commitment stage and the constrained economic dispatch stage. The proposed ... software package which addresses energy management systems.
IEEE Transactions on Power Systems, Vol. 11, No. 1, February 1996
254
A PRACTICAL RESOURCE SCHEDULING WITH OPF CONSTRAINTS K.H. Abdul-Rahman. S.M. Shahidehpour Department of Electrical and Computer Engineering Illinois Institute of Technology Chicago, IL 60616
1:;
M. Aganagic S. Mokhtari Empros Power Systems Control Siemens Energy & Automation Brooklyn Park, MN 55428
Abstract--This paper presents an efficient approach to short term resource scheduling based on the augmented Lagrangian relaxation method. The problem is divided into two stages, the commitment stage and the constrained economic dispatch stage. The proposed mathematical model incorporates optimal power flow (OPF) constraints in the unit commitment stage. By OPF constrains, we refer to the relevant active power constraints that are incorporated in the constrained economic dispatch stage (i.e. transmission capacity constraints, fuel and various r e g lated emission requirements). The inclusion of OPF constraints in the commitment stage will improve the feasibility of the constrained economic dispatch solution. Other unit commitment constraints such as spinning and operating reserve requirements, power balance as well as other relevant local constraints (i.e. unit ramping rates, upper and lower generation limits, minimum up and down times) are taken into account in the proposed model. As we deal with a larger number of constraints, a more rigorous method is introduced for updating Lagrange multipliers to improve the solution convergence. A software package which addresses energy management systems requirements is developed and tested.
dual problem is decoupled into unit based subproblems which are easier to solve. Due to the non convexity of the resource scheduling problem, difficulties were reported in obtaining a feasible solution. Many modifications were made to the basic algorithm to achieve better performance and/or accommodate more practical constraints [l-91.
Keywords: Resource Scheduling, Augmented Lagrangian Relaxation, Fuel Constraints, OPF Constraints
In our previous study [4], a mathematical model for short term resource scheduling was developed. The augmented Lagrangian relaxation method was used to overcome the difficulties encountered in resource scheduling. The convexity of the problem and convergence properties of the solution were improved by adding quadratic penalty terms to the objective function. The auxiliary problem principle was utilized to make the quadratic terms separable. The optimization problem was decomposed into N subproblems, each corresponding to the optimal unit commitment for individual units over the entire time span. Dynamic programming was used to solve each subproblem. The objective in that study was to present an approach for improving the resource scheduling solution.
1. INTRODUCTION The short term resource scheduling problem is one of the critical issues in the economical operation of power systems. Resource scheduling determines the commitment of generating units for minimizing the total power production costs (i.e. operation, maintenance and start up costs). Resource scheduling must ultimately satisfy generation unit constraints as well as transmission, emission and other relevant system constraints. Resource scheduling represents a large mixed integer programming problem, involving thousands of variables and a large number of equality and inequality constraints. Due to the existing difficulties associated with the non convexity of the objective function and the size of the problem, a number of solution methodologies were proposed for resource scheduling. Namely, heuristic methods (priority list type), dynamic programming methods, branch and bound methods, Benders partitioning methods and Lagrangian relaxation (LR) methods. The LR approaches are characterized by their ability to handle various constraints and to estimate the optimality of the solution in practical applications. The Lagrangian dual function is formed by adjoining a set of coupling constraints to the unit commitment primal objective function via Lagrange multipliers. The * Currently an employee of Siemens/Empros
This paper was presented at the 1995 IEEE Power Industry Computer Applications Conference held in Salt Lake City, Utah,May 7-12, 1995.
In order to manage the size of the resource scheduling problem in real time applications, the transmission network was reduced to one node in unit commitment which could lead to infeasible solutions with overloaded transmission lines in the constrained economic dispatch [3,4]. In compliance with the Clean Air Act of 1990, utilities are faced with additional problems for minimizing the cost of power systems operation. The current approach for modeling emission constraints in constrained economic dispatch (rather than embedding them in the commitment stage) may lead to an infeasible or more costly solution. As we introduce additional constraints (i.e. fuel constrained units) a more comprehensive approach is needed for updating Lagrange multipliers and reducing the duality gap in resource scheduling
PI.
This paper addresses more practical issues of utmost concern to power utilities which were not considered in [4] such as ramp rate limits, multiple emission requirements as well as fuel constraints. Moreover, a detailed description of the various aspects of the augmented Lagrangian relaxation applied to resource scheduling in an energy management system is discussed in this paper. The resource scheduling problem is divided into two stages, the commitment stage and the constrained economic dispatch stage. The OPF constraints that are relevant to the active power such as transmission capacity constraints, different types of emission requirements (i.e. S O 2 and N e ) , emission caps for certain areas of the system and the total system emission,as well as fuel constraints are considered in the formulation of the commitment stage to ensure the feasibility of the constrained economic dispatch stage. In the constrained economic dispatch, constraints corresponding to transmission capacity, load and reserve requirements as well as generating unit limits are incorporated. To obtain fast and efficient solutions, the constrained economic dispatch problem is decomposed into Nt subproblems, each corresponding to constrained economic dispatch of committed units at a given period. Since we use a piecewise linearized objective function, linear programming (LP) is applied to constrained economic dispatch.
0885-8950/96/$05.000 1995 IEEE
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2. PROBLEM FORMULATION In order to describe the optimization problem, a set of symbols is introduced in the Appendix. The objective of the short term generation resource scheduling is to minimize the power system operating cost which is the sum of the production cost and the start up cost,
8) Transmission line capacity limits Lm 5 Lm(t) 5 L , where
N
PC=
E C [ J ' ( W ( q ( t ) ) si(')] /=I t = l
(1)
+
The production fuel cost Fi(Pi(t)) is calculated as the product of the heat rate curve (MBTU/Hr) and the unit's fuel cost ($/MBTU). The start up cost of a unit depends on how long the unit was off line and is defined as,
sj(t)= qt)[l-I,.(t-l)]
[ (
ai+Bi 1-exp-
the sensitivity coefficients k , and k , can be determined based on DC or AC power flow corresponding to normal or contingency conditions.
9) System emission limit 4
N
c c H,(P,(t)>JXt) EMS
-4;ty
t-1
5
(11)
1-1
where several emission types (e.g. S02, NOx) are considered. 10) Area emission limit
The constraints of the optimization problem are:
NI
q.(P,.(t))J.(t) 5 EMA
1) System power balance equation
c 4(t)pi(t) D(t)
f-1
N
=
t=1,...,N,
(2)
i= 1
(12)
/EA
where i EA corresponds to all units in the constrained emission area A and several emission types can be considered.
2 ) System spinning reserve requirements
3. UNIT COMMITMENT PROBLEM
N
rsj(t)J.(t)2 R,(t) t=1,...,N,
(3)
3.1 Augmented Lagrangian For Unit Commitment
i= 1
where, rsi ( t ) = min {lO*MSRi , - P i ( t ) } .The spinning reserve requirement, R , ( t ) , is typically defined as a base component plus a fraction of the load requirement and a fraction of the high operating limit of the largest on-line unit. 3) System operating reserve requirements N
fOi(t) J.(t)2 RO(t) t=1,...,N,
(4)
i-1
Interruptible load are usually added to the operating reserve capacity of units (left hand side of (4)). The operating reserve requirement, R , ( t ) , is commonly defined similar to R , ( t ) as a function of a base component plus a fraction of the load requirement and a fraction of the high operating limit of the largest on-line unit.
The quality of the final LR solution depends on the sensitivity of the commitment to Lagrange multipliers. Unless a proper modification of multipliers is ensured in every iteration, unnecessary commitment of generating units may occur, which may result in higher operation costs. The difficulties are often explained by the non-convexity of this type of optimization problems. The difficulties can be overcome by the augmented Lagrangian method and the auxiliary problem principle technique [6], in which quadratic penalty terms associated with power demand are added to the objective function to improve the convexity of the problem. Applied to our unit commitment problem, the augmented Lagrangian function is, a (C(tl94 A(t)>PS(t)>P,>(4Y m(t)+,Ym(t)-?vs>va) ( 9 9
N
Nt
=E i-1 t-1
[F,(pi(O)4(t)+'i(t)]
4) Unit generation limits
(5) 5) Thermal unit minimum start up/shut down times (qO"(t-1)q.on)*(J(t-l)-J.(r)) 2 0 (Ay([-1)- 7 y ) * ( J ( t ) - J ( t - l ) )
2 0
(6) (7)
with constraints expressed by (5), (6), (7), (8) and (9). 6) Ramp rate limits
