Optimal short-term hydrothermal scheduling using decomposition approach and Linear Programming method

M R Mohan, K Kuppusamy and M Abdullah Khan School of Electrical and Electronics Engineering, College of Engineering, Anna University, Madras 600025. India

The operational planning problem of hydrothermal scheduling for the next day's demand in a power system is concerned with the determination of generation schedules for hydro and thermal plants to meet the daily system demand so that the total fuel cost of the thermal plants over the day is minimized subject to the operating constraints associated with the thermal and hydro plants as well as the network security constraints. This paper presents an effective algorithm which decomposes the problem into hydro and thermal subproblems and solves them alternatively. While the hydro subproblem is solved using a search procedure, the local variation method, the thermal subproblem is solved using the participation factors~Linear Programming method. The algorithm is very effective in enforcing security constraints and gives an optimal generation schedule which can be readily implemented for the next day. Results obtained from a 9-bus system and a 66-bus utility system demonstrate the effectiveness of the proposed algorithm.

Keywords." optimal power flows, scheduling, mathematical programming

I. I n t roduct ion The short-term hydrothermal scheduling problem is concerned with optimization over a time horizon of a day or a week. The solution to this problem, if the time horizon is a day, gives a plan for optimal withdrawal of water over the day from the hydro reservoirs for power generation and the corresponding thermal generation such that the total fuel cost of the thermal plants over

Received 1 August 1991

the day is minimized subject to the operating constraints of the hydro and thermal plants as well as network security constraints.

In many of the available methods 1-8, the solution process is simplified by discretizing the time span into small intervals and decomposing the full problem into hydro and thermal subproblems. The two subproblems are then solved through an alternating solution approach. While the hydro subproblem is solved by the incremental dynamic programming method 1'2, the dynamic programming method 3 or the local variation method 4, the thermal subproblem is solved using the nonlinear programming (NLP) method 5. Employing the NLP method for thermal optimization at each interval is quite complex and requires a large amount of computation time and the imposition of inequality network security constraints require a judicious selection of penalty factors which are system-dependent.

Brannlund et al. 6 have used a reduced gradient method to solve the hydro subproblem. The thermal subproblem is solved using two priority lists, one for unit commitment and the other for economic dispatch. Only non-network type constraints to take care of certain transmission limitations were included. Transmission loss and line flow security constraints were not considered. Calderon and Galiana 7 have formulated the hydrothermal problem as a parametric (load) variation problem and solved it by the continuation method which is an analytical method. The line flow constraints as well as transmission losses were ignored in this formulation.

Yang and Chens have used multipass dynamic programming for solving the hydro subproblem and the participation factor method 9 for solving the thermal subproblem. The line flow constraints and transmission losses were not included in the thermal scheduling problem.

Vol 14 No 1 February 1992 0142-0615/92/010039-06 1992 Butterworth-Heinemann Ltd 39

In this paper, an effective algorithm is proposed which uses the decomposition approach and solves the hydro subproblem using search through local variation method and the thermal subproblem using participation factor9/Linear Programming 1. The use of P-Q decoupling, state-independent matrices [B'] and [B"] and variable bounding in the LP formulation 1~ makes the algorithm very effective in implementing network security constraints.

II. Notation Yq water storage in ith hydro reservoir in cubic

metre per sec per hour (CMS-hour) at end ofjth time interval

Y~,(j+t) water storage in ith hydro reservoir at end of (j + 1)th time interval

Dij discharge of ith hydro reservoir in jth time interval in CMS

ALq inflow into ith hydro reservoir during jth time interval

Hoi basic head of ith hydro reservoir in meters Ci correction factor for head variation in any time

interval at ith hydro plant G constant used in determination of hydro power

generated in MW Yij,min minimum water storage in ith hydro reservoir at

end ofjth time interval Y~j .... maximum water storage in ith hydro reservoir at

end ofjth time interval D~j,mi, minimum discharge of ith hydro reservoir in jth

time interval Dgj .... maximum discharge of ith hydro reservoir in jth

time interval PH~j hydro power generation of ith hydro plant in

MW at jth time interval Pij thermal power generation of ith thermal plant in

MW at jth time interval N number of time intervals NH number of hydro plants in system NT number of thermal plants in system NB number of buses in system NL number of lines in system qS~j phase angle of ith transmission line in jth time

interval ~b~,min minimum phase angle rating of ith transmission

line qS~ .... maximum phase angle rating of ith transmission

line X state vector of system of dimension utmost 2 NB U general control vector Rij fuel cost function of ith thermal plant injth time

interval

III. P rob lem s ta tement The operating period of one day is subdivided into 24 equal intervals and the load is considered as constant over each interval. The reservoir inflows, correction factors for head variations and generating units available for scheduling for each interval are assumed to be deterministically known. Evaporation and spill over of water in the hydro reservoirs are neglected.

The one-day hydrothermal scheduling problem with NT thermal plants and NH independent hydro plants is stated as a nonlinear dynamic optimization problem :

To determine the water discharge Dij for each of the NH reservoirs (i = 1,2 . . . . . NH) during each N discrete time intervals (j = 1, 2 . . . . . N) and the corresponding generation schedule of the hydro and thermal plants so as to minimize the total fuel cost during the day

NT N R= ~, ~_, Ro(P,j ) (1)

i= l j= l

subject to the following constraints

the power flow equations Fi(X, U) = 0 j = 1,2 . . . . . N (2)

the characteristic equations of the hydro plants Yi,j+ 1 = Yij -t- ALI j - D o

i= 1,2 . . . . . NH; j= 1,2 . . . . . N (3)

H' [ c' 1 PHq=~- 1 +~(Y i j+ Yij+l) Dq (4)

the storage level in reservoirs Y~,mi ~< Yi~ ~ Y~j ....

i= 1,2 . . . . . NH; j= 1,2 . . . . . N (5) the water discharge

D~,mi. ~< D~j ~< D~j . . . . i= 1,2 . . . . . NH; j= 1,2 . . . . . N (6)

the hydro power generations eHij,min

perturbation is continued for other time instants and other hydro plants. This constitutes one hydrothermal iteration at the end of which the water storage trajectories of all the plants are better than the starting trajectories.

The participation factor method 9 is a fast but approximate method for optimal scheduling of thermal units since it does not consider network constraints and transmission loss. On the other hand, the LP method 1 can very effectively enforce any type of security constraints but requires more time than the participation factor method. In order to speed up the solution process the participation factor method is used to solve the thermal subproblems arising out of the perturbation of trajectories of all the hydro plants except the last hydro plant in which the use of the LP method effectively enforces the security constraints.

V. Development of models for thermal subproblem

V.1 Participation factors method The change in hydro generation, APH~j of the ith hydro plant due to perturbation at the end ofjth interval may be considered as an equivalent change in system demand at that bus and can be distributed optimally (for minimizing fuel cost) to the thermal plants over and above their initial schedule using participation factors 9.

The participation factor of the kth thermal plant is defined as

(1/R';~) PFkJ- Nr

(1/Rmj) ra= l

The updated thermal generation

Pkj = Ptk] + PFki"APHq k = 1,2 . . . . . NT

where Pt) Pki generation of kth thermal plant prior to/after k j ,

perturbation of trajectory at end ofjth interval APH o change in generation of ith hydro plant due to

trajectory perturbation at end ofjth interval R' j second derivative of cost curve of mth thermal

plant during jth interval PFkj participation factor of kth plant in jth interval

+ for decrease in hydro plant generation, i.e. APH o < 0

- for increase in hydro plant generation, i.e. APH o > 0

V.2 LP method While perturbing the storage trajectory of a hydro plant at the end of the jth time interval, the hydro plant constraints (5) and (6) are enforced for the time intervals j and (j + 1 ) and the hydrothermal scheduling problems for these two intervals reduce to pure thermal optimization subproblems. The problem for the jth time interval may be stated as

NT

min: Rj= ~ Ru(Po) (10) i=1

subject to the power flow equations F~(X,U)=O (11)

the transmission line flow constraints ~) i ,min

VI. Proposed algorithm The algorithm comprises two phases. In the first phase an initial feasible hydrothermal schedule is obtained and in the second phase the schedule is improved iteratively to obtain an optimal hydrothermal schedule. The computational steps of the two phases of the algorithm are as follows:

VI.1 Initial feasible water storage trajectory and hydrothermal schedule (1) Choose an initial trajectory for the ith hydro plant,

i = 1, 2 , . . . , NH. The discharge, Dii and the hydro plant output, PHq for i = 1,2 . . . . . NH and j = 1, 2 . . . . . N are determined using equations (3) and (4).

(2) For the jth interval, j = 1, 2 . . . . . N compute the difference between the system demand and the total hydro generation. Distribute this difference to the thermal units proportional to the rating of the units and obtain a base case power flow solution for each interval. Compute the total fuel cost for the day.

(3) Improve the base case solution by optimizing the thermal schedules in each interval using SLP. Compute the improved total fuel cost for the day.

VI.2 Improving the water storage trajectory (1) Set the hydro plant index, i = 1. (2) Set the time interval index, j = 1. (3) Perturb the storage level of the hydro plant at the

end ofjth interval by +AY. (4) Compute the discharge D u and hydro generation,

PHi~ for the ith plant in the jth interval. (5) Corresponding to the hydro generation PHil,

compute the optimal thermal generation schedule, P,,j for the thermal plants m = 1, 2 . . . . . N T using (i) participation factors method for the first

(NH - 1 ) hydro plants (ii) SLP method for the last hydro plant.

(6) Repeat Steps 4 and 5 for 0 + 1)th interval. (7) Compute the total cost for jth and ( j+ 1)th

intervals. Check for cost reduction by comparing this cost with the cost of these intervals in the pre-perturbed trajectory. If cost is less, proceed to Step 8. Otherwise repeat Steps 4 to 7 with a perturbation of - A Y. If the cost is less, go to Step 8. Otherwise, retain the pre-perturbed storage level and go to Step 8.

(8) Increment the interval index j = j + 1. If j < N go to Step 3. Otherwise go to Step 9.

(9) Increment the hydro plant index, i= i+ 1. If i < NH go to Step 2. Otherwise go to Step 10.

(10) Compute the difference in the total thermal generation costs corresponding to the pre- perturbed and perturbed trajectories of the hydro plants. If the cost difference is less than the convergence tolerance specified, the solution is reached. Otherwise, go to Step 1.

VII. Computational details The crucial part of the proposed algorithm is the solution of the thermal subproblem using Linear Programming in each time interval to perturbate each hydroplant. Certain computational techniques are introduced into the SLP part of the algorithm to make it fast and reliable.

VII.1 Variable bounds for control variables The SLP method is susceptible to oscillatory conver- gence. This drawback is overcome by applying restricted bounds on control variables in the LP instead of allowing the full limits as in equation (16). The limits to be applied for the kth variable in equation (16) are chosen as

Upperlimit min{cP k . . . . ;(Pk . . . . _ _ po)}

Lowerlimit max{ --CPk . . . . ; (Pk ,min - - po)} where the factor c lies between 0 and 1 and its value is reduced in subsequent LP moves to ensure smooth convergence.

VII.2 Elimination of phase I computation of LP solution Since the state-independent matrix [B'] is used to derive the constraint matrix of the LP model, the matrix [A] in equation (19) remains the same in all the LP moves. This enables the elimination of phase I computation in all the LP moves. The basic inverse matrix of the last iteration of the previous LP move is used to obtain the initial basic solution in the next LP move. If the initial basic solution is feasible the revised simplex procedure is continued. If the solution is not feasible, the revised dual simplex procedure is used to correct the infeasibility and then the revised simplex procedure is continued.

VI 1.3 Further improvement of LP solution p~o) being the original generation schedule (for thermal plants) and AP being the optimal correction given by an LP move, instead of using the updated schedule, p(1) = p~O~ + AP for the next LP move, a better schedule P* as defined below is used

p , = piO~ + ~ AP (20)

The optimum value of ~ to give the best value P* is obtained by substituting equation (20) in the objective function R (P) and solving the equation

dR(P*) -0

d~

The use of P* instead ofP tl) results in a faster convergence of the SLP method.

VIII. Numerical example and results The proposed algorithm has been tested on two sample systems the first one consisting of 9 buses, l l lines, 3 hydro plants and 4 thermal plants and the second one an Indian utility system comprising 66 buses, 93 lines, 16 hydro plants and 6 thermal plants.

A number of trial studies were made on both systems to choose the best initial incremental step size for A Y and its subsequent reductions during trajectory perturba- tions from the convergence point of view. The results presented in Tables 1 and 2 show the effect of choosing different step sizes for AY. It is observed from the results that an initial value of AY equal to approximately 1/3 of the average of the discharge of all hydro plants is the best choice, also its value should be reduced by 50% in the second and third iterations and thereafter maintained constant.

As proposed in Reference l l a value of 0.4 is chosen for the parameter 'c' to enforce the variable bound on the control variables in the first LP move and its value

42 Electrical Power & Energy Systems

is reduced by 50% during subsequent LP moves for both systems studies.

The SLP has taken, on an average, 4 LP moves in each time interval while optimizing the base case thermal schedule for the initial water storage trajectory. During the trajectory perturbation, the thermal optimization has taken only one LP move which is due to the incremental change caused by the perturbation. A cost difference of 1000 rupees/day is taken as the convergence criterion for the hydrothermal iteration for both systems.

The cost convergence pattern for the 9-bus system is presented in Figure 1. The first four iterations are very effective. The reduction in total fuel cost achieved through the optimization is 12.53%.

In the utility system, out of the 16 hydro plants, 11 are storage plants and the remaining 5 are run-of-river plants supplying fixed generation. Among the 6 thermal plants, 2 are under the control of the central sector and they are treated as fixed generation plants in practice while the remaining 4 plants belonging to the utility are considered as controllable plants. An import of 280 MW from the neighbouring utility system is accounted for in the modelling as fixed generation. A day in the summer month, March 1991 is taken for the study. The hydro

Table 1. Effect of step size A Y for 9-bus system

CPU Step- time in size AY No. of WIPRO- Optimum cost

Serial in CMS- itera- 386 rupees no. hour tions system (s) ( x 106)

1 140 10 55 .5 5.360368 2 100 10 56 .2 5.348376 3 70 11 62.6 5.353292

6.05~

Z" 5.95~ \

5.55 - k

5.45

5.3%

0 T I I I I I I I I I 0 I 2 5 4 5 6 7 8 9 I0

Hydrothermol iteration

Figure 1. Convergence pattern for 9-bus system

3050

295O

~2850

~2750 J

2650

2550

I I I I I i I I I I I 2 4 6 8 I0 12 14 16 18 20 22 24

Time interval, (h)

Figure 2. Load curve for 66-bus system

Table 2. Effect of step size A Y for 66-bus system

Stepsize Optimum A Y in No. of CPU time in cost in

Serial CMS- itera- WIPRO-386 rupees no. hour tions system ( x 106)

1 4 7 12 min 15.099925 32 s

2 3 9 14 min 15.095982 54 s

3 2 10 15 min 15.119778 35 s

14.0

12.0

I0.0

6.0 6.7

~6.0 /5

4.0

Optimal

~ ~j~' iol

2.0

2

Figure 3. system

1 L I I I 4 6 8 I0 12 14 16 18 20 22 24

Time interval, (h)

Discharge trajectory for hydro-1 66-bus

Table 3. Hydro plant characteristics Description 1 2 3 4 5 6 7 8 9 10 11

Maximum storage in CMS-hour 12000 21000 21000 21000 10000 30000 15000 8000 13000 13500 4500 Initial storage in CMS-hour 10072.8 19946.1 19956.1 19956.1 8652 27530.3 13835.7 6333.5 11873.6 12888.7 3980.1 Final storage in CMS-hour 9912.5 19662.4 19670.6 19650.8 8401.7 27414.7 13694.4 6135.5 11668.7 12077.9 3901.9 Maximum allowable discharge in CMS 17.0 56.6 28.9 52.4 68.3 56.7 9.6 49.5 21.0 76.9 16.9 Minimum allowable discharge in CMS 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Maximum hydro generation in MW 40 160 175 180 48 100 70 140 60 65 60 Minimum hydro generation in MW 0 0 0 0 0 0 0 0 0 0 0 Inflow (assumed constant throughout in CMS) 0.00 1.40 0.00 6.76 0.22 1.42 0.00 2.59 1.70 4.70 0.076

Vol 14 No 1 February 1992 43

and thermal plant characteristics are given in Tables 3 and 4, respectively, Figure 2 shows the daily load curve of the utility system. A set of 12 limiting lines are chosen for observing line flow constraints. Figure 3 displays the initial and optimal discharge trajectories of the hydro

'"T 18J6 k

Q 15.23 x

m G)

=~ )5.2i

~ 15.19

2

15,15

15.13

15.11

'%: I W I I i i I i I I 0 I 2 3 4 5 6 7 8 9

Hydrolhermol iterotion

Figure 4. Cost convergence pattern for 66-bus ut i l i ty system

Table 4. Thermal plant characteristics

Thermal plants

Description 1 2 3 4

Constant aj in Rs./MW-hr 50.0 52.0 56.0 40.0

Constant b i in Rs./MW2-hr 1.0 1.2 1.6 1.6

Maximum thermal generation (MW) 630 120 330 840

Minimum thermal generation (MW) 60 20 30 80

Table 5. Line flow in certain critical lines

plant-1 for the chosen step size, AY = 3 CMS-hour. The cost convergence pattern is presented in Figure 4. The first 3 iterations are very effective. The reduction in total fuel cost achieved through optimization is 16.98 %. Table 5 gives the details of the line flow in certain critical lines monitored during the optimization. The LP has been very effective in correcting the overload and containing the flow in these lines within the limits.

IX . Conc lus ions

The proposed algorithm based on the decomposition approach and LP method has been tested using two different systems: a 9-bus system and a 66-bus utility system. The results obtained reveal that the convergence is achieved within 9 to 10 hydrothermal iterations with the first 3 to 4 iterations being very effective. Network flow constraints have been effectively enforced. Use of participation factors and other computational techniques have made the algorithm fast enough for large utility systems.

X. Re ferences 1 Bernholtz, B and Graham, L J 'Hydrothermal economic

scheduling: part I: solution by incremental dynamic program- ming' AIEE Trans (PAS) Vol 79 (1960) pp 921-932

2 Bernholtz, B and Graham, L J 'Hydrothermal economic scheduling: part I1: extension of the basic theory" AIEE Trans (PAS) Vol 80 (1962) pp 1089-1096

3 Bonaert, A P, EI-Abiad, A H and Koivo, A J 'Optimal scheduling of hydrothermal power systems' IEEE Trans Power Appar & Syst Vol PAS-91 No 1 (1972) pp 263-271

4 Prakasa Rao, K S, Prabhu, S S and Agarwal, R P 'Optimal scheduling in hydrothermal power systems by the method of local variations' IEEE PES Winter Meeting New York, paper No. C 74025-3 (1974)

5 Dommel, H W and Tinney, W F "Optimal power flow solutions" IEEE Trans Power Appar & Syst Vol PAS-87 (1968) p 1876

6 Brannlund, H, Sjelvgren, D and Anderson, N 'Optimal short term operation planning of a large hydrothermal power system based on a nonlinear network flow concept' IEEE Trans Power Syst Vol PWRS-1 No 4 (1986) pp 75-82

7 Calderon, L R and Galiana, F D 'Continuous solution simulation in the short-term hydrothermal coordination problem" IEEE Trans Power Syst Vol PWRS-2 No 3 (1987) pp 737-743

8 J in-Shyr Yang and Nanming Chen 'Short-term hydro- thermal coordination using multipass dynamic programming" IEEE Trans Power Syst Vol PWRS-4 No 3 (1989) pp 1050-1056

9 Wood, A and Wollenberg, B Power Generation, Operation and Control John Wiley (1984)

10 Khan, M A and Kuppusamy, K 'Optimum load curtailment under emergency conditions using constant matrices' IEEEPES Winter Meet. New York, Paper A 7911 3-2, Feb 1 979

11 Sadasivam, G and Abdul lah Khan, M 'A fast method for optimal reactive power flow solution' Electrical Power and Energy Syst Vol 12 No 1 (1990) pp 65-68

Rating

Serial E~ivalent no. Line no. MVA MVA

Initial trajectory Optimal trajectory

Base case After LP optimization (~) MVA (~) MVA ((~)

1 7 280 2.44 310 2 52 160 7.76 145 3 67 80 5.23 78 4 81 320 5.63 300 5 85 160 5.33 145

2.71 140 1.02 134 0.97 7.03 149 7.25 134 6.49 5.10 69 4.49 61 3.93 5.28 300 5.09 308 5.16 4.83 158 5.47 152 5.28

44 Electrical Power & Energy Systems