Application of grey linear programming to short-term hydro scheduling

Electric Power Systems Research 41 (1997) 159-165

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Application of grey linear programming to short-term hydro scheduling Ruey-Hsun Liang Department of Electrical Engineering, National Yunlin Institute of Technology, Touliu, Yunlin 640, Taiwan, ROC

Received 21 June 1996; accepted 8 July 1996

Abstract

An approach using linear programming based on grey constrained interval is applied to solve the short-term hydro scheduling problem. A characteristic feature of this approach is that the errors in the forecast hourly loads and natural inflows can be taken into account by using grey number notation, making the approach superior to the conventional linear programming method in which the hourly loads and natural inflows are assumed to be exactly known and there are no errors in the forecast loads and natural inflows. To reach an optimal scheduling under the uncertain environment, a grey linear programming model in which the hourly loads, the hourly natural inflows are all expressed in grey number notation, is developed. The developed grey linear programming approach is applied to schedule the generation in the Taiwan power system. It is found that the proposed approach is very effective in obtaining proper hydro generation schedules in uncertain conditions. © 1997 Elsevier Science S.A. Keywords: Grey linear programming; Grey number; Hydro scheduling

I. Introduction

The major objective of short-term hydro scheduling in a power system is to minimize the total fuel cost of thermal units by utilizing the limited water resource. It is a typical optimization problem in which the total operating cost over the study period is minimized subject to load and all system constraints. Numerous approaches have been proposed for solving the short-term hydro scheduling problem [1-9]. In the conventional approaches to the hydroelectric generation scheduling problem, the power generation-load balance equations and the water balance equations must be maintained at each hour over the study period. The system load demands and natural inflows in the power generation-load balance equations and water balance equations respectively must be known in advance, before the short-term hydro scheduling problem can be tackled. Load demands and natural inflows can only be known through short-term forecasting. Since load demands and natural inflows depend on the social behaviour of customers, weather variables, etc. there are always errors in the forecast system loads and natural inflows. This raises the question of how to 0378-7796/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0378-7796(96)01 128-5

tackle the short-term hydro scheduling problem when the load demands and natural inflows are imprecise. In this paper, an approach using grey linear programming is proposed to reach the desired hydro generation schedules based on uncertain load demands and natural inflows. Grey systems [10-14] are systems which lack information, such as architecture, parameters, operation mechanism and system behaviour. Grey system theory was established by professor Deng Julong of Huazhong University of Science and Technology in 1982 [10]. It is a recent theory that deals with poor, incomplete, or uncertain problems of systems. Currently the grey system theory has many successful applications in China, in areas such as economics, agriculture, medicine, geography, earthquakes, industry, etc. More details on the grey system theory and its applications can be obtained in Refs. [11-13]. The idea is to replace the concept that each variable has a precise value by the grey concept that each variable is assigned a degree of membership for each possible value of the variable. In the hydro scheduling problem, to reach an optimal generation schedule under an uncertain environment, load demands and natural inflows are

R.-H. Liang / Electric Power Systems Research 41 (1997) 159 - 165

160

all expressed in grey number notation. Grey linear programming [13] is then used to obtain the desired generation schedule. The effectiveness of the proposed grey linear programming approach is demonstrated by the hydro scheduling of Taiwan power system.

Yit+ l = Yit ~- E X~t - fl(it-~- E S l t - Sit-~ Rit ]ENi l~Ni

i = 1, 2,..., I; t = l , 2 ..... 24 (3) bounds on water releases Xi, min ~ S/t ~ Xi . . . .

and

Si.min ~ Sit ~ Si . . . .

i = 1, 2 ..... I

2. Problem formulation

24

C = Y~ C O S T , ( G T H E R M A L , )

(4)

(4) bounds on reservoir storage

Scheduling hydro generation is well known to be coupled with its thermal counterpart. Under present Taiwan power Company (TPC) system operation conditions, short-term commitment changes are not allowed for most of the base-load and medium-load thermal units. The scheduling of hourly thermal generation is thus reduced to an economic dispatch problem, in which the thermal unit commitment aspect can be ignored. To meet the spinning reserve requirements, TPC keeps a certain percentage of the available thermal capacities (committed units and all the peaking units) as the spinning reserve. According to the aforementioned operation of the TPC system, the coupling between hydro and thermal generation scheduling is through the constraint that total generation should meet the system load. We decouple the hydro scheduling from the thermal part by first establishing the generation cost function of meeting the system load by purely thermal generation. For each given load level, the lambda-iteration method is performed to solve the economic dispatch over the set of available units [1] and to evaluate the thermal generation cost in order to meet the load demand. In other words, we aggregate all the available thermal units into one equivalent unit and construct its generation cost function. Then, in hydro scheduling, we try to find the best way of substituting hydro for thermal energy based on this function so that the system generation cost is minimized. In the TPC hydro system, there is no significant delay relative to the 1 h time increment for water to flow from one reservoir to its immediate downstream neighbour. To do this, the study period (1 day for the present work) is divided into N stages (N = 24 in the present case) and the hydro scheduling problem is then formulated as follows: minimize

(3)

(1)

Yi,minK-YitK-Yi

.....

i = 1 , 2 ..... I

(5)

where C = system generation cost over the study period; L, = system load at hour t; COSTt(.) = generation cost function at hour t which is approximated by a second order polynomial; G T H E R M A L , = total generation from thermal units at hour t; Y, = the water volume of reservoir i at the beginning of hour t; X, = the volume of water released from reservoir i for generation during hour t; S, = the spillage from reservoir i during hour t; Pi(') = t h e water-to-energy conversion function of the power plant associated with reservoir i; R, = the volume of natural inflow to reservoir i during hour t; N, = the set of the immediate upstream reservoirs of reservoir i. Note that Yi~ and Yi25 are the given initial and final water volume of reservoir i. To deal with the optimization problem, both dynamic programming and linear programming can be employed. In the present work, the linear programming approach is used to tackle the hydro scheduling problem. Since the convex objective function in Eq. (1) is a nonlinear function of the variable G T H E R M A L , it is necessary to linearize this function first in the linear programming formulation. Fig. 1 depicts a piecewise

F~ r.,o ©

t=l

subject to

/

(1) the generation-load balance equations G T H E R M A L , + Y, P i ( X . ) = L, i

(2) the water balance equations

(2)

G T H E R M A L (thermal generation) Fig. 1. Thermal generation cost function approximated by a multisegment piecewiselinear curve.

161

R.-H. Liang /Electric Power Systems Research 41 (1997) 159-165

linear approximation of the cost curve which is originally represented by a second-order polynomial. The hydro scheduling problem c a n now be formulated as a general linear programming (LP) problem. min C = C~X

(6)

Cho-Shui River

Sun-Moon

t

Lake Ta-Kuan 1

subject to AX < b

(7)

x > 0

(8)

Ta-Kuan2

\

Chu-Kung /

where C = total cost to be minimized; _Xt = [x~..... x,] = vector of control variables; C t = [cx..... c,] = vector of cost parameters; b t = [b~..... bin] = constraint vector; A = [a0] = constraint matrix. In conventional linear programming formulation, the elements of constraint vector _/2 are all crisp numbers. The crisp inequality constraints can be replaced by grey numbers since we consider uncertain load demands and natural inflows. In this paper, the grey linear programming formulation to be described in the following section must be used instead of the conventional linear programming formulation.

(~)

: NaturalInflow : Reservoir ] : PowerHouse

Ta-Chia River

Te-Chi

3. The proposed grey linear programming approach In this section, a grey linear programming (GLP) model [13] is developed based on grey constraint. Grey number formulation is used because we want a good solution to a model that represents the uncertainties inherently existing in a practical optimization problem. In the proposed G L P approach, the power generationload balance equations and water balance equations in Eqs. (2) and (3) are treated as grey constraints since they are related to the imprecise hourly loads L and natural inflows R. The hydro scheduling problem originally formulated in Eqs. (6)-(8) under crisp conditions must now be reformulated under uncertain environments. maxf(_X) = - C ' X

(9)

Chin-Shan

Ku-Kuan

/

\

F-

\

Tien-Lun /

T Fig. 2. Schematic diagram of Cho-Shui River and Ta-Chia River.

subject to AX < ®(b)

(lO)

X>0

(11)

~(b~)e[b~,/~]

i = 1,2 ..... m

(12)

where ® indicates grey number which represents a number with incomplete information and Eq. (10) is a grey constraint. The elements of the grey constraint vector in Eq. (12) represent the range of forecast hourly load and natural inflow under uncertain environments. To solve the above-mentioned hydro scheduling

problem formulated in Eqs. (9)-(12) under grey conditions, grey linear programming is developed. 3. I. Make the membership function of the grey constraint Define that 0
i = 1, 2 ..... 264

(13)

where/~i(d~) is strictly monotonous increasing function, and 0 < dr < ~. - b / ( i = 1..... 264) which consists of the

R.-H. Liang /Electric Power Systems Research 41 (1997) 159 165

162 Table 1 Hydro system data Reservoir

Sun-Moon Storage Pond Chu-Kung Te-Chi Chin-Shan Ku-Kuan Tien-Lun Li-Wu Lung-Chen I-Hsing

24 h o u r l y

Storage lower bound

Storage upper bound

(kin 3)

(kin 3)

13 269 1565 1.6 89 996 26 101 90 0 0 0

155 685 9407 105 243 120 647 6563 560 340 202 1343

loads

L, ( t = 1 . . . . . 24)

Plant

Ta-Kuan 2 Ta-Kuan 1 Chu-Kung Te-Chi Chin-Shan Ku-Kuan Tien-Lun Li-Wu Lung-Chien I-Hsing

and

240

(24 x 10)

Water release lower bound

Water release upper bound

(m3/'s)

(m3/s)

-249 0 0 0 0 0 0 0 0 0

380 50 45 217.5 174.8 133.6 68 36.7 13.2 31.7

d7 = max{Cld, eN,(x,)}

(21)

n a t u r a l i n f l o w s R,, (i = 1. . . . . 10,; t = 1. . . . . 24).

N~(x,) = {d,l#,(d, ) > c~}

i = 1, 2 . . . . . 264

(22)

3.2. Normalize the grey objective 3.4. Solution to the grey linear programming The

grey

objective

is

normalized

in

the

interval R e l a t e d t o t h e level set C~, s u p X c ( x ) c a n b e s o l v e d

[0, 1], t h a t is,

by LP

f(x)

and

by solving the GLP

with the following

(14)

steps.

V = max - C'X

(15)

Table 3 Results from the linear programming (LP) approach and the grey linear programming (GLP) approach

k = {_XIA_X< b*, _X_> 0}

(16)

Hour (h)

b * = b_,.+ d *

(17)

V

where .v~k

d * = m a x { d i e s u p p,(di)}

i = 1, 2 . . . . . 264

(19)

where

(20)

b 7 = b, + d~' Table 2 Forecast hourly loads Hour

Load (MW)

Hour

Load (MW)

1 2 3 4 5 6 7 8 9 l0 11 12

7624.3 7340.2 7096.7 6925.7 6818.0 6761.7 6916.0 7466.9 9461.0 10 192.0 10 531.0 10 619.6

13 14 15 16 17 18 19 20 21 22 23 24

9567.0 10 743.7 10 982.3 10 822.9 10 560.2 9726.0 9683.1 9897.0 9690.0 9294.0 8968.2 8475.8

Hydroelectric generation (MW)

LP

GLP

LP

GLP

7624.3 7340.2 7096.7 6925.7 6818.0 6761.7 6916.0 7466.9 9461.0 10 192.0 10 531.0 10 619.6 9567.0 10 743.7 10 982.3 10822.9 10 560.2 9726.0 9683.1 9897.0 9690.0 9294.8 8968.2 8475.8

7616.0 7332.2 7088.9 6918.1 6810.5 6754.3 6908.4 7458.7 9450.6 10 180.8 10 519.0 10 608.0 9556.5 10 7 3 1 . 9 1'0 970.3 10811.0 10 5 4 8 . 6 9715.3 9672.5 9886.2 9679.4 9284.6 8958.4 8466.5

- 100.2 -677.6 - 693.3 - 749.6 - 749.6 -749.6 - 749.6 - 119.1 323.3 788.5 1468.7 1851.1 877.0 1851.1 1851.1 1851.1 1849.8 755.8 629.6 327.0 263.6 203.0 10.6 -709.6

-88.9 -677.6 - 686.8 - 749.6 - 749.6 -749.6 -749.6 - 127.3 324.7 788.6 1478.0 1851.1 866.5 1851.1 1851.1 1851.1 1851.1 759.6 665.5 327.5 277.3 204.3 24.1 -709.6

(18)

3.3. Define ~ level set, C~ C~ = {_XIAX_Q}

Load demand (MW)

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

:~ = 0.963496. Total cost from LP equals 123 163 560 NT$ = 1.0 pu. Total cost from GLP equals 122 881 088 NT$ = 0.997707 pu.

R.-H. Liang / Electric Power Systems Research 41 (1997) 159 165

80

163

i

70

60

~40

30

20

10

0

15

20

25

time (hour)

---o--- : LP

*

: GLP

Fig. 3. Water release for Te-Chi hydro plant f r o m LP and GLP.

Step I: Let n = 1, and e, el be given; Step 2." Solve X ~ ) = supx~c XG(x); Step 3: Solve 6 = c~,,- X~),~]f [e,,I > e, and go to Step 4, otherwise, go to Step 5; Step 4: Find ~,, + ~ = c~. - 7.e,,, 5, ~ [0, 1], let n = n + 1, and go to Step 2; Step 5: Let ~ = ~,,, solve sup.,.~c, XG(x)= XG()() to obtain )?, and STOP. Note that ~ must lie within the range [0, 1] because all membership functions are within this range.

4. Example To demonstrate the effectiveness of the proposed grey linear programming approach, hydroelectric generation scheduling is performed on the Taiwan power system which consists of four Ta-Chia River cascaded plants, three Cho-Shui River plants and three hydraulically independent plants. A schematic diagram of the hydro plants along both the Cho-Shui River and the Ta-Chia River is shown in Fig. 2. The hydro system data used for the present work are presented in Table 1. The forecast hourly loads are listed in Table 2. The computation results are summarized in Table 3 and Figs. 3 and 4. Table 3 gives the hourly load

demands, the total power from the hydro units and the production costs for the optimal hydro generation schedules reached by using the linear programming and the proposed grey linear programming. It is observed from Table 3 that different hourly load demands have been assumed in the two approaches. In the LP approach the forecast hourly loads are directly employed. But these load demands have been modified in the GLP approach by taking the uncertainties in the load demands into account. By comparing the total costs from LP and GLP in Table 3, it is observed that the total cost for the generation schedule from the proposed GLP approach is less than for the generation schedule from conventional (crisp) LP. The main reason for the difference in the resultant production cost is that the uncertainties in load demands and natural inflows are taken into account in the proposed grey linear programming approach. The reduction in production cost is achieved by taking advantage of the uncertainties in load demands and natural inflows. The observation is confirmed by the different hourly loads and hourly hydro generation in Table 3. The negative hydroelectric generations indicate the pumping operations for the Ta-Kuan 2 pumped-storage plant during the light load period (a low fuel-cost period). The hydro plants are operated to

164

R.-H. Liang /Electric Power Systems Research 41 (1997) 159 165

5O 45 40 35 ul (...)

30

~25

~ 20 15 10

0 0

5

15

10

20

25

time (hour) Fig. 4. Water release for Ta-Kuan 1 hydro plant from LP and GLP.

serve the peak load (a high fuel cost load) with hydro energy, which is usually referred to as the peak shaving operation. Figs. 3 and 4 illustrate the water release schedules for two of the ten reservoirs over the 24 h scheduling periods. It is observed from Figs. 3 and 4 that different water release schedules have been obtained by using the two approaches since different hourly inflows are assumed by the two approaches. In the LP approach, the forecast hourly inflows are directly employed. But these natural inflows have been modified in the GLP approach by taking the uncertainties in the natural inflows into account.

system which consists of ten hydro plants. Results demonstrate that the proposed grey linear programming is very effective in reaching an optimal hydro generating schedule when the imprecision in the hourly loads and natural inflows in considered.

Acknowledgements Financial support given to this work by the National Science Council of ROC under contract NSC 85-2213E-224-016 is appreciated.

References 5. Conclusions A technique using linear programming based on a grey constrained interval has been proposed for short term hydro scheduling of a power system. A standard LP algorithm can be applied to the grey formulation. To take the errors in both forecast hourly loads and inflows into account, membership functions are derived for the load demand and the natural inflow. With these membership functions at hand, an algorithm for grey linear programming is presented to reach the optimal hydro generation schedule under an uncertain environment. The developed algorithm is applied to the hydroelectric generation scheduling of the Taiwan power

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[7] S.C. Chang, C.H. Chen, 1.K. Fong and P.B. Luh, Hydroelectric generation scheduling with an effective differential dynamic programming algorithm, IEEE Trans. Power Syst., P W R S - 5 (1990) 737 743. [8] R.H. Liang and Y.Y. Hsu, Hydroelectric generation scheduling using self-organizing feature maps, Electr. Power Syst. Res., 30 (1994) 1-8. [9] R.H. Liang and Y.Y. Hsu, Scheduling of hydroelectric generations using artificial neural networks, lEE Proc.-Gener. Transm. Distrib., 14 (1994) 452 458.

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