levels of the various power plants.
Electric power generation expansion."planning with multiple objectives
303
Objective functions of the model
Three objectives are considered in the model as follows: (i)
to minimize the present worth of all investments and operating costs over the planning horizon, i.e. T
(1 + r)-~(CP~tP~t + CE~Ejt )
Minimize F 1 = ~ ' ~ ' j~d
(1)
t = 1
(ii) to minimize the environmental impact, i.e. T
Minimize jeJ
l= 1
(iii) to minimize the risk or damage potential, i.e. T 1~----~ ~-----~
7" ,~7---~
(3)
Minimize g,,,,,,,,,,,,,d g,,,,,,,,,,,,d jed t=l
¢ d k=O
where: J = total n u m b e r of different types of plants; T = total n u m b e r of periods (planning horizon); Pit = added power capacity for p l a n t j during period t in G W (decision variable); Ejt = energy generated for plant j during period t in T W h (decision variable); CPj, and CEjt = unit costs of power and generated energy for plant j during period t in $ / M W and $ / T W h respectively; 1i, = environmental impact coefficient associated with electricity generation in p l a n t j during period t; R~, = risk coefficient associated with installed power capacity for plant j during period t; r = rate of interest. Constraints for the model
Six different sets of constraints are considered in the model as follows' (i)
the energy d e m a n d s (EDt) have to be satisfied at all times; for all t~ T, y j~J
Ej, >_ED t
(4)
304
M. A. Quaddus, T. N. Gob
(ii) the power plants in operation must be sufficient at all times to meet the instantaneous power demand (PD,); t
forallt6T,~ZPjk>PD
t
(5)
jc:d k = 0
(iii) energy that can be generated from any type of plant cannot exceed the existing production capacity multiplied by the appropriate availability factor (Ji); t--1
for all j, t, Ej, < j j ) '
Pjk
(6)
k=0
(iv) the addition of new capacity cannot exceed a certain rate of growth, aj, for any type of power plant; for all j, t, P~, < ajPj,,_ 1
(7)
(v) renewable energy is restricted by existing energy potential (EPj) at the time in question; for all t, and j = renewable resources, Er, < EP r
(8)
(vi) the available reserve (ERr) cannot be exceeded for depletable resources; t
for all t, and j = depletable resources, ~ ' Er~ <_ER r
(9)
k=l
The model, as presented above, has been trimmed to suit the electricity supply system in Singapore. In doing so, some variables and the last two sets of constraints were considered irrelevant. The specific mathematical model for Singapore is provided in the next section. A CASE STUDY Singapore has no natural energy resource and has to depend totally on imported oil to meet its energy requirements. Approximately 50 ~ of this oil is used for electricity generation. * Thus, careful planning is necessary for the generation and expansion of electric power.
GWh
] 1962
I 1964
~
I 1966
I 1968
j...C..-.":/~
I 1970
/~:/.
Produ
/
Electricity eWh
1 1976
I 1978
I 1980
I 1982
Consumption,
Power Demand, M W .~
........ Peak
I 1974
/ / ~ "
I 1972
Electricity
~" .~'="
,~
~
~ 1000
~
_-3°°°2ooo ~~" ~=.=.~
--l.00o MW
- 50o0
--600o
Fig. 1. Historical growth of electricity production, consumption and peak power demand. (Source: Reference 7.)
1960
1000 --
2ooo3°°°--
z.O00--
5000--
6000--
70(20 - -
7000--
/
-- 8000
8000-
306
M. A. Quaddus, T. N. Goh
Along with its economic growth, Singapore has experienced sustained demand in electricity consumption. This increasing demand has been met by new power additions, thereby increasing the rate of electricity production. Figure 1 shows these growth patterns. Currently, there are three power stations in Singapore, with a total installed capacity of 2586 MW serving 600000 customers. The peak demand is 1452 MW. Table 1 shows the existing power stations, which are oil-fired (industrial diesel oil for gas turbines and light fuel oil for steam turbines). The TABLE 1 Existing Power Stations (as at 1983) Power stations
Units
MW ratings
Total capacity (MW)
Installation year
Pasir Panjang B (PPB) Jurong (JS) Stage ! Stage 11 Senoko (SS) Stage I Stage II Stage III
4 4 3 3 3 2
60 60 120 120 250 250
240 240 360 360 750 500
1965 1969 1974 1976 1979 1983
Gas turbines at PPB Senoko
1 2
96 20
96 40
existing plants are sufficient to meet the current demands. However, due to the retirement of older, smaller, and less-efficient generating units and the increasing electricity demand, expansions of existing power stations, as well as the construction of new power stations, are necessary. Power-generation planning basically involves forecasting the local demand and then installing new power plants to meet such demand. Long lead-times are necessary from the decision to build a new plant to its final operation. Table 2 shows the development plans embarked upon by the Public Utilities Board (PUB) of Singapore to meet the future load demands. A number of coal-fired plants have been planned for the future. This was due to the 1979 oil crisis and further escalation in unit oil prices. However, coal may not be an attractive alternative now, during this period of oil glut. In this study, coal-fired, natural-gas fired and nuclear thermal power
Electric power generation expansion." planning with multiple objectives
307
TABLE 2 D e v e l o p m e n t Plans for Power G e n e r a t i o n (Source: P U B Official s )
Power stations
J u r o n g Power Station
Power capacity (MW)
Type of plant
Installation year
2 x 100
Gas Turbines
1986
Pulau Seraya
Stage Stage Stage Stage
I I II II
2 x 1x 1x 2 ×
250 250 250 250
Oil-fired Oil-fired Oil-fired Oil-fired
1987 1988 1990 1991
Pulau T e k o n g A
Stage Stage Stage Stage Stage Stage
I I II lI III II|
1× 1x 1x 1x 1x 1×
350 350 350 350 350 350
Coal-fired Coal-fired Coal-fired Coal-fired Coal-fired Coal-fired
1992 1993 1994 1995 1995 1996
Pulau Seraya
Stage Stage Stage Stage
III III II1 IlI
1× 1x 1× 1×
350 350 350 350
Coal-fired Coal-fired Coal-fired Coal-fired
1997 1998 1999 2000
Pulau T e k o n g B
Stage Stage Stage Stage
I I I II
1× 1x 1× 2 x
350 350 350 350
Coal-fired Coal-fired Coal-fired Coal-fired
2000 2001 2002 2003
generation plants have been considered, along with traditional oil-fired plants. While coal and nuclear processes pose some environmental and reliability problems, natural gas seems to be the most viable alternative to fuel oil in Singapore. By the year 1988, there will be a Malaysian gas pipeline reaching Singapore, and by 1990 natural gas is expected to account for 30-50 ~o of fuel requirements for power stations. 9 The power-generation planning model has been formulated into a multi-objective linear program described previously. Two models have been developed. Model A explicitly considers the future development plans of PUB (see Table 2) and thus determines only optimal powergeneration during each period of the planning horizon. On the other hand, Model B determines both the added power capacity and the power generation for the planning horizon. The purpose is to compare the
M. A. Quaddus, T. N. Goh
308
optimal power-generation expansion plan (from Model B) with that of PUB (Model A). Model A
The PUB has already planned its generation-expansion programme to the year AD2003. In this model, the added power capacities (Pit) are given the values as planned by the PUB. Thus, the only decision variable is Lit. Seven different types of plant have been considered in this model, as shown in Table 3. The planning horizon has been taken as 18 years, with 6 periods of 3 years' duration. The base year is 1982. From the general model presented previously, Model A can be represented as follows: 7
Minimize F1 = 3
6
3
j=l
t=l
7
6
(l+r)-tCEjtEjt
Minimize F 2 = 3 ~ I j t E j t j=l
(10)
(11)
t=l
subject to: 7
E jr >__ED,
for t - 1 . . . . , 6
(12)
for j = 1 , . . . , 7 t = l . . . . ,6
(13)
j=l t-1
Ejt ' P j k k=O
Note that we now have two objectives (the third objective is a constant). The model has 40 constraints and 34 legitimate variables (some Ejt values are zeros). The process of data collection and estimation (CEil, Ijt, ED~, etc.) will be described in the next section. Model B
In this model Pit will be given values as planned by the PUB for the first three periods. This is because plans are underway to install power stations during this time; also any new power additions generated by the model
Electric power generation expansion." planning with multiple objectives
309
TABLE 3 P l a n t D e s c r i p t i o n for M o d e l s A a n d B
Model A J
Model B
Type
1
R
1 PPB
1
1 0.79
2 3 4 5 6 7
1 1 1 1 1 5
1 1 1 1 1 1
JS (60 M W Size) JS (120 M W Size) SS Gas turbines F u t u r e oil-fired F u t u r e coal-fired
f
0.78 0.76 0-76 0.93 0.75 0.70
J
Type
1 Existing oil-fired (PPB, JS, SS) 2 Gas turbines 3 F u t u r e oil-fired 4 F u t u r e n a t u r a l gas-fired 5 F u t u r e coal-fired 6 Future nuclear
I
R
1
1 0"77
1 1 1 5 7
1 1 2 1 10
f
0.93 0.75 0.80 0.70 0.77
would need a lead time of 5-9 years for planning and installation. As before, a planning horizon of 18 years has been taken. Six different types of plant have been considered in this model, as shown in Table 3. The specific m o d e l can be represented as follows: 6
6
MinimizeFl=~(l+r)-t(CPjtPjt+CEjtEjt) j=l 6
t=l 6
MinimizeFz= ~ ~ IjtEjt j=l
t=l
6
6
t=l
(15)
6
MinimizeF3=~Rjt~Pj~ j=l
(14)
(16)
k=0
subject to" 6
~ Ejt> EDt 6
for t = 1 , . . . , 6
j=l t
~Pjk>PDtfort=4,...,6 j=l
(17)
k=O
(18)
M. A. Quaddus, T. N. Goh
310
t-1
or;=l, ,:},=1,,
19,
f ° r J = 3"'"66}
(20)
k=0
Pj~ "< a j P j a
-
1
t = 4,...,
The model has 3 objectives, 47 constraints and 38 legitimate variables. (Some values of Pj, and Ej, are either constants or zeros.) S O U R C E S OF D A T A Various data, from load demand to interest rate, are needed in this study. This section provides an account of how these data have been collected and/or estimated. The main sets of data needed are on energy demand (ED) and power GWh
MW
36000-
-36000
33000-
-
30000
--30000
27000
-27000
24000--
--24000 Energy
21000
Demand,
GW
~1~0 0
--
18000 --
~
1.~000 --
r
- 18000
~
- 15000
12000 --
- 12000
0000 - - ~
-
6000 --
. ~
Peak 3000
[~mand, MW .......~. '~ . ___---- • ___---. ~
- -
0000 6000
Power
--
- -
O--
3000
-0 I
1983 1984
Fig. 2.
33000
I
I
I
Rl~8
1992
1906
I
2000
2004
Forecast of energy and peak power d e m a n d in the period 1983 to 2003.
Electric power generation expansion: planning with multiple objectives
31 l
d e m a n d (PD). Various approaches are available in the literature to forecast E D and PD. lo In this study, however, the PUB source has been used to collect E D and PD.8 Figure 2 shows the forecasted data on E D and P D to the year 2003. Data on C P and C E have been estimated based on a number of sources. 8' l ~'l 2 These data have been inflated somewhat for future time periods. The availability factor,j; is a fraction of time (in a year) a plant is available for the generation of electricity. These data have been estimated from the preventative maintenance schedule and forced outage rate of different power plants. Table 3 shows the availability factors of different plants. The environmental impact coefficient, I, and risk coefficient, R, are not easily available as these are qualitative in nature. In this study a wellknown approach, the 'Analytic Hierarchy Process', ~3 has been used to estimate I and R. A survey is conducted to obtain judgements on relative weights for various power plants in the context of environmental impact and risk. Saaty's ~3 method is then used to estimate the coefficients l a n d R (see Table 3).
RESULTS AND DISCUSSION As mentioned earlier, a new interactive method has been used to solve the models. The method finds an initial efficient solution by inputting positive weights for different objectives and then explores other efficient solutions in a systematic way (see the appendix). In this study a number of efficient solutions (for both models) have been generated and analyzed. Thus a satisfactory solution may be selected by the decision maker for implementation. Results of Model A
This model uses the power additions as planned by the PUB and determines only the optimal operating levels of the plants. Individual optimal values of the objectives are as follows: F 1 (economic cost) = S$13.8 F 2
x 10 9
(environmental impact) = 365-4 (dimensionless) F 3 (risk) = 29 910 (dimensionless, constant)
Table 4 provides the s u m m a r y of solutions from Model A. It is noted that,
312
M. A. Quaddus, T. N. Goh
with an input weight of (8, 1), four solutions have been generated. Furthermore, the per cent deviation of F 3 is zero as F 3 is a constant. Some of the solutions from Table 4 deserve attention. With a weight of (10, 1) a solution with (0, 93.9, 0) per cent deviations is generated, which is obvious because objective F 1 is given maximum emphasis here. However, if the weighting process is reversed, i.e. (1, 10), then objective F 2 is not fully achieved: it is still 9 ~o away from the optimum. In fact, the solution with TABLE 4 Summary of Results for Model A
Input weights of objective functions
Per cent deviation j r o m the optimal value
F1
F2
F1
F2
F~
8
1
O 0"3 0.4 0"8
93.9 90.8 89.5 84.7
0 0 0 0
10
3
0 0.3
93-9 90.8
0 0
8
3
0
93-9
0
1
1
0
93"9
0
1
10
f8"3 7-2
9.0 19-6
0 0
10
1
0
93.9
0
f
(8.3, 9.0, 0) per cent deviations seems to be the best compromise solution for Model A. To show the trade-off between F 1 and F 2, a trade-off function has been generated by starting with the weight (10, 1), then gradually improving F 2 at the expense of objective F 1. Figure 3 shows the trade-off curve. Because the power additions are constant in this model (see Table 2), different solutions only imply different operating levels with the corresponding objective values. The measures of goodness of the PUB plan are the optimal objective values, which will be compared later with those of Model B.
Electric power generation expansion: planning with multiple objectives
313
2t .E e~
o E
o
) 8 '~1,5
.~_ .~ 6
)
~ ~
>
parentheses represent wei g ~ctively htsassigned Numbers
in the
u
"~01 } I
I
20
~.0
F2,
60
I
I
I
80
I00
120
environmental impact (*/, deviation
from optimum)
Fig. 3. Trade-off function of cost (FI) and environmental impact (F2) in Model A.
Results of Model B This model shows the optimal power additions and operating levels of various power plants. Optimal values of the individual objective functions are: F 1 (economic cost) = S$13.66 x 109 F 2 (environmental impact) = 310.5 (dimensionless) F 3 (risk) = 22 220 (dimensionless) A comparison of these optimal values with those of Model A shows that the power addition plans by the P U B are not quite optimal. However, in terms of cost and environmental impact, the P U B plan is very good. The s u m m a r y of results for Model B is shown in Table 5. As before a trade-off function o f cost versus environmental impact has been developed and this
314
M . A . Quaddus, T. N. Goh
TABLE 5 Summary of Results for Model B
Input weights o f objecti~,e functions r~
F2
F3
10
3
4
Per cent deviation Jrom
the
optimal value F1
F2
F3
[" 11"3 '~ 13.5 k 11.5
66"4 40.6 52.3
2"6 0"2 2.3
5.7 5.6 5.4 5'9 6"0 6-3 6"8
120.1 116.1 116.1 71.3 70.5 70.5 55.1
11.2 11.9 12.9 19.3 19.9 18.2 19.0
0
220.1
184.3
|
/
f
8
2
2
1
0
0
0
1
0
21-7
0
25.3
0
0
1
23-3
20.2
0
is shown in Fig. 4. Unlike the trade-off function of Model A, this function has two distinct zones. It can be observed from Table 5 that there are two groups of solutions. If it is desired to minimize the environmental impact and risk (not forgetting cost), then the solution with (11.5, 52.3, 2-3) per cent deviations seems to be the best compromise solution. On the other hand, the best compromise solution for minimizing cost and environmental impact (not forgetting risk) appears to be (6.8, 55.1, 19) per cent deviations. By analyzing the power additions lor these two solutions, it is observed that for (11.5, 52.3, 2.3) per cent deviations, the plants to be introduced are 1050 MW and 337 MW coal-fired during periods 5 and 6 respectively and 1050MW new oil-fired during period 6. For (6.8, 55-1, 19) per cent deviations, 1050MW and 362 MW natural-gas fired plants are to be introduced during periods 5 and 6 respectively, and a 820 MW coal-fired plant is to be introduced during period 4. In Model B, the power additions for the first three periods are taken as planned by the PUB. These are 500 MW expansion of the existing oil-fired plant during period 1,200 MW
Electric power generation expansion." planning with multiple objectives
315
22 ~, (0, 1,0)
,0_l\
E
o=
~.
(10, 3,4 }
~2-{
Numbers
~>
represent
in the parentheses weights
assigned
to F 1, F 2 & F 3 respectively 18, 2,21
ca
(8,1,2
}
~-... (I,0,0) 0 0
I 40
F2,
I 80
environmental
I 120
I 160
impact (°/o deviation
I 200
from
220
optimum)
Fig. 4. Trade-off function of cost (F~) and environmental impact ( F 2) in Model B.
gas turbine and 750MW new oil-fired plants during period 2, and a 750 MW new oil-fired plant during period 3. Nuclear power plant is not introduced at all when all three objectives are considered. This is due to high environmental impact and risks associated with the nuclear plant. However if cost is considered as the sole criterion, then 1050 MW and 4 6 4 M W nuclear plants are introduced during periods 4 and 5 respectively. Sensitivity analysis has been performed to determine under what circumstances of environmental
316
M. A. Quaddus, T. N. Goh TABLE 6 Effects of Environmental Impact and Risk on Nuclear Power Plant (Blank, not tested; Y, tested and nuclear plants are introduced; N, tested and nuclear plants are not introduced) R
I 1
1 2 3 4 5 6 7
Y N
N
2
3 N N N N
4
5
6
Y N
7 Y N
8
9
10 Y
N
N
N
N
N
N
8 9 10
N
N
impact(I) and risk (R) nuclear plants will be introduced in the threeobjective model. Table 6 shows this analysis. CONCLUSI ON Multi-objective modelling of power-system planning is a difficult task. This study has shown that a considerable amount of information can be obtained from multi-objective analysis. It has generated alternative efficient solutions and has recommended some compromise solutions. Existing plans of Singapore's PUB have been compared with the 'ideal' plans generated by the model and have been found to be near optimal in terms of achievements. The model recommends new oil-fired, coal-fired and natural-gas fired plants, but does not recommend nuclear plants. In the context of Singapore, nuclear plant may not be feasible on account of its environmental impact and associated risks. However, nuclear plant is economically attractive as it is recommended by a cost criterion model. Sensitivity analysis has shown that any risk factor R exceeding unity (on a scale from 1 to 10) would not recommend nuclear power plants. A natural extension of the model presented here would be to reduce the planning period to one year, and thus incorporate a load-duration curve into the model on a yearly basis. In doing so, however, the model size would be greatly increased.
Electric power generation expansion: planning with multiple objectives
317
REFERENCES 1. I. Kavrakoglu (Ed.), Mathematical modelling of energy systems, Sijthoff & Noordhoff, The Netherlands, 1981. 2. I. Kavrakoglu and G. Kiziltan, Multi-objective strategies in power systems planning, Eur. J. Operat. Res., 12 (1983), pp. 159-70. 3. M.A. Quaddus and A. G. Holzman, A man-machine interactive system for multi-objective decision making, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, 1983, pp. 873-7. 4. B.A. Bayraktar, E. A. Cherniavsky, M. A. Laughton and L. E. Ruff (Eds), Energy policy planning, Plenum Press, New York, 1981. 5. D. Anderson, Models for determining least-cost investments in electricity supply, Bell J. Econ. Manag. Sci., 3(1) (1972), pp. 267-99. 6. E. T. Lee, Energy conservation in Singapore, Energy, 8(1) (1983), pp. 137-41. 7. Yearbook of statistics, Singapore, 1967-1982, Department of Statistics of the Government of Singapore. 8. PUB Singapore, Private communication with the Acting Senior Engineer, Planning Department, October 1983. 9. The Strait Times (English), Press reports on natural gas and power generation, Singapore, August-December 1983. 10. T. N. Goh, S. S. Choi, C. H. Tan and K. C. Tan, A comparative study of short-term forecasting of energy and peak power demand, Electric Power Systems Res., 5 (1982), pp. 63-71. 11. H. Paul and T. N. Goh, A study of electric power-generation expansion plannning, Energy Manag., 6(2) (1982), pp. 109-18. 12. B. K. Tan, An economic comparison of alternative power generation strategies in Singapore, Unpublished MSc Dissertation, University of Singapore, 1978. 13. T. L. Saaty, The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill, New York, 1980.
APPENDIX: THE SOLUTION APPROACH A multiple objective linear optimization model can be represented as follows: Maximize ~J1(2) (Minimize)l)).()?) subject to
2~X
The interactive method to be presented concentrates on the efficient
318
M. A. Quaddus, T. N. Goh
surface and finds the efficient (pareto-optimal, non-dominated) solutions. A solution Y* is efficient if 2 " ~ X and there exists no other 2~ X such that ji(2) _>Ji(:~*)
i = 1,..., L
and L(2) ¢ j i ( 2 " )
for at least one i
Because the efficient solution is at least as good as any other solution, it is obvious that one of the efficient solutions should be the satisfactory (optimal) solution for the DM. The method is presented briefly. For a detailed exposition see Quaddus and Holzman. 3 The method finds a small sub-set of efficient solutions at a time by interacting with the D M and thus explores the efficient surface in a systematic way. It starts with a set of positive weights, w i values, supplied by the D M and then develops the following composite objective: L
F = ~ ' wi/'i(x ) i=1
The function F is maximized subject to the original set of constraints which then gives the starting efficient extreme point 21 . Objective function values at 21 are then presented to the D M for his evaluation and recommendation, If the D M is satisfied, the solution process terminates with 21 as the satisfactory solution. However, if the D M is not satisfied then he is required to provide the local trade-off information regarding which objectives he wants to 'sacrifice' in order to 'gain' with respect to achieving the other objectives. Note that the D M is not required to specify the exact a m o u n t by which the objectives are to be sacrificed or gained. This is the only information required to direct the search procedure. F r o m the above information the algorithm finds a sub-set of adjacent efficient extreme points which match exactly with the DM's trade-off information. F r o m this small sub-set of efficient points, the D M chooses his next satisfactory solution (say 22). The solution process then starts all over again with 22 as the new starting efficient point. However if the D M is completely satisfied with 22, then he can terminate the solution process. It is to be noted that the method finds a new sub-set of efficient points every time, i.e. rejected efficient points are not reconsidered. The main feature of the method is its simplicity in searching for the satisfactory solution.
Electric power generation expansion: planning with multiple objectives
319
The method is exploratory in nature, requires very little response from the DM, and relies on the D M for the search procedure and its termination. The solution procedure has been coded in F O R T R A N for the IBM 3033 system at the National University of Singapore. The present version can handle 50 variables, 50 constraints, and 10 objectives.
Electric Power Generation Expansion: Planning with Multiple Objectives
M. A. Q u a d d u s and T. N. G o h Industrial and Systems Engineering Department, National University of Singapore, Kent Ridge, Singapore 0511 (Singapore)
SUMMARY A linear multi-objective modeljor power generation expansion is studied in this paper. The model is applied to the power system planning of Singapore and solved by a newly developed interactive method. The generation alternatives considered are oil-fired, coal-fired, natural-gas fired and nuclear power plants. The multi-objective model does not recommend nuclear plant jbr Singapore because of the high risks and environmental impact, though the single objective version considers it attractivejor economic reasons. Results obtained are analyzed and some prejerred expansion plans for Singapore are recommended.
INTRODUCTION Since the oil embargo of 1973, energy planning has been the most emphasized area of national activities for non-oil producing countries. One of the worst hit energy sources in these countries is electric power, owing to its heavy dependence on oil. Generally, careful study and research are necessary for the generation and expansion of electric power.~ Besides oil embargoes, planning for electric power generation 301 Applied Energy 0306-2619/85/$03.30 (© Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain
302
M. A. Quaddus, T. N. Goh
stems from other reasons: (i) high capital investment, (ii) risks associated with power generating plants, and (iii) environmental considerations for various plants. Planning for electric power generation expansion has multiple dimensions, and, as such, a traditional single-objective (or measure of performance) based approach will be inadequate to model the planning process in a realistic way. A comprehensive planning model would have multiple objectives, be highly non-linear and stochastic, 2 but such a model would also be very complex to understand and apply. Thus a workable model for power generation planning has to consider all the dimensions realistically, yet be easy enough for the Decision Maker (DM) to understand and apply, and to interpret the solutions. This paper studies a multi-objective model for power generation expansion planning which has been applied previously to the Turkish electrical system. 2 The model has been adapted for application to power-system planning in Singapore. A new interactive approach has been adopted to solve the model in a man-machine interactive environment. 3 This approach allows a DM to communicate with a computer and thus solve the model on-line. A brief description of the solution approach is included in the Appendix. In the following sections, the multi-objective model for powergeneration planning is described. The actual application, results and discussions are presented in detail.
MULTI-OBJECTIVE MODEL FOR POWER GENERATION Optimization-based methodologies are widely used for power-systems planning. 1,4 Anderson 5 provided a comprehensive state-of-the-art survey of mathematical modelling in electricity supply. However, very few applications of multi-objective methods to energy-policy planning are available in the literature. These include Kavrakoglu and Kiziltan, 2 some papers in Kavrakoglu, 1 and Bayraktar e t al. 4 The model due to Kavrakoglu and Kiziltan 2 has been selected in this study because of its application in the Turkish electrical system. The model is linear in nature and determines, for each period in the planning horizon, the types and sizes of generating plants to be installed and the operating levels of the various power plants.
Electric power generation expansion."planning with multiple objectives
303
Objective functions of the model
Three objectives are considered in the model as follows: (i)
to minimize the present worth of all investments and operating costs over the planning horizon, i.e. T
(1 + r)-~(CP~tP~t + CE~Ejt )
Minimize F 1 = ~ ' ~ ' j~d
(1)
t = 1
(ii) to minimize the environmental impact, i.e. T
Minimize jeJ
l= 1
(iii) to minimize the risk or damage potential, i.e. T 1~----~ ~-----~
7" ,~7---~
(3)
Minimize g,,,,,,,,,,,,,d g,,,,,,,,,,,,d jed t=l
¢ d k=O
where: J = total n u m b e r of different types of plants; T = total n u m b e r of periods (planning horizon); Pit = added power capacity for p l a n t j during period t in G W (decision variable); Ejt = energy generated for plant j during period t in T W h (decision variable); CPj, and CEjt = unit costs of power and generated energy for plant j during period t in $ / M W and $ / T W h respectively; 1i, = environmental impact coefficient associated with electricity generation in p l a n t j during period t; R~, = risk coefficient associated with installed power capacity for plant j during period t; r = rate of interest. Constraints for the model
Six different sets of constraints are considered in the model as follows' (i)
the energy d e m a n d s (EDt) have to be satisfied at all times; for all t~ T, y j~J
Ej, >_ED t
(4)
304
M. A. Quaddus, T. N. Gob
(ii) the power plants in operation must be sufficient at all times to meet the instantaneous power demand (PD,); t
forallt6T,~ZPjk>PD
t
(5)
jc:d k = 0
(iii) energy that can be generated from any type of plant cannot exceed the existing production capacity multiplied by the appropriate availability factor (Ji); t--1
for all j, t, Ej, < j j ) '
Pjk
(6)
k=0
(iv) the addition of new capacity cannot exceed a certain rate of growth, aj, for any type of power plant; for all j, t, P~, < ajPj,,_ 1
(7)
(v) renewable energy is restricted by existing energy potential (EPj) at the time in question; for all t, and j = renewable resources, Er, < EP r
(8)
(vi) the available reserve (ERr) cannot be exceeded for depletable resources; t
for all t, and j = depletable resources, ~ ' Er~ <_ER r
(9)
k=l
The model, as presented above, has been trimmed to suit the electricity supply system in Singapore. In doing so, some variables and the last two sets of constraints were considered irrelevant. The specific mathematical model for Singapore is provided in the next section. A CASE STUDY Singapore has no natural energy resource and has to depend totally on imported oil to meet its energy requirements. Approximately 50 ~ of this oil is used for electricity generation. * Thus, careful planning is necessary for the generation and expansion of electric power.
GWh
] 1962
I 1964
~
I 1966
I 1968
j...C..-.":/~
I 1970
/~:/.
Produ
/
Electricity eWh
1 1976
I 1978
I 1980
I 1982
Consumption,
Power Demand, M W .~
........ Peak
I 1974
/ / ~ "
I 1972
Electricity
~" .~'="
,~
~
~ 1000
~
_-3°°°2ooo ~~" ~=.=.~
--l.00o MW
- 50o0
--600o
Fig. 1. Historical growth of electricity production, consumption and peak power demand. (Source: Reference 7.)
1960
1000 --
2ooo3°°°--
z.O00--
5000--
6000--
70(20 - -
7000--
/
-- 8000
8000-
306
M. A. Quaddus, T. N. Goh
Along with its economic growth, Singapore has experienced sustained demand in electricity consumption. This increasing demand has been met by new power additions, thereby increasing the rate of electricity production. Figure 1 shows these growth patterns. Currently, there are three power stations in Singapore, with a total installed capacity of 2586 MW serving 600000 customers. The peak demand is 1452 MW. Table 1 shows the existing power stations, which are oil-fired (industrial diesel oil for gas turbines and light fuel oil for steam turbines). The TABLE 1 Existing Power Stations (as at 1983) Power stations
Units
MW ratings
Total capacity (MW)
Installation year
Pasir Panjang B (PPB) Jurong (JS) Stage ! Stage 11 Senoko (SS) Stage I Stage II Stage III
4 4 3 3 3 2
60 60 120 120 250 250
240 240 360 360 750 500
1965 1969 1974 1976 1979 1983
Gas turbines at PPB Senoko
1 2
96 20
96 40
existing plants are sufficient to meet the current demands. However, due to the retirement of older, smaller, and less-efficient generating units and the increasing electricity demand, expansions of existing power stations, as well as the construction of new power stations, are necessary. Power-generation planning basically involves forecasting the local demand and then installing new power plants to meet such demand. Long lead-times are necessary from the decision to build a new plant to its final operation. Table 2 shows the development plans embarked upon by the Public Utilities Board (PUB) of Singapore to meet the future load demands. A number of coal-fired plants have been planned for the future. This was due to the 1979 oil crisis and further escalation in unit oil prices. However, coal may not be an attractive alternative now, during this period of oil glut. In this study, coal-fired, natural-gas fired and nuclear thermal power
Electric power generation expansion." planning with multiple objectives
307
TABLE 2 D e v e l o p m e n t Plans for Power G e n e r a t i o n (Source: P U B Official s )
Power stations
J u r o n g Power Station
Power capacity (MW)
Type of plant
Installation year
2 x 100
Gas Turbines
1986
Pulau Seraya
Stage Stage Stage Stage
I I II II
2 x 1x 1x 2 ×
250 250 250 250
Oil-fired Oil-fired Oil-fired Oil-fired
1987 1988 1990 1991
Pulau T e k o n g A
Stage Stage Stage Stage Stage Stage
I I II lI III II|
1× 1x 1x 1x 1x 1×
350 350 350 350 350 350
Coal-fired Coal-fired Coal-fired Coal-fired Coal-fired Coal-fired
1992 1993 1994 1995 1995 1996
Pulau Seraya
Stage Stage Stage Stage
III III II1 IlI
1× 1x 1× 1×
350 350 350 350
Coal-fired Coal-fired Coal-fired Coal-fired
1997 1998 1999 2000
Pulau T e k o n g B
Stage Stage Stage Stage
I I I II
1× 1x 1× 2 x
350 350 350 350
Coal-fired Coal-fired Coal-fired Coal-fired
2000 2001 2002 2003
generation plants have been considered, along with traditional oil-fired plants. While coal and nuclear processes pose some environmental and reliability problems, natural gas seems to be the most viable alternative to fuel oil in Singapore. By the year 1988, there will be a Malaysian gas pipeline reaching Singapore, and by 1990 natural gas is expected to account for 30-50 ~o of fuel requirements for power stations. 9 The power-generation planning model has been formulated into a multi-objective linear program described previously. Two models have been developed. Model A explicitly considers the future development plans of PUB (see Table 2) and thus determines only optimal powergeneration during each period of the planning horizon. On the other hand, Model B determines both the added power capacity and the power generation for the planning horizon. The purpose is to compare the
M. A. Quaddus, T. N. Goh
308
optimal power-generation expansion plan (from Model B) with that of PUB (Model A). Model A
The PUB has already planned its generation-expansion programme to the year AD2003. In this model, the added power capacities (Pit) are given the values as planned by the PUB. Thus, the only decision variable is Lit. Seven different types of plant have been considered in this model, as shown in Table 3. The planning horizon has been taken as 18 years, with 6 periods of 3 years' duration. The base year is 1982. From the general model presented previously, Model A can be represented as follows: 7
Minimize F1 = 3
6
3
j=l
t=l
7
6
(l+r)-tCEjtEjt
Minimize F 2 = 3 ~ I j t E j t j=l
(10)
(11)
t=l
subject to: 7
E jr >__ED,
for t - 1 . . . . , 6
(12)
for j = 1 , . . . , 7 t = l . . . . ,6
(13)
j=l t-1
Ejt
Note that we now have two objectives (the third objective is a constant). The model has 40 constraints and 34 legitimate variables (some Ejt values are zeros). The process of data collection and estimation (CEil, Ijt, ED~, etc.) will be described in the next section. Model B
In this model Pit will be given values as planned by the PUB for the first three periods. This is because plans are underway to install power stations during this time; also any new power additions generated by the model
Electric power generation expansion." planning with multiple objectives
309
TABLE 3 P l a n t D e s c r i p t i o n for M o d e l s A a n d B
Model A J
Model B
Type
1
R
1 PPB
1
1 0.79
2 3 4 5 6 7
1 1 1 1 1 5
1 1 1 1 1 1
JS (60 M W Size) JS (120 M W Size) SS Gas turbines F u t u r e oil-fired F u t u r e coal-fired
f
0.78 0.76 0-76 0.93 0.75 0.70
J
Type
1 Existing oil-fired (PPB, JS, SS) 2 Gas turbines 3 F u t u r e oil-fired 4 F u t u r e n a t u r a l gas-fired 5 F u t u r e coal-fired 6 Future nuclear
I
R
1
1 0"77
1 1 1 5 7
1 1 2 1 10
f
0.93 0.75 0.80 0.70 0.77
would need a lead time of 5-9 years for planning and installation. As before, a planning horizon of 18 years has been taken. Six different types of plant have been considered in this model, as shown in Table 3. The specific m o d e l can be represented as follows: 6
6
MinimizeFl=~(l+r)-t(CPjtPjt+CEjtEjt) j=l 6
t=l 6
MinimizeFz= ~ ~ IjtEjt j=l
t=l
6
6
t=l
(15)
6
MinimizeF3=~Rjt~Pj~ j=l
(14)
(16)
k=0
subject to" 6
~ Ejt> EDt 6
for t = 1 , . . . , 6
j=l t
~Pjk>PDtfort=4,...,6 j=l
(17)
k=O
(18)
M. A. Quaddus, T. N. Goh
310
t-1
or;=l, ,:},=1,,
19,
f ° r J = 3"'"66}
(20)
k=0
Pj~ "< a j P j a
-
1
t = 4,...,
The model has 3 objectives, 47 constraints and 38 legitimate variables. (Some values of Pj, and Ej, are either constants or zeros.) S O U R C E S OF D A T A Various data, from load demand to interest rate, are needed in this study. This section provides an account of how these data have been collected and/or estimated. The main sets of data needed are on energy demand (ED) and power GWh
MW
36000-
-36000
33000-
-
30000
--30000
27000
-27000
24000--
--24000 Energy
21000
Demand,
GW
~1~0 0
--
18000 --
~
1.~000 --
r
- 18000
~
- 15000
12000 --
- 12000
0000 - - ~
-
6000 --
. ~
Peak 3000
[~mand, MW .......~. '~ . ___---- • ___---. ~
- -
0000 6000
Power
--
- -
O--
3000
-0 I
1983 1984
Fig. 2.
33000
I
I
I
Rl~8
1992
1906
I
2000
2004
Forecast of energy and peak power d e m a n d in the period 1983 to 2003.
Electric power generation expansion: planning with multiple objectives
31 l
d e m a n d (PD). Various approaches are available in the literature to forecast E D and PD. lo In this study, however, the PUB source has been used to collect E D and PD.8 Figure 2 shows the forecasted data on E D and P D to the year 2003. Data on C P and C E have been estimated based on a number of sources. 8' l ~'l 2 These data have been inflated somewhat for future time periods. The availability factor,j; is a fraction of time (in a year) a plant is available for the generation of electricity. These data have been estimated from the preventative maintenance schedule and forced outage rate of different power plants. Table 3 shows the availability factors of different plants. The environmental impact coefficient, I, and risk coefficient, R, are not easily available as these are qualitative in nature. In this study a wellknown approach, the 'Analytic Hierarchy Process', ~3 has been used to estimate I and R. A survey is conducted to obtain judgements on relative weights for various power plants in the context of environmental impact and risk. Saaty's ~3 method is then used to estimate the coefficients l a n d R (see Table 3).
RESULTS AND DISCUSSION As mentioned earlier, a new interactive method has been used to solve the models. The method finds an initial efficient solution by inputting positive weights for different objectives and then explores other efficient solutions in a systematic way (see the appendix). In this study a number of efficient solutions (for both models) have been generated and analyzed. Thus a satisfactory solution may be selected by the decision maker for implementation. Results of Model A
This model uses the power additions as planned by the PUB and determines only the optimal operating levels of the plants. Individual optimal values of the objectives are as follows: F 1 (economic cost) = S$13.8 F 2
x 10 9
(environmental impact) = 365-4 (dimensionless) F 3 (risk) = 29 910 (dimensionless, constant)
Table 4 provides the s u m m a r y of solutions from Model A. It is noted that,
312
M. A. Quaddus, T. N. Goh
with an input weight of (8, 1), four solutions have been generated. Furthermore, the per cent deviation of F 3 is zero as F 3 is a constant. Some of the solutions from Table 4 deserve attention. With a weight of (10, 1) a solution with (0, 93.9, 0) per cent deviations is generated, which is obvious because objective F 1 is given maximum emphasis here. However, if the weighting process is reversed, i.e. (1, 10), then objective F 2 is not fully achieved: it is still 9 ~o away from the optimum. In fact, the solution with TABLE 4 Summary of Results for Model A
Input weights of objective functions
Per cent deviation j r o m the optimal value
F1
F2
F1
F2
F~
8
1
O 0"3 0.4 0"8
93.9 90.8 89.5 84.7
0 0 0 0
10
3
0 0.3
93-9 90.8
0 0
8
3
0
93-9
0
1
1
0
93"9
0
1
10
f8"3 7-2
9.0 19-6
0 0
10
1
0
93.9
0
f
(8.3, 9.0, 0) per cent deviations seems to be the best compromise solution for Model A. To show the trade-off between F 1 and F 2, a trade-off function has been generated by starting with the weight (10, 1), then gradually improving F 2 at the expense of objective F 1. Figure 3 shows the trade-off curve. Because the power additions are constant in this model (see Table 2), different solutions only imply different operating levels with the corresponding objective values. The measures of goodness of the PUB plan are the optimal objective values, which will be compared later with those of Model B.
Electric power generation expansion: planning with multiple objectives
313
2t .E e~
o E
o
) 8 '~1,5
.~_ .~ 6
)
~ ~
>
parentheses represent wei g ~ctively htsassigned Numbers
in the
u
"~01 } I
I
20
~.0
F2,
60
I
I
I
80
I00
120
environmental impact (*/, deviation
from optimum)
Fig. 3. Trade-off function of cost (FI) and environmental impact (F2) in Model A.
Results of Model B This model shows the optimal power additions and operating levels of various power plants. Optimal values of the individual objective functions are: F 1 (economic cost) = S$13.66 x 109 F 2 (environmental impact) = 310.5 (dimensionless) F 3 (risk) = 22 220 (dimensionless) A comparison of these optimal values with those of Model A shows that the power addition plans by the P U B are not quite optimal. However, in terms of cost and environmental impact, the P U B plan is very good. The s u m m a r y of results for Model B is shown in Table 5. As before a trade-off function o f cost versus environmental impact has been developed and this
314
M . A . Quaddus, T. N. Goh
TABLE 5 Summary of Results for Model B
Input weights o f objecti~,e functions r~
F2
F3
10
3
4
Per cent deviation Jrom
the
optimal value F1
F2
F3
[" 11"3 '~ 13.5 k 11.5
66"4 40.6 52.3
2"6 0"2 2.3
5.7 5.6 5.4 5'9 6"0 6-3 6"8
120.1 116.1 116.1 71.3 70.5 70.5 55.1
11.2 11.9 12.9 19.3 19.9 18.2 19.0
0
220.1
184.3
|
/
f
8
2
2
1
0
0
0
1
0
21-7
0
25.3
0
0
1
23-3
20.2
0
is shown in Fig. 4. Unlike the trade-off function of Model A, this function has two distinct zones. It can be observed from Table 5 that there are two groups of solutions. If it is desired to minimize the environmental impact and risk (not forgetting cost), then the solution with (11.5, 52.3, 2-3) per cent deviations seems to be the best compromise solution. On the other hand, the best compromise solution for minimizing cost and environmental impact (not forgetting risk) appears to be (6.8, 55.1, 19) per cent deviations. By analyzing the power additions lor these two solutions, it is observed that for (11.5, 52.3, 2.3) per cent deviations, the plants to be introduced are 1050 MW and 337 MW coal-fired during periods 5 and 6 respectively and 1050MW new oil-fired during period 6. For (6.8, 55-1, 19) per cent deviations, 1050MW and 362 MW natural-gas fired plants are to be introduced during periods 5 and 6 respectively, and a 820 MW coal-fired plant is to be introduced during period 4. In Model B, the power additions for the first three periods are taken as planned by the PUB. These are 500 MW expansion of the existing oil-fired plant during period 1,200 MW
Electric power generation expansion." planning with multiple objectives
315
22 ~, (0, 1,0)
,0_l\
E
o=
~.
(10, 3,4 }
~2-{
Numbers
~>
represent
in the parentheses weights
assigned
to F 1, F 2 & F 3 respectively 18, 2,21
ca
(8,1,2
}
~-... (I,0,0) 0 0
I 40
F2,
I 80
environmental
I 120
I 160
impact (°/o deviation
I 200
from
220
optimum)
Fig. 4. Trade-off function of cost (F~) and environmental impact ( F 2) in Model B.
gas turbine and 750MW new oil-fired plants during period 2, and a 750 MW new oil-fired plant during period 3. Nuclear power plant is not introduced at all when all three objectives are considered. This is due to high environmental impact and risks associated with the nuclear plant. However if cost is considered as the sole criterion, then 1050 MW and 4 6 4 M W nuclear plants are introduced during periods 4 and 5 respectively. Sensitivity analysis has been performed to determine under what circumstances of environmental
316
M. A. Quaddus, T. N. Goh TABLE 6 Effects of Environmental Impact and Risk on Nuclear Power Plant (Blank, not tested; Y, tested and nuclear plants are introduced; N, tested and nuclear plants are not introduced) R
I 1
1 2 3 4 5 6 7
Y N
N
2
3 N N N N
4
5
6
Y N
7 Y N
8
9
10 Y
N
N
N
N
N
N
8 9 10
N
N
impact(I) and risk (R) nuclear plants will be introduced in the threeobjective model. Table 6 shows this analysis. CONCLUSI ON Multi-objective modelling of power-system planning is a difficult task. This study has shown that a considerable amount of information can be obtained from multi-objective analysis. It has generated alternative efficient solutions and has recommended some compromise solutions. Existing plans of Singapore's PUB have been compared with the 'ideal' plans generated by the model and have been found to be near optimal in terms of achievements. The model recommends new oil-fired, coal-fired and natural-gas fired plants, but does not recommend nuclear plants. In the context of Singapore, nuclear plant may not be feasible on account of its environmental impact and associated risks. However, nuclear plant is economically attractive as it is recommended by a cost criterion model. Sensitivity analysis has shown that any risk factor R exceeding unity (on a scale from 1 to 10) would not recommend nuclear power plants. A natural extension of the model presented here would be to reduce the planning period to one year, and thus incorporate a load-duration curve into the model on a yearly basis. In doing so, however, the model size would be greatly increased.
Electric power generation expansion: planning with multiple objectives
317
REFERENCES 1. I. Kavrakoglu (Ed.), Mathematical modelling of energy systems, Sijthoff & Noordhoff, The Netherlands, 1981. 2. I. Kavrakoglu and G. Kiziltan, Multi-objective strategies in power systems planning, Eur. J. Operat. Res., 12 (1983), pp. 159-70. 3. M.A. Quaddus and A. G. Holzman, A man-machine interactive system for multi-objective decision making, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, 1983, pp. 873-7. 4. B.A. Bayraktar, E. A. Cherniavsky, M. A. Laughton and L. E. Ruff (Eds), Energy policy planning, Plenum Press, New York, 1981. 5. D. Anderson, Models for determining least-cost investments in electricity supply, Bell J. Econ. Manag. Sci., 3(1) (1972), pp. 267-99. 6. E. T. Lee, Energy conservation in Singapore, Energy, 8(1) (1983), pp. 137-41. 7. Yearbook of statistics, Singapore, 1967-1982, Department of Statistics of the Government of Singapore. 8. PUB Singapore, Private communication with the Acting Senior Engineer, Planning Department, October 1983. 9. The Strait Times (English), Press reports on natural gas and power generation, Singapore, August-December 1983. 10. T. N. Goh, S. S. Choi, C. H. Tan and K. C. Tan, A comparative study of short-term forecasting of energy and peak power demand, Electric Power Systems Res., 5 (1982), pp. 63-71. 11. H. Paul and T. N. Goh, A study of electric power-generation expansion plannning, Energy Manag., 6(2) (1982), pp. 109-18. 12. B. K. Tan, An economic comparison of alternative power generation strategies in Singapore, Unpublished MSc Dissertation, University of Singapore, 1978. 13. T. L. Saaty, The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill, New York, 1980.
APPENDIX: THE SOLUTION APPROACH A multiple objective linear optimization model can be represented as follows: Maximize ~J1(2) (Minimize)l)).()?) subject to
2~X
The interactive method to be presented concentrates on the efficient
318
M. A. Quaddus, T. N. Goh
surface and finds the efficient (pareto-optimal, non-dominated) solutions. A solution Y* is efficient if 2 " ~ X and there exists no other 2~ X such that ji(2) _>Ji(:~*)
i = 1,..., L
and L(2) ¢ j i ( 2 " )
for at least one i
Because the efficient solution is at least as good as any other solution, it is obvious that one of the efficient solutions should be the satisfactory (optimal) solution for the DM. The method is presented briefly. For a detailed exposition see Quaddus and Holzman. 3 The method finds a small sub-set of efficient solutions at a time by interacting with the D M and thus explores the efficient surface in a systematic way. It starts with a set of positive weights, w i values, supplied by the D M and then develops the following composite objective: L
F = ~ ' wi/'i(x ) i=1
The function F is maximized subject to the original set of constraints which then gives the starting efficient extreme point 21 . Objective function values at 21 are then presented to the D M for his evaluation and recommendation, If the D M is satisfied, the solution process terminates with 21 as the satisfactory solution. However, if the D M is not satisfied then he is required to provide the local trade-off information regarding which objectives he wants to 'sacrifice' in order to 'gain' with respect to achieving the other objectives. Note that the D M is not required to specify the exact a m o u n t by which the objectives are to be sacrificed or gained. This is the only information required to direct the search procedure. F r o m the above information the algorithm finds a sub-set of adjacent efficient extreme points which match exactly with the DM's trade-off information. F r o m this small sub-set of efficient points, the D M chooses his next satisfactory solution (say 22). The solution process then starts all over again with 22 as the new starting efficient point. However if the D M is completely satisfied with 22, then he can terminate the solution process. It is to be noted that the method finds a new sub-set of efficient points every time, i.e. rejected efficient points are not reconsidered. The main feature of the method is its simplicity in searching for the satisfactory solution.
Electric power generation expansion: planning with multiple objectives
319
The method is exploratory in nature, requires very little response from the DM, and relies on the D M for the search procedure and its termination. The solution procedure has been coded in F O R T R A N for the IBM 3033 system at the National University of Singapore. The present version can handle 50 variables, 50 constraints, and 10 objectives.