Reservoir operation using grey fuzzy stochastic dynamic
programming
Fi-John Chang,* Shyh-Chi Hui and Yen-Chang Chen
Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, R.O.C
Abstract:
This paper presents an optimal regulation programme, grey fuzzy stochastic dynamic programming (GFSDP), for reservoir operation. It is composed of a grey system, fuzzy theory and dynamic programming. The grey system represents data by covering the whole range without loss of generality, and the fuzzy arithmetic takes charge of the rules of reservoir operation. The GFSDP deals with the multipurpose decision-making problem by fuzzy optimization theorem. The practicability and effectiveness of the proposed approach is tested on the operation of the Shiman reservoir in Taiwan. The current M5 operating rule curves of this reservoir also are evaluated. The simulation results demonstrate that this new approach, in comparison with the M5 rule curves, has superior performance with regard to the total water deficit and number of monthly deficits. Copyright 2002 John Wiley & Sons, Ltd.
KEY WORDS fuzzy arithmetic; grey system; reservoir operation; stochastic dynamic programming
INTRODUCTION
Taiwan’s annual average rainfall is about 2600 mm. Compared with other countries, Taiwan should have plentiful water, but owing to the characteristics of Taiwan’s watersheds (erodible soil, uneven rainfall, steep slopes and high mountains), most of the water runs directly into the sea; only 20 per cent water can be stored for use. Water resources are becoming important issues in Taiwan because of the continuous increase in water demand accompanying economic growth, opposition to the construction of new reservoirs by the heightened desire for environmental protection, and dwindling number of suitable dam sites. At the same time, the serious reduction in storage capacity of existing reservoirs caused by sedimentation makes water resources management even more complicated. Unless reservoir operation can be improved drastically, water shortages will occur more frequently. It is thus natural that reservoir operation has emerged as an urgent and difficult task to be studied for the sustainable utilization of water resources in Taiwan.
In general, the vast majority of studies of reservoir operation technique rely on simulation and optimization approaches. Because of the rapid progress in computers, the optimization approach can now be carried out easily. Dynamic programming (DP) was developed originally by Bellman (1957). Dynamic programming consists of an objective function, which is assumed to be a set of decisions, and of the constraints, which represent the financial, physical and institutional limitations of the system. The objective function may be either maximized or minimized to meet the goal. In Taiwan’s reservoir operation, the objective function usually is the optimal amount of release from a reservoir. Dynamic programming offers a relatively flexible means of modelling and has been applied to solving problems of reservoir operation (Hall and Buras, 1961). However, the computation of DP was time consuming. Many methods were proposed to improve DP for reservoir operation. Bras et al. (1983) used non-stationary control stochastic dynamic programming (SDP) to present a real-time stream-flow forecast in reservoir operation. Datta and Houck (1984) combined DP and
* Correspondence to: F.-J. Chang, Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. E-mail: [email protected].
forecast error to form chance-constrained SDP for real-time forecasts. Stedinger et al. (1984) used SDP to calculate the expected benefits for future reservoir operation. Trezos and Yeh (1987) developed differential SDP for large-scale reservoir operation systems. Kelman et al. (1990) presented sampling stochastic dynamic programming (SSDP) by using a large number of sample stream-flow sequences to improve the reservoir operating policy. Karamouz and Vasiliadis (1992) developed Bayesian stochastic dynamic programming (BSDP), which continuously updated the probabilities with new incoming information to generate optimal operating rules. Archibald et al. (1996) developed an aggregate stochastic dynamic programming model to determine the policy of multireservoir operating systems.
The present study develops a grey fuzzy stochastic dynamic programming (GFSDP) model that consists of fuzzy arithmetic and a grey system for determining reservoir operating strategies. The method proposed is different from the mathematical programming pursuing the deterministic solution. It is not meaningful to have an optimal solution because of the ambiguities and uncertainties in objectives and constraints. The GFSDP model deals with the representative intervals of inflow and outflow using grey arithmetic and handles the multipurpose decision-making problem by fuzzy arithmetic. Grey arithmetic can be used to deal with uncertainties and to address the imprecision of the input and output, and the fuzzy arithmetic provides a suitable way to deal with ambiguity and uncertainty. Instead of black and white logic for grey relationships, the GFSDP holds that the information for reservoir operation is a matter of degree. All the possible hydrological and operating conditions also are considered when the model is established. Thus, GFSDP can behave as an adaptive system and create a set of more flexible optimal reservoir operating rules. In the following sections, the theories and formulae of the model are described first. It is then implemented for the Shiman Reservoir, Taiwan.
GREY SYSTEM
The grey system was introduced originally by Deng (1984). In this system, the information falls into three categories: white with completely certain information, grey with insufficient information, and black with totally unknown information. The grey system deals with the information (data) belonging to the grey category. Owing to insufficient information, most of the statistical characteristics of the system cannot be identified. However, the data available reveal the range of information. A grey number for the system can be expressed mathematically as
ašDaC, a
Dt 2 ašjat aC 1 where ašis a grey number; t is information; aCand aare the upper and lower limits of the information. Thus, the uncertainty in the parameters can be expressed by the grey number. The flexible solutions of the system can be obtained by transforming uncertainty by the grey number through stochastic dynamic programming processes. The operation of the grey number is defined as follows
ašbšD[mina, b,maxaC, bC]
where is an operator. The operation of the grey function can be defined as
[fa1š, aš2, . . . , ašn]šD f[faš1, aš2, . . . , ašn],[fa1š, aš2, . . . , ašn]Cg 3
where [faš1, a2š, . . . , aš
n]šis the value of the grey function; [fa š 1, a š 2, . . . , ašn]and [fa š 1, a š 2, . . . , ašn]C are the minimum and maximum values of the function. Equation (3) shows that faš1, aš2, . . . , aš
n is not a grey number, but aš1, aš2, . . . , aš
FUZZY ARITHMETIC
The notion of fuzzy sets was first proposed by Zadeh (1965) to represent vagueness in linguistics in a mathematical way. Fuzzy arithmetic provides a good approach to dealing with ambiguity and uncertainty. It combines rule base and fuzzy control to describe non-linear characteristics in nature. The rule base is the collection of rules. A rule contains two statements, the premise and the conclusion. It is a logical implication: IF premise THEN conclusion. For example, classic logic theory can represent only one colour in black and white. This logic does not accord well in grey. Fuzzy logic, which can be used to represent vague concepts, lets elements be represented by degrees of membership. The degree of membership is a positive real number in the interval [0,1]. A membership function assigns a degree of membership to an element and can be any shape. Using the centroid defuzzification method, the fuzzy control output can be determined as follows
IF a is Mi THEN yI 4 O Y D n iD1 Miayi n iD1 Mia 5
in which OYis the fuzzy control output, Mia is the degree of membership of the ith rule, a is the input, yi is the output of the ith rule and n is the number of the rule.
GREY FUZZY STOCHASTIC DYNAMIC PROGRAMMING
The forward-moving solution procedure is used for grey fuzzy stochastic dynamic programming. Figure 1 illustrates sequential reservoir operation processes. In each stage t, the inflow and release of the reservoir are It and Rt. The goal of the reservoir operation is to create the maximum benefit,
ft. The capacity of the reservoir at stage t, St, is the capacity at the end of stage t 1. Thus, the state transformation equation of non-invertible form for stage t is defined as
St DSt1CIt1Rt1Et1ASt1, St 6 where Et1 is the rate of evaporation and A(St1, St) is the water surface of the reservoir.
A stochastic dynamic programming with grey variables is called grey stochastic dynamic programming (GSDP). The recursive equation of GSDP that includes h objectives can be written
[FiSši , C š ]šD maximum(or minimum) f[fiSši , x š i , C š ]šC[FiC1SšiC1, C š ]šg 7 t=1 t=2 t S1 S2 S3 St St+1 f1 f2 ft ... I1 I2 It R1 R2 Rt Benefit Inflow Release State
in which fižis the objective function at stage i, xiis a decision variables C is a constant, such as temperature or water surface of reservoir, that can be input with an intervals SšiC1 is grey state variable at tC1 stage, and can be defined as SiC1š D[TiSši , x š i , C š ] 8
where Tižis a state transformation function that represents the storage of the reservoir at the end of stage i. Equation (7) can be split into the following two equations
[FiSi, Ii1]CD max (or min)
z
kD1
p[Ii,kjIi1][[fiSši , R š i , I š i,k] CC FiC1[SiC1š , I š i,k] C ] 9
[FiSi, Ii1]D max(or min)
z
kD1
p[Ii,kjIi1][[fiSiš, R š i , I š i,k] C FiC1[SšiC1, I š i,k] ] 10
where p[Ii,kjIi1] is the probability of Ii,k if Ii1 also occurs, p(Ii,k) is the probability of inflow in the ith stage given inflow condition k and z is the number of inflow conditions. In this study, z D 3 (high, normal and low water levels). The state transformation function of Equations (9) and (10) is
Rši,kDSši SšiC1CIši,k 11
where the evaporation loss, Et, is ignored to simplify the model. Equations (9) and (10) are the optimal solutions subjected to the best and worst conditions. The recursive relationship equation of the pth objective is as follows
pFiSi, Ii1 D max(or min)
z
kD1
p[Ii,kjIi1][pfiSi, Ri, Ii,k CpFŁiC1SiC1, Ii,k]
12
wherepFŁiC1SiC1, Ii,kis the optimal solution of objective p with SiC1 at stage i C 1 and p D 1, 2, . . . , h. For every objective of reservoir operation, each stage could have n possible states. Then n decisions for each stage would be created and represented by Chen’s fuzzy theory (Chen and Jiang, 1990). The pth objective with n possible decisions that can be used for determining water release becomes
pF1iSi, Ii1, S1iC1,pF2iSi, Ii1, S2iC1, . . . ,pFmi Si, Ii1, SmiC1, . . . ,pFniSi, Ii1, SniC1 13
where pFmi Si, Ii1, SiC1m DpFmi Si, Ii1 is the net benefit during period i when the decision m is made. Thus all the objectives of the model that can be the possible water release in stage i will be
Decision 1 Decision 2 . . . Decision m . . . Decision n
Objective 1 1F1iSi, Ii1 1F2iSi, Ii1 1Fmi Si, Ii1 1FniSi, Ii1 Objective 2 2F1iSi, Ii1 2F2iSi, Ii1 2Fmi Si, Ii1 2FniSi, Ii1
.. .
Objective p pF1iSi, Ii1 pF2iSi, Ii1 pFmiSi, Ii1 pFniSi, Ii1 ..
.
The weight of the mth decision of the pth objective in stage i can be calculated as primD pFmi Si, Ii1 n max mD1pF m i Si, Ii1 14 primD1 pFmiSi, Ii1 n max mD1pF m iSi, Ii1 15
Equation (14) should be used when the objective is to find the maximum benefit, such as hydropower generation. For finding the minimum benefit, i.e. the water deficit, Equation (15) is used in this model. Then, the best and worst solutions during period i and starting in state Si are defined by
Gi D n mD1 1rim, n mD1 2rim, . . . , n mD1 hrmi 16 D1 gi,2 gi, . . . ,hgi Bi D n mD1 1rim, n mD1 2rim, . . . , n mD1 hrmi 17 D1 bi,2 bi, . . . ,hbi
where [ and \, which are the union and intersection operators, are the max and min, respectively. Thus the fuzzification matrix with decision m during period i at state Si represents the Chen’s degree of membership of reservoir operation based on the best and worst solutions that can be derived
UiD u1,1 u1,2 . . . u1,m. . . u1,n u2,1 u2,2 . . . u2,m. . . u2,n 18
Ui is all of the Chen’s degree of membership based on the best and worst solutions during the ith stage. u1,m is the Chen’s degree of membership of the mth decision based on the best solution that is derived by calculating the fuzzification matrix by minimizing the distances between prmi and Gi and prim and Bi. The distance needed to be determined can be calculated by differentiating the equation, and is given by
min H D n mD1 u1,mdprmh, Gi2Cu2,mdprhm, Bi2 19
Equation (19) can be written
min H D n mD1 u2i,m h pD1 Wpprmi pgi2 C1 u1,m2 h pD1 Wpprimpbi2 20
where Wp is the weight for objective p;
h
pD1Wpprimpgi2 and
h
pD1Wpprmi pbi2 are the sum of squares of the distances between the mth decision and the best and the worst solutions. They should always be positive. Solving Equation (20) yields
u1,mD 1 1 C h pD1 Wpprimpgi2 h pD1 Wpprimpbi2 D 1 1 C dprim, Gi dprmi , Bi 2 21
where dprim, Gidenotes the distance between prim and Gi. The details of the solution from Equation (20) are given in the Appendix. The Chen’s degree of membership of the mth decision during period i at state Si based on the worst solution, u2,m, can be calculated by replacing ui,m by 1 ui,m.
After the optimal solutions of every stage based on z different inflows are obtained, fuzzy arithmetic is used to determine the release of the reservoir. Figure 2 describes the optimal solutions at different stages with k D 3 (low, normal and high inflows). Parameters G1, G2 and G3 are the initial gauge height at stage i C1 depending on the inflows of stage i 1 and the initial gauge height at stage i. Figure 3 shows the Gaussian membership functions for reservoir operation with three different inflows. The Gaussian function is given as Mk Dexp Qkk 2 k2 22
where k is the widths k is the centres Qk is the inflow and 1 k z. The degree of membership is determined by the inflow of the last month. Thus, the release of reservoir at stage i is determined by
GrD z kD1 MkGk z kD1 Mk 23
Initial gage height at stage i (m)
200 205 210 215 220 225 230 235 240 245
Initial gage height at stage
i+ 1 (m) 200 205 210 215 220 225 230 235 240 245
Low inflow at stage i-1 Normal inflow at stage i-1 High inflow at stage i-1
G1
G2
G3
Inflow 0 100 200 300 400 500 600 700 800 Membership at inflow 0.0 0.2 0.4 0.6 0.8 1.0
Membership function for low inflow Membership function for normal inflow Membership function for high inflow
M3
M2
M1
Figure 3. Membership at inflow
SDP Grey input
variable
Fuzzy arithmetic for grey solutions and multiobjectives
Optimal solution Best and worst solutions for
each objective
t=t+1
Figure 4. Structure of grey fuzzy stochastic dynamic programming
where Mk is the degree of membership with different inflows. Figure 4 shows the structure of GFSDP. It represents that the best and worst solutions of each objective are obtained through SDP and grey input variables. The fuzzy arithmetic is then used to deal with the grey solutions within each objective, and find the optimal solution of multi-objectives.
CASE STUDY
The Tanshui River, the third largest river in Taiwan, is situated near the metropolis of Taipei (Figure 5). The catchment area, which extends from high mountains (elevation 3530 m) to sea level, drains an area of 2762 km2 to the Taiwan Strait. As present water demands are continuously expanding, and probably will continue to do so, two reservoirs, Feitsui and Shiman, have been constructed. The Shiman Reservoir is located on the Tahan Creek, which is the upstream of the Tanshui River. Within the Tahan Creek catchment area the river is 135 km long and the average slope is 2.7%. The catchment receives an average (1964– 1988)
Taiwan Strait Pacific Ocean Tanshui River Taipei Feitsui Reservoir Shiman Reservoir N
Figure 5. Locations of the Tanshui River basin and the Shiman Reservoir
of 2409 mm of rainfall; however, the rainfall is uneven—over 70% occurs between May and October. The Shiman Reservoir was built not only for municipal and industrial water supply but also for irrigation, flood control and power generation. The operation plan of the Shiman reservoir is based on the M5 rule curve, as shown in Figure 6. The lowest, lower and upper curves are used for irrigation, power generation and food control, respectively. A careful evaluation of water use shows that the water needed to the year 2010 appears to increase drastically. Therefore, the authority of the Shiman Reservoir will need to develop an efficient operation policy for the challenge of future water demands.
The monthly inflow, release, gauge height and storage of the Shiman Reservoir measured from 1964 to 1993 are used for verifying the grey fuzzy stochastic dynamic programming. Table I summarizes the water demands in every month for the year of 2001. Table II describes the mean, standard deviation, and historical maximum and minimum of the monthly inflows of the Shiman Reservoir. Table III shows the parameters used for determining the membership functions. The boundaries of low, normal and high inflows are defined by containing the 20th percentile of the smaller historical data set, the middle 80% data, and the 20th percentile of the larger data set, respectively. Parameters 1 and 3 are set to be the lower and upper limits of their data, and 2 is the centre of the normal inflow data. Parameters 1 and 3 are set to be the 90% interval of the data, and 2 is 45%.
Two objectives, i.e. storage and release, are concerned in this case study. The objective of storage is relative to flood control, hydropower generation and recreation purposes, whereas the objective of release deals mainly
Time
Dec. Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
Gage height (m) 195 200 205 210 215 220 225 230 235 240 245 250 Upper cure Lower curve Lowest curve
Bottom of inactive pool at gage height 195 m Maximum design water surface at gage height 245 m
Buffer pool Conservation pool Flood control pool Surcharge pool
Figure 6. Rule curve for the Shiman Reservoir
Table I. Water demands in every month for the year 2001
Month Municipal and Irrigation Total
industrial supply (million m3) (million m3)
(million m3) January 484 206 690 February 453 515 968 March 518 738 1256 April 525 603 1128 May 580 568 1148 June 597 548 1145 July 644 752 1396 August 689 696 1385 September 606 576 1182 October 592 580 1172 November 519 303 822 December 508 32 540
with water supply for both public and agricultural users. The inflow, release and goal of storage are designed as grey variables. The objectives and constraints are shown as the follows
1. Objective equations
min[1FiSi, Ii1]C, min[1FiSi, Ii1] 24 and
min[2FiSi, Ii1]C, min[2FiSi, Ii1] 25 2. Recursive equations of the storage objective
[1FiSi, Ii1]CDmin
3
kD1
pIi,kjIi1
[1fi]CC[1FŁiC1SiC1, I1,k]C 26 [1FiSi, Ii1]Dmin 3 kD1
pIi,kjIi1
[1fi]C[1FŁiC1SiC1, I1,k]
Table II. Statistics of inflow data
Month Maximum Minimum Mean Standard
monthly inflow monthly inflow (million m3) deviation
(million m3) (million m3) (million m3)
January 964 223 446 179 February 3343 204 701 804 March 4732 254 946 891 April 3165 163 852 609 May 2049 318 1015 441 June 5316 793 1985 1110 July 3808 433 1441 1001 August 9067 360 2681 2269 September 8535 396 2934 2292 October 6719 401 1810 1392 November 2355 313 739 426 December 823 247 449 143 and [1fi]š Dmin SiXši 2 28
3. Recursive equations of the release objective:
2FiSi, Ii1 C Dmin 3 kD1 ! p Ii,kjIi1 " 2fi C C2FŁiC1 SiC1, I1,k C#$ 29 2FiSi, Ii1 Dmin 3 kD1 ! p Ii,kjIi1 " 2fi C2FŁiC1 SiC1, I1,k #$ 30 and 2fi š Dmin ri,kDi 2 š 31
4. The state transformation equations
ri,kC DSiSiC1CICi,k 32
and
ri,k DSiSiC1CIi,k 33
5. Constraints Dead storage Si XCi 34 SiCICi,kX C i < r C
i,k< SiCICi,kdead storage 35 and
SiCIi,kX i < r
i,k< SiCIi,kdead storage 36 where XCi and Xi are the storage of the upper and lower curves in rule curve M5 at stage i, respectively; Di is the water demand at stage i, i D 1, 2, . . . ., T; T is the total number of the operating stage, which is equal to 12 in the case study. The definition of other variables can be found in previous sections.
T able III. The p arameters in the case study Month L ow inflow N ormal inflow High inflow Minimum M aximum 1 1 Minimum M aximum 2 2 Minimum M aximum 3 3 (10 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3)( 1 0 6 m 3) January 0 292 0 263 220 697 459 413 622 923 923 271 February 0 303 0 273 189 1251 720 648 1088 1445 1445 321 March 0 446 0 401 219 1777 998 898 1553 2055 2055 452 April 0 439 0 395 206 1560 883 795 1359 1809 1809 405 May 0 681 0 613 483 1609 1046 941 1447 1952 1952 455 June 0 1178 0 1060 804 3182 1993 1794 2835 4075 4075 1116 July 0 716 0 644 392 2587 1490 1341 2248 2992 2992 670 August 0 1377 0 1239 785 4754 2770 2493 4028 5582 5582 1399 September 0 1495 0 1346 814 5511 3163 2846 4737 6180 6180 1299 October 0 922 0 830 552 3334 1943 1749 2931 3880 3880 854 November 0 419 0 377 250 1283 767 690 1141 1564 1564 381 December 0 337 0 303 277 626 452 406 562 813 813 226
The performances of the proposed algorithm and M5 rule curve for the operation of the Shiman Reservoir are evaluated using the generalized shortage index (GSI) (Hsu, 1995), given by
GSI D 100 N N iD1 % DPYi 100 ð DYi &2 37
where N is number of sample years, DYi is number of the ith year (365 or 366) and DPYi is the deficit per cent year index of the ith year, given by
DPY D 12
jD1
MDR ð NMC 38
where MDR is the monthly deficit rate (%) and NMC is number of months in a continuous deficit. The GSI is an indicator of the social tolerance limits to water shortage that measures the monthly deficit rate. Under this criterion the model that produces the minimum GSI is the one with a better performance.
Figure 7 compares the monthly deficit resulting from the operation under the M5 rule curve with GFSDP. It shows that most of the points fall below the agreement line and the points that fall on the y-axis are less than those on the x-axis. The monthly deficit of GFSDP usually is observed to be smaller than the M5 rule curve. By using GFSDP the number of monthly deficits is reduced from 127 to 90. The GSI obtained by GFSDP is 0Ð0001, which is much smaller than 0Ð278 obtained by the M5 rule curve. It also indicates that GFSDP appears to provide better results. Figure 8 illustrates the yearly deficit obtained by both GFSDP and rule curve M5 and the difference between the two methods. The yearly deficit amount obtained by using GFSDP is much less than by rule curve M5 in most of the 29 years. There are only 4 years where the yearly deficit obtained by using the rule curve M5 is slightly less than by GFSDP. During the study period, the total deficit will be reduced significantly from 54Ð48 billion m3 to 32Ð42 billion m3 if GFSD is used. Both Figures 7 and 8 and GSI indicate that the performance of GFSDP is much better than the actual operation in reducing the total deficit volume, and the number of monthly and yearly deficit volumes.
Deficit by rule curve M5 (million m3)
0 200 400 600 800 1000 1200 Deficit by GFSDP (million m 3) 0 200 400 600 800 1000 1200
1964 1968 1972 1976 1980 1984 1988 1992 Deficit (million m 3) 0 1000 2000 3000 4000 5000 6000 Rule curve M5 GFSDP Year 1964 1968 1972 1976 1980 1984 1988 1992
Deficit reduction (million m
3) −3000 −2000 −1000 0 1000 2000 3000
Figure 8. Comparison of model results and deficit reduction
CONCLUSIONS
A real-world reservoir operation can be very complex. It has to meet various demands without violating the constraints of the system. In addition, the uncertainties and ambiguities always arise from the system. Fortunately, uncertainty and ambiguity can be dealt with more efficaciously using grey and fuzzy arithmetics. In Taiwan, water shortage is a major socio-economic problem. The grey and fuzzy arithmetics are combined with the optimization approach to develop GFSDP for the improvement of reservoir operation. The grey stochastic dynamic program presents an interval of the solution, and the fuzzy arithmetic carries out the solution through the membership function. They can successfully communicate the uncertainty and ambiguity of the complex natural system to the optimization process. Making a careful choice of and for input variables (inflow) is important, because they are sensitive to the model. The states of release for each stage can be divided according to the objectives and as estimated by Chen’s fuzzy theory. The parameters can be estimated easily. However, the model will be influenced by the number and boundary of states. To verify the proposed model, GFSDP is used to simulate the operation of the Shiman Reservoir. The results show that GFSDP is an efficient and flexible tool for reservoir operation. It can significantly improve on the water release and reduce the water deficit values compared with the existing M5 rule curve.
ACKNOWLEDGEMENTS
This paper is based on partial work supported by National Science Council, Taiwan (Grant number NSC 89-2313-B-002-239).
APPENDIX
In this study the parameters are determined by differentiating the following equation
H D n mD1 u21,m h pD1 wp primpgi 2 C 1 u1,m 2 h pD1 wp primpbi 2 A1
Differentiating Equation (A1) yields
dH du1,m D2u1,m h pD1 wp primpgi 2 C h pD1 wp primpbi 2 2 h pD1 wp prmi pbi 2 A2 d2H du2 1,m D2 h pD1 wp primpgi 2 C h pD1 wp primpbi 2 2 A3
Equation (A3) is a convex function. The minimum point is always found when dH du1,m D0. Thus we obtain u1,mD 1 1 C h pD1 Wpprimpgi2 h pD1 Wpprimpbi2 A4 REFERENCES
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