Md.S.Salam K.M. Nor A.R. Hamdan

Indexing terms: Hydrothermal scheduling, Lagrangian velaxation, Hydvothemal co-ordination

Abstract: The authors present a comprehensive hydrothermal co-ordination algorithm where a new Lagrangian relaxation based hydrothermal co-ordination algorithm is integrated into an expert system. In this algorithm, the problem is decomposed into the scheduling of individual units by relaxing the demand and reserve requirements using Lagrangian multipliers. Dynamic programming is used for solving the thermal subproblems without discretising generation levels. Instead of solving the hydro subproblems independently as in the standard Lagrangian relaxation approach, hydrothermal scheduling is used to solve the output levels of hydro units. Hydrothermal scheduling uses the commitment status of thermal units obtained from the solutions of the thermal subproblems. The expert system takes care of constraints that are difficult or impractical for implementation in the Lagrangian relaxation based hydrothermal co-ordination algorithm, such as cycling of gas and steam turbine units, etc. It is also applied to check the feasibility of the solution. Extensive constraints such as power balance, spinning reserve, minimum up/down time, must run, capacity limits, ramp rate and hydro constraints are considered. Accurate transmission losses are incorporated. Nonlinear cost function is used, and the hydrothermal scheduling is implemented using a fast and efficient algorithm. Numerical results based on a practical utility data show that this new approach provides feasible schedules within a reasonable time.

List of symbols

M , H = number of thermal and hydro units, respec-

T = number of periods for dividing the scheduling tively

time horizon 0 IEE, 1997 IEE Proceedings online no. 19970819 Paper first received 25th March 1996 and in final revised form 1st March 1997 Md. S. Salam and K. M. Nor are with the Electrical Engineering Department, University of Malaya, 50603 Kuala Lumpur, Malaysia A.R. Hamdan is with the Faculty of Information Science & Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

i = index of the thermal unit h = index of the hydro unit t = time index P = MW power output of a generating unit P = maximum MW power of a generating unit P = minimum MW power of a generating unit CL(.), SI(.) = production cost and start-up cost func-

-

x x = 2

x = -3

U

D, R, r r -

Ai

Tu mini Td mini

4h(.)

qtoth

si,

Sh

h

tions of the ith thermal unit, respectively

unit has been in service for 2 successive periods unit has been shutdown for 3 successive periods

= commitment state, when U = 1, unit is on when U = 0, unit is off

= demand in period t = MW spinning reserve in period t = spinning reserve contribution = maximum spinning reserve contribution of

= ramp rate maximum permitted for the ith

= minimum up time of the ith thermal unit = minimum down time of the ith thermal

= water flow rate function for the hth hydro

= prespecified volume of water available for

= initial volume of water of the reservoir of

= final volume of water of the reservoir of

= water inflow rate of the hth hydro unit

= state variable e.g.

a generating unit

thermal unit

unit

unit

the hth hydro unit

the hth hydro unit

the hth hydro unit

B, B l , BO = loss coefficients Ploss, 60, bl , b2 = coefficients for thermal generation cost

function h, ,ii = vector of Lagrangian multipliers associ-

ated with power balance and spinning reserve constraints, respectively

= tth component of A, E respectively

= transmission loss in period t

At, ,ut

1 introduction

The hydrothermal co-ordination (HTC) problem deter- mines the thermal unit commitments and generation

IEE Proc-Gener Transm Distvib., Vol. 144, No. 5, September 1997 482

dispatch, as well as the hydro schedules, to meet the forecasted demand and other operating constraints at minimum thermal production cost.

Until recently, the HTC problem for large-scale power systems has been solved in practice using heuris- tic methods to make the problem computationally fea- sible. The Lagrangian relaxation method offers a new approach for solving such problems [l-31. It eliminates the additional assumptions imposed by those heuristic methods. By using Lagrangian multipliers, the Lagrangian relaxation method relaxes demand and reserve requirements and thus generates a new problem known as the dual problem. The dual problem is then decomposed into the scheduling of individual thermal and hydro units. To maximise the dual function, the multipliers are adjusted iteratively. A gap, known as a duality gap, is created between the solutions of primal and dual problems by the relaxation of constraints, so the dual optimal solution infrequently satisfies the power balance and reserve constraints. Therefore, a suboptimal feasible solution is usually searched near the dual optimal point. The search method is an itera- tive process where relaxed subproblems are solved and Lagrangian multipliers are updated according to the extent of violation of the power balance and the spin- ning reserve constraints.

In the standard Lagrangian relaxation method (SLR), the thermal subproblem is solved to obtain the commitment states as well as the tentative output levels of thermal units, whereas the hydro subproblem is solved to obtain the tentative output levels of hydro units [l-31. During the updating process of Lagrangian multipliers, these tentative output levels of thermal and hydro units are used. The hydrothermal scheduling determines the final output levels of thermal and hydro units. Another improvement was suggested by Guan et al. [4] where the individual thermal subproblem is solved independently using dynamic programming without discretising generation levels to obtain the commitment states and the output levels of that partic- ular thermal unit at all intervals.

So far, all proposed formulations have ignored trans- mission losses and used linear or piecewise linear ther- mal cost functions. In normal problem formulation, the transmission losses are usually estimated and incorpo- rated in the demand value. If the primal problem is solved using variable losses, the duality gap will widen. The weak co-ordination between hydro and thermal generation has led to suboptimal solutions. Solving hydro and thermal subproblems independently seems to contribute to this weak co-ordination.

Since the Lagrangian relaxation technique handles only constraints expressable in mathematical form, some constraints cannot be included in the problem formulation; and to keep the execution time require- ment within a reasonable range, all complex operating constraints, although they are not violated frequently, are generally relaxed in the problem formulation. The relaxation of these constraints may lead to infeasible solutions in some cases [ 5 ] . Hence, a number of ques- tions concerning the results are habitually raised by power system operators. Analysis by an HTC program- ming expert will be required to answer these questions. Some of the data involved is heuristic in nature, and a small change in this data often makes a significant change to the result. Experienced operators using the HTC program results as a starting point could adjust

IEE ProcGener. Trunsm. Distrib., Vol. 144, No. 5, September 1997

heuristic data and input parameters to improve them. An expert system which contains the knowledge of both an HTC programming expert and an experienced power system operator can be used as a post-processor to the Lagrangian relaxation based HTC program to check and modify commitment results. By iterating between the expert system and the Lagrangian relaxa- tion technique, this approach may help to obtain a fea- sible and/or preferable optimal schedule [6-8].

The authors report a new Lagrangian relaxation approach. Nonlinear functions are used for thermal generation cost and water discharge characteristics. An efficient hydrothermal scheduling algorithm is used to solve the output levels of hydro units [9]. A refinement algorithm has been developed and used to fine tune the schedule. An exact variable loss equation is used in the primal problem. In the dual problem, the loss formula- tion is variable except for the cross-coupling terms between units. This allows the duality gap to be nar- rowed, and more accurate schedules are obtained. An expert system similar to that presented in the authors’ previous work [7, 81 has been developed by combining the knowledge of experienced power system operators and HTC programming experts. The developed expert system is used as a preprocessor as well as a postproc- essor to the Lagrangian relaxation based HTC pro- gram to check and modify results. The constraints that are difficult or impractical for implementation in the Lagrangian relaxation based HTC algorithm are han- dled by this expert system. Results of the implementa- tion on a practical utility data show that the new approach produces feasible schedules within reasonable time.

2 Problem formulation

2. I Objective function and constraints The objective of HTC is to minimise the system’s total operating cost. This includes fuel cost and start-up cost and is given as:

T / M

t=l \i=1

(i) Power balance constraints The constraints to be considered are as follows:

M H

i=l h=l (ii) Spinning reserve constraints

M H

z = 1 h=l

(iii) Capacity limits of generating units

et L Pz,t I p, 0 I Ph,t 5 Ph

(P,,t+l - A,) L P,,t I (PZ,t+l + A,)

(4)

(5)

pz,t = e, (6)

x,,t 2 Tumin, (7 )

x,,t 2 Tdmin, ( 8 )

(iv) Ramp rate constraints

(v) Minimum generation for the 1st and last hour

(vi) Minimum up time constraints

(vii) Minimum down time constraints

483

(viii) Hydro constraints T

t=l where

C, and q12 are considered as quadratic functions of power output, and Si is a function of down time. The transmission loss is represented using a general trans- mission loss formula [lo] whose expression has a simi- lar quadratic form to the B matrix loss formulation as follows:

qtoth = SZh - S f h f h * T (10)

M+H M+H M+H

s=l k = l s=l

The unknown variables are the commitment states of thermal units and the output powers of thermal and hydro units.

3 Proposed approach

3. I Lagrangian dual problem In the Lagrangian relaxation approach, the HTC prob- lem defined by eqns. 1 to 11 is known as the primal problem, and the unknown variables are denoted as the primal variables. In the primal problem, only the power balance constraints, (eqn. 2) and the spinning reserve constraints, (eqn. 3) are coupling constraints that link the operation of the generating units. These coupling constraints are relaxed by adjoining them onto the cost function using two sets of Lagrangian multipliers h and ,E with the components of ,ti (i.e. ,ut) confined to be positive. If we use the transmission loss represented by eqn. 11 in eqns. 2 and 3, we will find difficulty in separating the Lagrangian dual problem into thermal and hydro subproblems because of the off-diagonal loss coefficients. We can overcome this problem by carrying out the following modifications.

M+H M+H M+H

The transmission loss:

s= l k=l s=l

M+H M+H M+H

s= 1 k = l s=l

M+H r~ M+H 1

s=l Lk=l k=M+l

M+H

+ B1,P,,t +BO s=l

M r M 1

M

s=M+l Lk=1 k=M+1

M-CH

Now we diagonalise the equation by neglecting all the terms with B,y,k, s # k coefficients. The resulting equa- tion for transmission loss becomes:

M M M+H

s=l s=l s=M+l

s=M+1

This results in the following Lagrangian dual problem:

where max F(X, I) with all put 2 0 (14)

/ M

\i=1 H \

h=l / / M

..

+ C[Fh - PEBh,h - PhBlh] 1 h=l 1

subject to eqns. 4 to 10. Although Bs,k, s z k coefficients have been neglected

in the formulation of the dual problem, the same has not happened in the primal problem i.e. during check- ing the violation of the power balance and the spinning reserve constraints.

3.2 Solution of the dual problem The solution process to the dual problem is an iterative that has the following two major steps: (i) F(h, ,ti) is determined by minimising the right-hand side of eqn. 15 for a fixed setting of Lagrangian multi- pliers. This proposes a solution for the primal varia- bles. (ii) The convergence test is performed. If all the conver- gence criteria are fulfilled, the dual optimal is found. Otherwise, the multipliers will be updated and step (i) will be performed.

The variable metric method has been used for updat- ing the Lagrangian multipliers while maximising the dual function [l].

If the constant terms are removed, eqn. 15 can be separated into two sets of subproblems where each sub- problem deals with only one generating unit. The clas- sification of the subproblems is based on the types of generating units, as follows: (a) Thermal subproblems:

min L

with T

s=M+l

484 IEE ProcGener. Transm. Distrib., Vol. 144, No. 5, September 1997

3.3 Solving thermal subproblems Guan et al. [4] have proposed an improved method that solves the thermal subproblem independently using dynamic programming without discretising generation levels. In our approach, the formulation is modified to take into account noiilinear functions for thermal gen- eration cost and variable transmission losses. Two types of thermal units, steam turbine and gas turbine, are considered. Some of the thermal units even have ramp rate constraints. This variation of thermal units calls for various solution methods as described below.

The solution method for a thermal (steam turbine unit) subproblem without ramp rate constraints is pre- sented first. For the objective function in eqn. 16 with a fixed setting of h and p, the non start-up cost is defined as:

.fi(pi,t,xi,i) ~ C i ( P i , t ) - Xt[Pi,t - P:tB,,, - R, tBl t ]

- Pt[G, l + pi,t - (G, t + Pi,t)%,Z

- ( h , t + E , t ) B L ] ( 1 8 )

(19)

Eqn. 16 can be rewritten as: T

Li = C[fi(Pi,t,.i,t) + Si(.i,t, Ui,t)I t=l

We know that Li is step-wise additive, there are no dynamics on the generation levels, and the start-up cost Si (xi,,, U,,) is independent of generation Pi,(. Hence, the optimal generation level PTt at time t for an up state > 0) can be obtained by minimisingJ1 (Pi,,, xi,,) subject to the first and last hour generation con- straint, (eqn. 6). That is,

P,Tt = argmin .fi(pi,t, q t ) (20 ) p, ,t

Substituting the cost curve by a quadratic function: provided eqn. 6 is not active. Otherwise, P,:, = _Pi.

Ci(Pi,t) = bOi + bliP,,t + b2iP$?, into eqn. 18, we obtain:

.fi(Pi,t,.i,t) =bOi + bliP,,t + b2iP;t

- U P i J - q t B i , i - Pi,tB12]

- / A t p i - PTB,,, - P,Bl,] -

(21) Using the condition for optimum dJlIdP,, = 0 and con- straining Pi,, between the minimum and maximum gen- eration levels, we obtain the solution to eqn. 20 as:

b l i + 2b2iP,,t - X t [ l - 2Pi,tB,,i - Bl,] = O

or, P2,t = {&(l - Bl;) - bli}/{Z(b2i + XtB;,;)}

(22 )

P:, = min{max{Pi,t,P,},Pi} ( 2 3 )

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 5, Septembeu 1997

For each unit, the spinning reserve contribution r Z t is given by:

- r,,t = P, - p;t

- &[RJ - PZtB,,, - P,,tBlZ]

- Pt[F, + p,,t - (T, + P,,t)2&,z

If r, > r,, then after fixing r, , = 7, eqn. 18 becomes:

fi(Pt,t, xt,t) =bo, + bl,p,,t +

~ (7% + P,,t)B1,] (24) Using the condition for optimum as before, we get

b l , + 2b2,P,,t - X t [ l - 2P,,tB,,, - Bl,] - P c [ l - a(?=, + P,,t)B,,% - a] = 0

or,

(25

P2:t = min{max{P,,t,P,},P,} (26 provided eqn. 6 is not active. Otherwise, Pl:t = p,.

Since the start-up cost is a linear function of down time, the number of down states required to describe the different start-up costs at a particular hour is equal to the cold start-up time Tc,. Again, a unit can be kept on and shut down after it is up for the minimum up time. Hence the required number of up states is the minimum up time plus one, where the extra one is needed to consider last hour generation. Using the above analysis for down and up states, the state transi- tion diagram can be depicted as in [4]. From this state transition diagram, the optimal commitment and gener- ation of unit i can be obtained by using dynamic pro- gramming that requires few states and well-structured state transitions every hour.

If we consider the ramp rate constraints (in eqn. 5) for a steam turbine unit, then the generation levels at two consecutive hours are coupled. Hence by using additional sets of Lagrangian multipliers v l , and v2, to relax the ramp up and ramp down constraints respec- tively, the optimal generation at hour t is obtained. These multipliers are updated at an intermediate level and the subproblem is solved at the low level as if there were no ramp rate constraints [4].

A gas turbine unit does not have ramp rate and min- imum upldown time constraints. It has negligible start- up cost. Hence its cost function calculation is the same as described above but neglecting the start-up cost. Since minimum upldown time is not considered here, only two states (up and down) are needed in the state transition diagram

3.4 Solving hydro subproblems In the hydro subproblem defined by eqn. 17, the unknown variables are the output levels of a hydro unit. Hydro units do not have minimum upldown time constraints. In the SLR method, the hydro subproblem is solved independently as described in [2]. The hydro- thermal scheduling can be used to solve the output lev- els of hydro units. Hence in the proposed Lagrangian relaxation approach, to achieve the output levels of hydro units the hydrolherind scheduling is perrormcd using an efficient algorithm [9] with a thermal unit Commitment schedule obtained by solving only thermal subproblems. This hydrothermal scheduling algorithm

485

data related driver

user operator

- - - J knowledge base (user I

L - - -

procedural knowledge (rules)

Fig. 1 Structure of the expert system

is able to handle nonlinear functions for water dis- charge characteristics, thermal cost and transmission loss constraints.

3.5 Framework of the developed expert system The structure of the expert system is shown in Fig. 1. The data and commitment result related knowledge are generated by the Lagrangian relaxation based HTC program. The procedural knowledge consists of the rules that direct the use of knowledge for yielding spe- cific recommendation. The driver co-ordinates the functions of the inference engine and user interface [7].

The expert system checks for the unit commitment and loading problems. Under the unit commitment problem, the following subproblems are considered: (i) Gas turbine unit’s cycling (ii) Combined commitment of gas and steam turbine units (iii) Steam turbine unit’s cycling (iv) Commitment for adequate voltage control (v) Commitment at a particular plant On the other hand, the following subproblems are con- sidered in the unit loading problem (i) Gas turbine unit’s loading (ii) Largest loaded unit’s loading (iii) Loadings of units for group constraints

3.6 Summary of the solution methodology To solve hydrothermal scheduling, we only need the commitment states of thermal units, not those of hydro units. In the Lagrangian relaxation approach, commit- ment states of thermal units can be obtained by solving thermal subproblems only. Hence the algorithm for the proposed approach follows the flow presented in Fig. 2.

In the initial iteration, the hydro units’ generations are set to zero, and thermal subproblems are solved. During maximisation of the dual function, Lagrangian multipliers are updated using the hydro units’ genera- tions and the solution of the thermal subproblems. The variable metric method as shown in the Appendix is used for updating the multipliers. The solution of hydrothermal scheduling [9] is used to reset the values of the hydro units’ generations. In the subsequent itera- tion, thermal subproblems and hydrothermal schedul- ing are solved as shown in the Figure. If the power balance and/or the spinning reserve constraints are not satisfied, a suboptimal feasible solution is searched where the Lagrangian multipliers are adjusted by the linear interpolation method.

A refinement algorithm similar to that proposed by Tong and Shahidehpour [2] is then used to fine tune the schedule. In the refinement algorithm [2], the linear

486

programming technique is applied to find the dispatch of units, and the cost function of the fictitious unit is considered linear. Tong and Shahidehpour introduced a linear relaxation that uses fictitious units to substitute the operation of candidate units. However, in the refinement algorithm presented in this paper, the hydrothermal scheduling algorithm [9] that has been used to find a feasible solution is used, and the nonlin- ear cost function for a fictitious unit has been used.

operator console (user)

set hydro units generation to zero

s dual optimal found”

4 Yes

perform hydro- thermal scheduling

I 1 set hydro outputs

to the values obtained from hydro- thermal

scheduling

are power balance and reserve constraints satisfied?

refine the schedule t ’I 1

expert system

t

[print schedule I Fig. 2 The proposed algorithm for HTC

The expert system recommends modifying specific input data for the Lagrangian relaxation based HTC program if the result is found operationally unaccepta- ble. The operator resets the input data and repeats the whole cycle until an operationally feasible and/or pref- erable solution is found.

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 5, September 1997

This proposed approach also allows us to use other nonlinear optimisation packages as hydrothermal scheduling solvers for considering complex hydro con- straints such as variable head, or the complex river net- work model.

4 Numerical results

The HTC program implementing the proposed algo- rithm excluding the expert system has been written in the C language. The expert system was developed in SICStus Prolog [l 11. The programs were run on a SUN SPARCstation 10 and tested on a practical utility sys- tem using the data of the generating units and system demands of 1989 and 1993. In 1989, the test system consisted of 32 thermal and 12 hydro generating units, of which seven were gas turbine units. The total ther- mal capacity of the system was 3640MW, and the total hydro capacity was 848MW. In 1993, the system had 36 thermal generating units, of which 11 were gas tur- bine units. The system also comprised 12 hydro units which were the same as in 1989. It had a total thermal capacity of 4345MW. All the constraints mentioned in Section 2.1 were considered in the test cases. Numerical results presented here are based on six data sets: Case 1, Wednesday’89; Case 2, Saturday’89; Case 3, Sun- day’89; Case 4, Wednesday’93; Case 5 , Saturday’93; and Case 6, Sunday’93. In all cases, the scheduling horizon was 24 hours. A summary of the system char- acteristics and parameters for these data sets is shown in Table 1.

Table 1: Summary of power system

System Number of Total capacity or characteristics units requirements, M W

Steam units 25 3 185-3360

Gas turbine units 7-1 1 280-1 160

Hydro units 12 848 All units 44-48 4488-51 93

Peak demand 2 1984830

Minimum demand 1563-2735

Maximum reserve 166-284

The results were monitored at two stages. First, the results of the proposed method excluding the expert system in terms of total production cost in Malaysian Dollars and CPU time requirements in seconds were recorded. They are shown in Table 2. The results show that the proposed method excluding the expert system yields schedules in reasonable time in all cases except in Case 5.

Table 2: Cost and CPU time

Data set Cost, $ CPU, s

Case 1 2 113 407 52

Case 2 1 983 504 51

Case 3 1 582 802 53

Case 4 4 075 242 63 Case 5 3661 546 125 Case 6 2 622 489 58

In Case 5, excessive iterations (a long computation time) were required to find the appropriate values for the Lagrangian multipliers while searching for a subop-

IEE Proc.-Gener. Transm. Distrib.. Vol. 144, No. 5, September 1997

timal feasible solution near the dual optimal point. This is because a slight modification of the multipliers did not change the schedule at all in successive itera- tions for some time. If a system consists of several groups of identical units as used in this work, then this type of problem, called the sensitivity problem, may arise. Even though the fuel costs of identical units were slightly modified so that small differences existed among cost characteristics, excessive iterations were still required to find the appropriate values of the mul- tipliers in this particular case.

At the second stage, the complete algorithm, which is iterative in nature, was applied. The complete solution process for Case 3 is given below: Step 1: Using the proposed method excluding the expert system, a schedule was obtained whose cost of operation is denoted as 1.Op.u. The expert system ana- lysed the results, and recommended keeping thermal unit 3 ‘must run’ at hour 13. It also gave a reasoning, shown in Fig. 3, stating violation of rules for the steam turbine unit’s cycling.

Keep thermal unit 3 must run at hour 13 was derived by rule36 as follows

unit 3 is not switched on again was derived by rule3 1 as follows

steam turbine unit 3 at plant 2 is switched off at 13th hour as extracted from the schedule and

steam turbine unit 3 at plant 2 is never switched on

steam turbine unit 7 at plant 2 is switched on at 13th hour as extracted from the schedule

unit 3 and unit 7 are not the same

13th hour is within 24 hours of 13th hour as found from calculation

unit 3 is not on scheduled outage at any hour as given by you

and

and

and

and

~

Fig. 3 Reasoning behind a recommendation

Step 2: The information to keep thermal unit 3 ‘must run’ at hour 13 was added to the data. The program implementing the proposed method excluding the expert system was rerun. The schedule obtained corre- sponds to an operating cost of 1.0002p.u. The must run constraint inclusion causes the increase in operat- ing cost in this step over the previous step. The expert system analysed the results and found no problem in unit commitment and unit loading. Thus it suggested that the schedule obtained in this step was operation- ally feasible.

The expert system has led to obtaining an operation- ally feasible solution by adjustment of the input data for the Lagrangian relaxation based HTC program. Each consultation with the expert system about a sub- problem took between five and 15 seconds.

5 Conclusions

A comprehensive algorithm for HTC has been pre- sented where an expert system is used as a postproces- sor as well as a preprocessor to an efficient Lagrangian relaxation based HTC program. Important operating constraints (power balance, spinning reserve, minimum upidown time, must run, capacity limits, ramp rate,

481

limited generation for the first and last hour and hydroconstraints) are considered. Transmission loss is included using a general transmission loss formula. Nonlinear functions are used for thermal generation cost and water discharge rate.

The hydro subproblem has not been solved inde- pendently. An efficient hydrothermal scheduling algo- rithm is used for solving the output levels of hydro units. Each thermal subproblem is solved independ- ently using dynamic programming without discretising generation levels.

The expert system enforces complex operating con- straints that are not violated frequently, and are not computationally viable to be included in the Lagrang- ian relaxation based HTC program. Since all the rele- vant data and results of the Lagrangian relaxation based HTC program were transferred to the expert sys- tem as a knowledge base by the HTC program itself, consultation with the expert system was very fast. The expert system obtained a feasible schedule with reason- ing, to increase the operator’s confidence in the results. Numerical results for practical utility data show that the proposed approach produces feasible schedules within a reasonable time.

6 References

1 AOKI,A., SATOH, T., ITOH, M., ICHIMORI,T., and MASEGI, K.: ‘Unit commitment in a large scale power system including fuel constrained thermal and pumped storage hydro’, ZEEE Trans. Power Syst., 1987, PS-2, (4), pp. 1077-1084

2 TONG, S.K., and SHAHIDEHPOUR, S.M.: ‘ An innovative approach to generation scheduling in large-scale hydro-thermal power systems with fuel constrained units’, IEEE Trans. Power

3 YAN, H., LUH, P.B., GUAN, X., and ROGAN, P.M.: ‘Sched- uling of hydrothermal power systems’, IEEE Trans. Power Syst.,

4 GUAN, X., LUH, P.B., YAN, H., and AMALFI, J.A.: ‘An opti- mization-based method for unit commitment’. Int. J. Elect. Power

SYS~. , 1990, PS-5, (2), pp. 665-673

1993, PS-8, ( 3 ) , pp. 1358-1365

Energy Syst , 1992, 14, (l), pp. 9-17 SAKAGUCHI, T., TANAKA, H., UENISHI, K., GOTOH, T., 5 and SEKINE, Y.: ‘Prospects of expert systems in power system operation’, Int. J. Elect. Power Energy Syst., 1988, 10, (2), pp.

6 MOKHTARI, S., SINGH, J., and WOLLENBERG, B.: ‘A unit commitment expert system’, IEEE Trans. Power Syst., 1988, PS-

7 SALAM, MD.S., HAMDAN, A.R., and NOR, K.M.: ‘Integrat- ing an expert system into a thermal unit commitment algorithm’, IEE Proc. C, Gener. Transm. and Distrib., 1991, 138, (6), pp. 553- 559 SALAM, MD.S., NOR, K.M., and HAMDAN, A.R.: ‘An expe- rience in developing an expert system for scheduling problem in electric power system’. 1 1 th international conference on Applica- ti0n.i’ of artficial intelligence in engineering - AIENG96, Tampa, Florida, USA, 1996, pp. 647-662

71-82

3, ( l ) , pp. 272-217

8

9 RASHID, A.H.A., and NOR, K.M.: ‘An efficient method for optimal scheduling of fixed head hydro and thermal plants’, IEEE Trans. Power Syst., 1991, PS-6, (2), pp. 632-636

10 ELGERD, 0.1.: ‘Electrical energy systems theory: An introduc- tion’ (McGraw-Hill, New York, 1971)

11 ‘SICStus Prolog Release 2.1’ (Swedish Institute of Computer Sci- ence, Sweden, October 1991)

7 Appendix

Variable metric method for dual optimisation Let de,! and Et,, be the subgradients for ht and pt, respectively, where the subscript 4 indicates the itera- tion counts. Then and are defined as follows:

H

de,t = Dt + Plosst - ($ pi,t + c (27) h= l

h=l (28)

By using and defined by eqns. 27 and 28, the Lagrangian multipliers A and p are updated to maxim- ise the dual function by the following variable metric updating rule [l]:

= l / ( ~ + b * e ) , U > 0, b > 0 (31)

& = E , O < y < l (33 )

(34)

where 6, and Et are vectors composed of 6t,t and Ee,t respectively. ‘T and ‘E’ represent the transpose opera- tion and the identity matrix.

Note that pe in eqn. 31 represents the step size satis- fying the following two conditions: (i) converges to zero, that is, lime,, (ii) does not converge to a point excepting the solution, that is, Ze (3e 4 CO.

= 0.

488 IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 5, September 1997