Hydrothermal scheduling by augmented lagrangian - Semantic Scholar

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001

Hydrothermal Scheduling by Augmented Lagrangian: Consideration of Transmission Constraints and Pumped-Storage Units Salem Al-Agtash, Member, IEEE

Abstract—This paper presents an augmented Lagrangian (AL) approach to scheduling a generation mix of thermal and hydro resources. AL presents a remedy to duality gap encountered with the ordinary Lagrangian for nonconvex problems. It shapes the Lagrangian function as a Hyperparaboloid associating penalty in the direction of the coupling constraints. This work accounts further for the transmission constraints. We use a hydrothermal resource model with pumped-storage units. An IEEE 24-bus test system is used for AL performance illustration. Computational models are all coded in C. The results of the test case show that the AL approach can provide better scheduling results as it can detect optimal on/off schedules of units over a planning horizon at a minimal cost with no constraint violation. It requires no iteration with economic dispatch algorithms. The approach proves accurate and practical for systems with generation diversity and limited transmission capacity. Index Terms—Augmented Lagrangian, hydrothermal scheduling, pumped-storage units, transmission constraints.

I. INTRODUCTION

I

N AN era of restructuring, the power industry migrates from a state of separate operating areas to a regional market-wide ISO (Independent System Operator) coordinated operation. ISO operation incorporates diverse generation resources. As public environmental concerns rise in importance, new regulations are imposed on transmission expansion. In consequence, systems would become more constrained by line flows as the market expands and competition increases. This poses a requirement to develop scheduling methods that accommodate generation diversity and line flow limitations and concurrently can produce accurate scheduling results. Numerous methods have been developed for electric power scheduling. These include dynamic programming, Lagrangian methods, and meta-heuristics, [2], [14]. Research results have pointed out the potential of Lagrangian methods to solve largescale problems and handle large number of constraints. The Lagrangian converts the scheduling problem into a min–max dual problem structure. Minimization is achieved by solving unit-wise subproblems. Solving the dual problem is, however, nontrivial. The problem is nondifferentiable. Strict application of subgradient algorithms is numerically inefficient, [3]. This has been resolved by modified and adaptive step-sizing subgradient, [16], [20] and bundle-type, [8], [10], [13] methods.

Generally, a dual gap exists because the problem is nonconvex. If the optimal solution of the problem exists in a dual gap, the standard Lagrangian fails to find such a solution, [4], [15]. In contrast, the augmented Lagrangian (AL) method may find the optimal solution, [1], [3], [9], [17]. The AL method shapes the Lagrangian function as a Hyperparaboloid associating penalty in the direction of the coupling constraints, [1]. In the reference, we presented to hydro-thermal scheduling an AL method that incorporates a penalty function with generation-load balance, spinning reserve and emission constraints. We used a simple hydro model with no pumped-storage units. Relaxing transmission constraints, however, does not guarantee a feasible generation schedule, especially in constrained transmission systems. To produce a feasible schedule, extra units are committed. The resultant schedule would, therefore, deviate from optimality. This paper presents an AL approach that incorporates as well, transmission constraints and pumped-storage units. The results of an IEEE-24 bus test case show that the approach can provide better scheduling results as it can detect optimal on/off schedules of units over a planning horizon at a minimal cost with no constraint violation. It requires no iteration with economic dispatch algorithms. The approach proves accurate and practical for systems with generation diversity and limited transmission capacity. The paper is organized as follows: Section II presents thermal and hydro resource models. Generation scheduling formulation and methodology is given in Section III. Section IV discusses the performance of the AL approach on an IEEE 24-bus test system. The paper is concluded in Section V. II. RESOURCE MODEL This paper considers a generation mix of thermal and hydro resources for scheduling. The set of thermal resources is . The . The total number of generation set of hydro resources is in the set . The generated electric power is resources is lines in , transmitted over a network. The network has interconnecting system busses in . A demand at load bus in , during hour is . The total demand is . We consider a planning horizon for scheduling. A. Thermal Resources

Manuscript received October 6, 1999; revised February 6, 2001. The author is with the Department of Computer Engineering, Yarmouk University, 21163 Irbid, Jordan. Publisher Item Identifier S 0885-8950(01)09433-0.

Thermal resources include steam and combustion turbine units. During each hour , a thermal unit has an operating (1 ON schedule defined by a commitment status

0885–8950/01$10.00 © 2001 IEEE

AL-AGTASH: HYDROTHERMAL SCHEDULING BY AUGMENTED LAGRANGIAN

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or 0 OFF) and a generation level . The costs for operating unit are: start-up, normal operational, shut-down , , and maintenance denoted respectively by and . We use the standard cost models of [19]. The operating constraints of thermal units, as defined in [1], are: generation bounds, minimum up/down times and ramp up/down limits.

. The objective is to minimize the generation cost and satisfy system total demand and operating constraints. We use or to denote the domain of commitment schedules of to denote the domain of generthermal units and denotes the ation level schedules of thermal and hydro units. denotes the number of thermal units number of hours in . in .

B. Hydro Resources

A. Optimization Problem

Hydro resources include conventional storage and pumped storage units. During hour , a conventional storage unit has . A pumped storage unit has a a generation schedule . We use and to generation/pumping schedule denote the sets of conventional and pumped storage units. Costs of the hydro power generation is considered negligible. We only include maintenance costs of hydro conventional and pumped storage resources denoted respectively by and . The operating constraints of conventional storage units, as defined in [1], are: generation bounds and total available hydro energy. The operating constraints of pumped storage units, as defined in [19], are: pond level and generation/pumping level constraints. C. System Operation

We formally set up an objective function as

(5) and . We, furthermore, denote the equality where and inequality system coupling and resource local constraints by real-valued vector functions and , respectively. The optimization problem is then

The entire system operation is subject to the following constraints: generation-load balance, spinning reserve, emission bounds, formally defined in [1], and line flow limits, (1) where

is a maximum power flow of line

subject to:

(6)

Equation (6) is a nonconvex mixed-integer nonlinear programming problem (MINLP).

, and (2)

is the injected power at bus and defines the system transfer admittance matrix with elements ; : resistance and reactance of line connecting buses and ; : matrix with a zero row and a zero column corresponding to the slack bus and a submatrix matrix with elements equal to the inverse of an if

B. Geometric Analysis We use geometric analysis on a simplified version of the scheduling problem to explain the performance of AL searching for the optimal solution. A perturbation function [15], [18], subject to: , st is defined for all that the optimal solution of (6) corresponds to define a set

(7) . Note . We next

(3)

(8)

if By reconstructing rewritten as

as

,

,

and

, (2) can be easily

Fig. 1 shows the geometry of (7) and (8) for a 3-bus system with two generators and one load, [19]. We consider the generation-load balance constraint for and two hours time span for . The standard Lagrangian of (6) is (9)

(4)

is said to minimize Then for some , the point over and if and only if a hyper-plane of the form (10)

III. GENERATION SCHEDULING We present an AL approach to solve for the optimal generation schedule of available thermal and hydro resources over

at the point . Therefore, supports the set can be reached if and only if the optimal solution there exists a supporting hyper-plane for the set at the point

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is convex in

and has a minimum at, [1]

(15)

(16)

(17)

Fig. 1.

Geometry of w (z ) for a 3-bus system.

. Since is nonconvex, the solution of (6) in the dual gap region cannot be obtained by a supporting hyperplane of the form of (10) as can be seen in Fig. 1. Moreover, by shaping the Lagrangian as a Hyperparaboloid of the form (11) then we can choose values for and to support the set at . The Lagrangian of this form is known the point as the augmented Lagrangian.

and denote the emission rate and fuel cost of where defines emission allowance, during hour . unit ; and Equations (15)–(17) can be easily achieved by taking the deriva. The result is then equated to zero tive of (26) with respect to . and solved for subject to Equation (6) is then equivalent to optimizing resource constraints. By Proposition 3 of [1], the optimal solution is at the saddle point of . This is achieved by a two-level hierarchical optimization structure. At the lower level, a dual functional is defined as the minwith respect to for a given set of values for imum of and .

subject to

(18)

C. Solution by AL Since (6) accounts for all constraints, there is no need to decompose the problem into separable unit commitment and economic dispatch problems. We apply the AL approach to solve (6). The AL, however, does not apply to the inequality constraint case. A trick by [4] is to come back to the equality constraint case by introducing slack variables. Formally, (6) is converted into the equivalent problem with (explicit) equality constraints subject to:

(12)

where denotes a vector of slack variables associated with the system spinning reserve, emission and line flow inequality con. At this stage, straints. The dimension of is resource constraints are relaxed. The augmented Lagrangian of (12) is defined as

(13) where is a positive constant and denotes a vector of the Lagrangian multipliers associated with all coupling constraints. . We then remove the slack The dimension of is variables by defining, [4] (14)

By [1, Corollary 1], decomposes into separable thermal [ of (32)], hydro [ of (33)] and pumped-storage [ of (34)] subproblems as

subject to

(19)

, and are vectors of dimension repwhere resenting schedules of thermal unit , conventional storage hydro unit and pumped-storage hydro unit over . The complete derivations are given in the Appendix. Solving individual thermal, hydro and pumped-storage subproblems is presented relatively well in the literature for different sets of unit constraints. See for example [2], [6], [16]. Even though this paper gives different formulation of the unit-wise objective functions by incorporating constraint penalties, the developed solution methods can still apply to this formulation. Thermal Subproblem: Dynamic programming has been a common approach to solving individual thermal subproblems. , a network is constructed. The network For each unit represents all possible on/off and ramp up/down states of the unit during all hours in . The cost of each path in the network is computed by (32). The unit optimal schedule is then given


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by the states of the minimal cost path. For more details, see [1], [16]. Hydro Subproblem: The optimal schedule of hydro unit is given by, [1] if (20) if

where the values of are obtained from the previous th itimplies generation and a negeration. A Positive value of ative value implies pumping. However, if this value is greater than the maximum allowable generation/pumping, then its value is set to maximum provided that the pond level constraints are met. The optimal water discharged/pumped can then be obtained using a quadratic approximation of the water-power conversion formula, [5]. It should be noted that the power generation also depends upon the head level of the reservoir. This dependency can practically be taken into account by considering for every head level an appropriate water–power conversion curve, [11]. The Dual Problem: At the higher level, a dual problem is defined as the maximum of with respect to

where

exists

(21) Provided that hydro generation is available. The total generation is then scheduled over using a merit order allocation method, are arranged in a descending order [1], [20]. is set with their associated . Following this order of , according to (20) until all available power is allocated. For the is set to zero. remaining hours, , Pumped-Storage Subproblem: Since (34) is convex in the optimal generation/pumping schedule of unit , during hour , is then given by

(23)

involves both integer and continuous variables, hence non differentiable at every point. A common approach to solving (23) is by iteratively updating and using subgradient algorithms. Simulation analysis showed that strict application of sub-gradient algorithms results in slow and often oscillatory convergence. This has also been noted in [7], [8]. It was shown that the algorithm often causes the multipliers to zigzag as illustrated graphically in [8]. Several methods have been proposed for improving convergence. These include modified and adaptive step-sizing subgradient, [16], [20] and bundle-type, [8], [10], [12], [13] methods. We use the criterion given in [1] to update the multipliers. It combines both modified and adaptive step-sizing subgradient methods. The penalty , on the other th iteration according to, [15] hand, is updated at (24) where is a constant representing certain physical characteristics of the optimization problem and

(25) is an estimate of the optimal where is a scaling factor, and and ; value of is a square of the elements of vector for as defined in the Appendix by (27)–(30). IV. NUMERICAL TESTING

(22)

The IEEE test system used in [1] is also used here to test the performance of the AL incorporating transmission constraints and pumped-storage units. We use the same load profile and constraint settings as described in [1]. The system data and line flow constraints are given in [21]. Computational models developed in this paper are all coded in C. The code is about 2100 lines, compiled and run on a Linux operating PC machine. We discuss the scheduling results of the test system while considering in addition transmission constraints for two cases: Case 1: The system has no pumped-storage units. We discuss the results of this system: Maximum generation of hydro units 27–30 is scheduled at peak hours until available power is allocated. Units 1–9 and 21–23 remained shut down and 24–26 are committed at all hours. Table I gives the commitment

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TABLE I COMMITMENT SCHEDULES OF UNITS 10–20

21–23 remained shut down and 24–26 are committed at all hours. The commitment schedule of the other units is given in Table I. The main difference is in the schedule of units 17. The unit is committed during off-peak hours. The total cost is about $497 366.2. The approach was also tested on different sets of load profiles with higher and lower demands. The results proved accurate and practical. V. CONCLUSION The augmented Lagrangian approach presented in this paper accommodates further for pumped-storage units and line flow limitations and concurrently can produce accurate scheduling results. The approach produces feasible schedules and requires no iteration with economic dispatch algorithms. Because of the unit-wise decomposition structure of the problem, a more complex hydrothermal model and more unit constraints can be easily incorporated in our approach. Moreover, convergence properties of the augmented Lagrangian approach may be improved further using bundle-type methods at the higher level. APPENDIX Substituting for

schedule of the remaining thermal units. The letters u and d are used to indicate a ramp-up and a ramp-down process for the corresponding unit. Note that cheaper units are committed for longer hours. Line 9 (Bus 5–Bus 10) is congested during peak hours. We compare these schedules with the schedules of AL2 which considers no line flow limits as given in [1]. Unit 16 is committed during peak hours but was decommitted with AL2. Other units are committed similarly with little deviation. The value of is 0.000 83. This value is fifth the value of used in AL2. This shows that as we include more constraints less penalty is required. In other words, the shaping of the hyperparaboloid is set more flat. The method detects necessary shut-downs and start-ups within different time intervals. The total cost incurred is $496 935.8. This cost is approximately equal to the cost incurred by AL2 ($496 771.6). The little cost deviation is due to the difference in the generation level of units to accommodate for transmission constraints. Results converged to the optimal values within seconds and in 71 iterations for this test case. Case 2: We assume the hydro units 31 and 32 of the test system are pumped-storage units. The units have a maximum generation/pumping of 50 MW. The initial pond level is assumed 1000 acr-ft. The values of the water–power conversion parameters are set for this example as in [19]. The results of this system are as follows: Pumped-storage units 31–32 are scheduled for maximum generation during peak hours and for maximum pumping load during off-peak hours. The hydro units 27–30 are scheduled for maximum generation during peak hours until available power is allocated. Units 1–9 and

and the constraints, (13) becomes

(26) where and

(27)

(28) (29)


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(30) , and Given the computed values of earlier, we obtain . Using the results of [1], as follows

as defined decomposes

(32)

(31) where

(33)

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(34)

(35)

(36)

REFERENCES [1] S. Al-Agtash and R. Su, “Augmented Lagrangian approach to hydrothermal scheduling,” IEEE Trans. Power Systems, vol. 13, no. 4, pp. 1392–1400, Nov. 1998.

[2] R. Baldick, “The generalized unit commitment problem,” IEEE Trans. Power Systems, vol. 10, no. 1, pp. 465–475, Feb. 1995. [3] J. Batut and A. Renaud, “Daily generation scheduling optimization with transmission constraints: A new class of algorithms,” IEEE Trans. Power Systems, vol. 7, no. 3, pp. 982–987, Aug. 1992. [4] G. Cohen and D. L. Zhu, “Decomposition coordination methods in large scale optimization problems: The nondifferentiable case and the use of augmented Lagrangian,” in Advances in Large Scale Systems: JAI Press Inc., 1984, vol. I, pp. 203–266. [5] X. Guan, P. Luh, H. Yan, and P. Rogan, “Optimization-based scheduling of hydrothermal power systems with pumped-storage units,” IEEE Trans. Power Systems, vol. 9, no. 2, pp. 1023–1031, May 1994. [6] X. Guan, E. Ni, R. Li, and P. Luh, “An optimization-based algorithm for scheduling hydrothermal power systems with cascaded reservoirs and discrete hydro constraints,” IEEE Trans. Power Systems, vol. 12, no. 4, pp. 1775–1780, Nov. 1997. [7] G. Guan, P. Luh, and L. Zhang, “Nonlinear approximation method in Lagrangian relaxation-based algorithms for hydrothermal scheduling,” IEEE Trans. Power Systems, vol. 10, no. 2, pp. 772–778, May 1995. [8] P. B. Luh, D. Zhang, and R. Tomastik, “An algorithm for solving the dual problem of hydrothermal scheduling,” IEEE Trans. Power Systems, vol. 13, no. 2, pp. 593–600, May 1998. [9] H. Ma and S. M. Shahidehpour, “Unit commitment with transmission security and voltage constraints,” IEEE Trans. Power Systems, vol. 14, no. 2, pp. 757–764, May 1999. [10] V. Mendes, L. Ferreira, P. Roldao, and R. Pestana, “Optimal short term resource scheduling by Lagrangian relaxation: Bundle type versus subgradient algorithms,” in 12th Power System Computation Conference, Dresden, 1996. [11] J. Medina, V. Quintana, A. Conejo, and F. Thoden, “A comparison of interior-point codes for medium-term hydro-thermal coordination,” IEEE Trans. Power Systems, vol. 13, no. 3, pp. 836–843, Aug. 1998. [12] F. Pellegrino, A. Renaud, and T. Socroun, “Bundle and augmented Lagrangian method for short term unit commitment,” in 12th Power System Computation Conference, Dresden, 1996. [13] N. Redondo and A. Conejo, “Short-term hydro-thermal coordination by Lagrangian relaxation: Solution of the dual problem,” IEEE Trans. Power Systems, vol. 14, no. 1, pp. 89–95, Feb. 1999. [14] G. B. Sheble’ and G. N. Fahd, “Unit commitment literature synopsis,” IEEE Trans. Power Systems, vol. 9, no. 1, pp. 128–135, Feb. 1994. [15] G. Stephanopoulos and A. W. Westerberg, “The use of Hestenes’ method of multipliers to resolve dual gaps in engineering system optimization,” Journal of Optimization Theory and Applications, vol. 15, no. 3, pp. 285–309, Mar. 1975. [16] C. Wang and S. M. Shahhidehpour, “Effect of ramp-rate limits on unit commitment and economic dispatch,” IEEE Trans. Power Systems, vol. 8, no. 3, pp. 1341–1349, Aug. 1993. [17] C. Wang, S. M. Shahhidehpour, D. S. Kirschen, S. Mokhtari, and G. D. Irisarri, “Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation,” IEEE Trans. Power Systems, vol. 10, no. 3, pp. 1294–1301, Aug. 1995. [18] N. Watanabe, Y. Nishimura, and M. Matsubara, “Decomposition in large systems optimization using the method of multipliers,” Journal of Optimization Theory and Applications, vol. 25, no. 2, pp. 181–193, 1978. [19] J. A. Wood and F. B. Wollenberg, Power Generation, Operation, and Control: John Wiley & Sons, 1996. [20] H. Yan, P. Luh, X. Guan, and P. Rogan, “Scheduling of hydrothermal power systems,” IEEE Trans. Power Systems, vol. 8, no. 3, pp. 1358–1363, Aug. 1993. [21] “IEEE reliability test system,” IEEE Trans. Power Apparatus and Systems, vol. PAS-98, no. 6, pp. 2047–2054, Nov./Dec. 1979.

Salem Al-Agtash (M’99) received the Ph.D. degree in electrical engineering from University of Colorado at Boulder in 1998. Dr. Al-Agtash is now an assistant professor of Computer Engineering at Yarmouk University. His research interests are in the areas of power systems, software applications, intelligent systems and expert control.