Dynamic economic dispatch: feasible and optimal solutions - Power ...

22

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 1, FEBRUARY 2001

Dynamic Economic Dispatch: Feasible and Optimal Solutions X. S. Han, H. B. Gooi, and Daniel S. Kirschen

Abstract—Dynamic economic dispatch is an extension of the conventional economic dispatch problem that takes into consideration the limits on the ramp rate of the generating units. This paper examines the factors that affect the feasibility and optimality of solutions to this problem. It proposes two new solution methods. The first is guaranteed to find a feasible solution even when the load profile is nonmonotonic. The second is an efficient technique for finding the optimal solution. The results obtained with these methods are compared with those obtained using previously published methods. Index Terms—Economic dispatch, optimization, power generation, ramping constraints.

I. INTRODUCTION

A

S COMPETITION intensifies in the electricity supply industry, generating companies try to further improve the operating efficiency of their portfolio of power plants. While the application of mathematical optimization techniques has a long history in power system operation, tangible improvements can still be achieved through a more rigorous formulation of the constraints and the application of more robust solution techniques. This paper discusses how such improvements can be achieved for the dynamic economic dispatch problem, i.e., the optimization of the production of a set of generating units when limits on the ramp rates of these units are taken into consideration. Since vertically-integrated utilities serving their own load are replaced by competing generating companies entering into standardized transactions, the nature of the dynamic economic dispatch problem is likely to change. It is no longer the natural daily load cycle but the net sum of a set of transactions that will determine the shape of a generating company’s production profile. Since standardized transactions have a rectangular shape, starting and stopping on the hour or the half-hour, the fronts of these production profiles will become steeper. In addition, since energy imbalances are settled on rather volatile spot markets, portfolio generators will try to manage their ramping constraints with their own resources. The importance of the dynamic economic dispatch problem is thus likely to increase. The next section of this paper formulates the dynamic economic dispatch problem and discusses various approaches that have been proposed to solve it either optimally or suboptimally

but efficiently. Section III discusses techniques for obtaining feasible but suboptimal solutions. It also proposes a new technique that is guaranteed to find a feasible solution even when the load profile is nonmonotonic. Section IV describes an efficient optimal solution method based on an adaptive look-ahead technique. In Section V, examples contrast the solution obtained using the proposed method with previously published results. II. PROBLEM FORMULATION The traditional economic dispatch (ED) problem assumes that the amount of power to be supplied by a given set of units is constant for a given interval of time and attempts to minimize the cost of supplying this energy subject to constraints on the static behavior of the generating units. Additional system constraints specifying the minimum amount of reserve capacity required are often added to this basic problem. Plant operators, to avoid shortening the life of their equipment, try to keep thermal gradients inside the turbine within safe limits. This mechanical constraint is usually translated into a limit on the rate of increase of the electrical output. Such ramp rate constraints distinguish the dynamic economic dispatch (DED) problem from the traditional, static economic dispatch. Since these ramp rate constraints involve the evolution of the output of the generators, the DED cannot be solved for a single value of the load. Instead it attempts to minimize the cost of producing a given profile of electricity demand. Formally, the objective function of the DED is: (1) where is the number of intervals in the study period considered; is the set of committed units; is the generation of unit during time interval ; is the cost of producing with unit . This optimization is subject to the following constraints: Load-generation balance: (2)

Manuscript received October 5, 1999. X. S. Han and H. B. Gooi are with Nanyang Technological University, Singapore (e-mail: [email protected]; [email protected]). D. S. Kirschen is with UMIST, Manchester, UK (e-mail: [email protected]). Publisher Item Identifier S 0885-8950(01)02303-3.

where

is the demand to be satisfied at the time interval

Maximum capacity:

0885–8950/01$10.00 © 2001 IEEE

(3)

HAN et al.: DYNAMIC ECONOMIC DISPATCH: FEASIBLE AND OPTIMAL SOLUTIONS

where and

is the reserve contribution of unit during time interval is the maximum capacity of unit

Minimum capacity: (4) where

is the minimum capacity of unit .

Maximum reserve contribution: (5) where capacity.

is the maximum contribution of unit to the reserve

Minimum system spinning reserve: (6) is the system spinning reserve requirement for where time interval . Ramp rate limits:

23

The formulation of the DED problem that has been adopted is traditional because it assumes that the objective is to minimize the cost of production for a given load profile. In a competitive environment, the load-generation balance constraint is relaxed and each generating company schedules its production to maximize its profits given a forecast of electricity prices for the scheduling period. As a first approximation, each generating unit could be optimized separately in this new problem because of the decoupling made possible by the availability of prices at each period. Dynamic constraints (such as ramp rates and minimum up and down time constraints) complicate the problem because a generating company that owns a portfolio of units must then decide whether to buy “flexibility” on the market or meet the dynamic constraints with its own resources. To facilitate comparisons with existing methods, it will be assumed that a generating company has contracted to provide a given load profile and decided to satisfy the ramp rate constraints using its own resources. III. FEASIBLE SOLUTIONS A. Limitations of the Existing Heuristic Method

(7) and are the maximum decreasing and inwhere is the duration of the time creasing ramp rates for unit and intervals into which the study period is divided. Besides some notational differences, this formulation of the DED problem is essentially identical to the one adopted in other papers on the same topic [1]–[7]. Other authors [8]–[11] also include security constraints, which will not be considered here. An interesting departure from this standard formulation is the approach proposed by Wang and Shahidehpour [12] who include in the objective function a term representing the reduction in the life of the turbine caused by excessive ramping rates. This flexible technique makes possible a tradeoff between the system operating cost and the life cycle cost of the generating units. Since the ramping constraints couple the time intervals, a units and time intervals direct solution of a DED with would require the solution of an optimization problem of size a considerably more complex task than the solution of ED problems, each with units. Fortunately, this intertemporal coupling is relatively weak and the problem lends itself to various forms of decoupled or relaxed solutions. Ross and Kim [1] solved the DED problem using dynamic programming. Waight et al. [2] combined dynamic programming with a Dantzig–Wolfe decomposition to solve a linearized form of the problem. Recently, Travers and Kaye [7] applied constructive dynamic programming to this problem. Lee et al. [6] have developed an interesting dual optimization technique whose principle is applicable not only to dynamic economic dispatch but also to the unit commitment problem. Other authors do not take direct advantage of the decoupling opportunities that the problem offers. For example, Somuah and Khunaizi developed an LP-based redispatch technique while ramp rate constraints have also been enforced by adding quadratic penalty terms [4] and barrier functions [11] to the objective function.

Because of its simplicity and despite the fact that it produces feasible but not necessarily optimal solutions, the heuristic technique developed early on by Wood [3] is probably the method that has been most widely implemented [14]. This technique decouples the DED into the solution of a backward sequence of conventional economic dispatches with artificial limits on the generators. The upper artificial limit on each generating unit is equal to the smallest of the following constraints: UL1: The unit’s mechanical or economical upper limit. UL2: The maximum output from which the unit could ramp down at its maximum rate and achieve the desired output at the next time interval. UL3: The output that this unit could achieve by ramping up from its initial condition at its maximum rate. Similarly, the lower artificial limit on each generating unit is equal to the largest of the following constraints: LL1: The unit’s mechanical or economical lower limit. LL2: The minimum output from which the unit could ramp up at its maximum rate and achieve the desired output at the next time interval. LL3: The output that this unit could achieve by ramping up from its initial condition at its maximum rate. Conditions UL2 and LL2 imply that the algorithm works backward from the final time interval where these limits on transitions do not apply. Fig. 1 illustrates this method using a simple example. Suppose that some of the units used to meet a certain load profile cannot increase or decrease their output at a combined rate greater than 50 MW per time interval. The curve labeled “unconstrained” shows the amount of power that these units would produce if their ramp rates were not limited. Moving backward from the end of the study period, the heuristic DED (the final interval in the study peworks as follows. At riod) the solution is not bound by ramp rate constraints as the and , value at the next period is unknown. Between the ramp-rate constraint is inactive because the output of these

24


for each time interval and to check each transition for ramp constraints violations. If such violations are detected, they usually cannot be corrected by modifying the solution at the time interval where they occur. The solution at some of the preceding intervals must also be modified. If the problem is solved progressively in a forward manner, the algorithm must “look ahead” to anticipate the difficulties. Formalizing this look-ahead procedure requires the definition of a few unit parameters and system variable. The number of time intervals necessary for unit to ramp up or down its entire operating ranges are respectively given by:

Fig. 1.

round

(8)

round

(9)

Relation between Heuristic and Unconstrained.

units does not change. At time , the maximum ramping down limit is translated into an artificial upper limit of 100 MW on the generation of these units. Since the unconstrained solution yields a larger value, it is clear that this constraint is binding. The same ramp rate considerations explain the solution at time and . At time , the heuristic soluintervals tion rejoins the unconstrained solution. The heuristic method of Wood in effect forces the solution to anticipate the ramp rate problem. A similar reasoning explains why the heuristic soluto tion starts deviating from the unconstrained solution at while satisfying the maximum ramping up rate rejoin it at constraint. The two triangular areas separating the unconstrained and heuristic solutions represent energy that has to be produced in an uneconomical manner because of the ramping constraints. and , the output of more efficient units has Between to be curtailed to make possible the early increase in output of the ramp-constrained units. On the other hand, between and , less efficient units have to produce more to compensate for the ramping down of the constrained units. Minimizing this displaced energy production will minimize the increase in operating cost resulting from the ramp-rate constraints. It is interesting to note that constraints UL1 and LL1 do not “look ahead” (or more exactly “look back”). On the other hand, constraints UL2 and LL2 look back one time interval while constraints UL3 and LL3 look back from the current time interval directly to the initial conditions. These triple-layered constraints guarantee that the heuristic solution will be feasible only when the unconstrained solution increases or decreases monotonically. When the unconstrained solution is nonmonotonous, constraints UL3 and LL3 are likely to be inactive. Under such conditions, applying constraints UL2 or LL2 to individual units may interfere with the system minimum and maximum capacity constraints and the method may fail to produce a feasible solution. The 5-unit system discussed in Section V provides an example of a situation where the application of Wood’s heuristic method [3] fails to produce a feasible solution. B. A New and Robust Heuristic Method The easiest way to determine whether a certain load profile requires the dynamic economic dispatch of a given set of committed units is obviously to perform a regular economic dispatch

time intervals ahead of the current time inWhen looking terval , four sets of units can be defined: (10) (11) (12) (13) It is important to keep in mind that these sets change for many units are rampeach value of . For small values of constrained and will therefore be classified as “slow.” As increases, more units will be classified as “fast.” The maximum amounts by which the system generation can increase or deare given by: crease between intervals and

(14)

(15) If, for any value of

, one of the following inequalities holds: (16) (17)

the dynamic economic dispatch problem does not have a solution and the unit commitment schedule must be revised. In the critical cases where (16) or (17) is a binding equality, the fast unit must generate their maximum (or minimum) capacity and the slow units must increase (or decrease) their output at the maximum rate. are smaller than the If the load changes for all values of maximum system capacity for increase or decrease in generation, there is some flexibility in the generation dispatch. However, this flexibility is limited by constraints that look ahead


25

for ramp rate difficulties. It is convenient to express these constraints differently for the “fast” and the “slow” units. The constraints for the “fast” units are derived directly from (14)–(17):

(25) (26)

(18)

6) Perform a conventional economic dispatch over the set of with: units (27) (28) (20)

(19) or . where Looking ahead from interval , these constraints might be violated for various values of m by amounts that will be denoted by . These violations can be corrected by adjusting the output downward of the fast units by the amount for violations of (18) or upward for violations of (19). The constraints for the “slow” units are based on the concept of “Unit Unavailable MW” (UMU for increasing output and UMD for decreasing output). This quantity represents the capacity of the unit that cannot be reached because of the unit’s ramp limitations. Formally: (20) (21) The unavailable MW of a unit at interval are thus dictated by the output of this unit at interval . The look-ahead constraints for the “slow” units can then be written as follows: (22) (23) or where Looking ahead from interval , these constraints might be violated for various values of by amounts that will be denoted by . These violations can be corrected by adjusting the output upward for of the slow units by the amount violations of (22) or downward for violations of (23). On the basis of these definitions, the “look-ahead” DED alto gorithm proceeds as follows for each interval from : 1) Perform a conventional economic dispatch for load 2) Calculate , and and identify the value of m for which this maximum violation occurs 3) Identify the sets of units whose output needs to be boosted ) and bucked ( ) to satisfy the ramping con( straints. The former are the slow units for an increasing load and the fast units for a decreasing load. The latter are the fast units for an increasing load and the slow units for a decreasing load. Note that these two sets depend on the values of and . 4) Calculate the power produced by these two groups of units according to the unconstrained dispatch ) ( 5) Perform a conventional economic dispatch over the set of with: units (24)

IV. OPTIMAL SOLUTION While the technique described in the previous section looks ahead to take into consideration all the constraints required to produce a feasible solution, it optimizes the operating cost of each interval separately. It is thus unable to make the intertemporal tradeoffs required of an optimal solution. While the results presented in Section V show that the difference between the cost of the feasible solution produced by this method and the cost of an optimal solution is, in most cases, almost negligible, there may be situations where this suboptimality may be unacceptable. Lee et al. [6] have proposed an elegant dual optimization approach where the interplay between the Lagrange multipliers used to relax the ramping constraints guides the inter-temporal tradeoffs. Unfortunately, like all applications of dual optimization to nonconvex problems, this method is not guaranteed to find the global optimum solution. One of the examples of Section V illustrates such a case. A direct but efficient approach to this optimization problem is thus needed. Such a technique has been developed on the basis of the look-ahead method described in the previous section. Its efficiency rests on the observation that, for , all units can cover their entire operating range. Ramping constraints therefore do not couple intervals . The optimal DED problem can separated by more than thus be formulated as follows: to , minimize For each interval from (30) subject to: (31)

(32)

(33) . where The reserve constraints have been omitted for the sake of clarity. This optimization problem can be solved using quadratic programming if the cost curves are given in quadratic form or using linear programming if they are given as piece-wise linear functions.

26


TABLE I UNIT 1 HOURLY MW GENERATION AS DETERMINED BY THE LOOK-AHEAD METHOD FOR THE 5-UNIT SYSTEM

TABLE III UNIT DATA FOR THE 10-UNIT SYSTEM

TABLE IV LOAD PROFILES FOR THE FOUR CASES FOR THE 10-UNIT SYSTEM

Fig. 2. Comparison of the output of unit 1 of the 5-unit system for the various solution methods.

COMPARISON

OF THE

TABLE II COSTS OF THE VARIOUS SOLUTIONS 5-UNIT SYSTEM

FOR THE


V. TEST RESULTS A. 5-Unit System The data for this example as well as the unconstrained and dual solutions can be found in [6]. The solution described in that reference as resulting from the application of Wood’s heuristic method [3] does not in fact result from the application of that algorithm. Wood’s heuristic algorithm as described in [3] works backward from hour 24 and terminates at hour 19 because the minimum generation that can be achieved exceeds the load at that hour by 50 MW. This is due to the facts that the load profile is not monotonous and that the algorithm does not take into consideration the difficulties created by the intermediate peaks and valleys. The look-ahead algorithm schedules the generation of unit 1 as shown in Table I. Unit 1 is the only ramp-constrained unit in this example. Fig. 2 illustrates the differences between the unconstrained, dual, look-ahead and optimal solutions for the first 12 hours of this example. With the exception of the initial conditions, the solution for the next 12 hours is identical to the solution for the first 12 hours. Table II gives a comparison of the costs of the various solutions for the 5-unit system.

B. 10-Unit System This system is based on the example of [3]. Unfortunately, since no cost data is provided in that reference, quadratic cost had to be defined. In curves of the form order to make the problem more realistic and in contrast with the 5-unit system, these cost curves have been constructed in such a way that the incremental cost curves of the various units overlap. Table III summarizes the unit data for this example and the load profiles of 4 different cases are shown in Table IV. Table V shows the unconstrained, optimal, heuristic [3] and look-ahead solutions for Case 1. Figs. 3 and 4 respectively show a comparison of the output of units 1 and 6 as determined by the various methods for Case 1. Since the heuristic method does not place any constraints on the solution at the last interval, at this point this solution matches the unconstrained solution. The look-ahead method on the other hand spreads the effect of the ramp constraint (which is particularly significant at the last time interval) on more intervals. This explains why it increases the output of unit 6, which is relatively cheap, and decreases the output of unit 1, which is more expensive. It is interesting to note that the optimal solution falls between the heuristic and


TABLE V UNCONSTRAINED, OPTIMAL, HEURISTIC, AND LOOK-AHEAD SOLUTIONS FOR CASE 1 OF THE 10-UNIT SYSTEM

27

TABLE VI COMPARISON OF THE COSTS AND COMPUTATIONAL REQUIREMENTS OF THE VARIOUS SOLUTIONS FOR THE 10-UNIT SYSTEM

heuristic method. The proposed method for finding the optimal solution requires two and a half times more computing time than is required to obtain a feasible but sub-optimal solution. When the load profile is steep, e.g., the load of 1391 in Case 1 of Table IV, then the cost difference between the heuristic and look-ahead methods does not change. When the load profile is such that the load profile is less steep, e.g., Cases 3–4 of Table IV, then the cost of the heuristic method is larger than that of the look-ahead method. VI. CONCLUSION Dynamic Economic Dispatch is a complex optimization problem whose importance may increase as competition in power generation intensifies. This paper has attempted to clarify the techniques that provide feasible solutions. It has also presented two new solution techniques. The first is guaranteed to find a feasible solution for all load profiles. The second is an efficient technique for finding the optimal solution. Tests results have been used to demonstrate the effectiveness of these techniques and to compare their results with those obtained using previously published methods. ACKNOWLEDGMENT D. Kirschen thanks Nanyang Technological University (NTU) for the Tan Chin Tuan Exchange fellowship that made this collaborative research possible. *O-Optional, U-Unconstrained, H-Heuristic, L-Look-ahead

REFERENCES


the look-ahead solutions. Table VI summarizes the cost and the computational requirements of these solutions. In this case, the heuristic method does not fail and delivers a feasible solution. The proposed look-ahead technique runs slightly faster than the

[1] D. W. Ross and S. Kim, “Dynamic economic dispatch of generation,” IEEE Trans. on Power Apparatus and Systems, vol. PAS-99, no. 6, 1980. [2] J. W. Waight, F. Albuyeh, and A. J. Bose, “Scheduling of generation and reserve mar-gin using dynamic and linear programming,” IEEE Trans. on Power Apparatus and Systems, vol. PAS-100, no. 5, 1981. [3] W. G. Wood, “Spinning reserve constrained static and dynamic economic dispatch,” IEEE Trans. on Power Apparatus and Systems, vol. PAS-101, no. 2, Feb. 1982. [4] P. P. J. Van den Bosch, “Optimal dynamic dispatch owing to spinning reserve and power-rate limits,” IEEE Trans. on Power Apparatus and Systems, vol. PAS-104, no. 12, 1985. [5] C. B. Somuah and N. Khunaizi, “Application of linear programming redispatch technique to dynamic generation allocation,” IEEE Trans. on Power Systems, vol. 5, no. 1, 1990. [6] F. N. Lee, L. Lemonidis, and K. C. Liu, “Price-based ramp-rate model for dynamic dispatch and unit commitment,” IEEE Trans. on Power Systems, vol. 9, no. 3, 1994. [7] D. L. Travers and R. J. Kaye, “Dynamic dispatch by constructive dynamic programming,” IEEE Trans. on Power Systems, vol. 13, no. 1, 1998. [8] W. R. Barcelo and P. Rastgoufard, “Dynamic economic dispatch using the extended security constrained economic dispatch algorithm,” IEEE Transaction on Power Systems, vol. 12, no. 2, 1997. [9] , “Control area performance improvement by extended security constrained economic dispatch algorithm,” IEEE Transaction on Power Systems, vol. 12, no. 1, 1997.

28

[10] C. W. Sanders and C. A. Monroe, “An algorithm for real-time security constrained economic dispatch,” IEEE Trans. on Power Systems, vol. 2, no. 4, 1987. [11] G. Irisarri, L. M. Kimball, K. A. Clements, A. Bagchi, and P. W. Davis, “Economic dispatch with network and ramping constrained via interior point methods,” IEEE Trans. on Power Systems, vol. 13, no. 1, 1998. [12] C. Wang and S. M. Shahidehpour, “Ramp-rate limits in unit commitment and economic dispatch incorporating rotor fatigue effect,” IEEE Trans. on Power Systems, vol. 9, no. 3, 1994. , “Optimal generation scheduling with ramping costs,” IEEE Trans. [13] on Power Systems, vol. 10, no. 1, 1995. [14] A. I. Cohen, “Modeling unit ramp limitations in unit commitment,” in Proceedings of the 10th Power Systems Computations Conference, Graz, Austria, Aug. 19–24, 1990, pp. 1107–1114. [15] A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control, Second ed: John Wiley & Sons, 1996.


X. S. Han is currently a Research Fellow in the School of EEE, NTU, Singapore.

H. B. Gooi is currently an Associate Professor in the School of EEE, NTU, Singapore.

Daniel S. Kirschen is currently a reader at UMIST in Manchester, UK.