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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH 2014
Fast Dynamic Economic Power Dispatch Problems Solution Via Optimality Condition Decomposition Abbas Rabiee, Behnam Mohammadi-Ivatloo, Member, IEEE, and Mohammad Moradi-Dalvand Abstract—This letter proposes an efficient real-time approach based on optimality condition decomposition (OCD) technique to solve dynamic economic dispatch (DED) problem. Ramp-rate limits, prohibited operation zones (POZs) constraints, along with transmission losses are considered in the studied DED model. In order to examine the effectiveness of the proposed approach, it is implemented on a 54-unit test system. The obtained results verify applicability of the proposed method for solving the DED problem in real-time environment.
Generating units may have certain restricted operation zone due to limitations of machine components or instability concerns. Power output of the generator must fall into one of the allowed operation zones. Hence, this problem can be formulated as a nonlinear programming (NLP) with vanishing constraints as follows [1]: (6)
Index Terms—Dynamic economic dispatch (DED), optimality condition decomposition (OCD), prohibited operation zones (POZs), real-time.
I. DED PROBLEM FORMULATION YNAMIC economic dispatch (DED) is one of the important optimization problems of the power system operation which looks for the best schedule of the committed units. The objective function of DED problem is to minimize the total production cost over the planning horizon, which can be written as
D
(1)
(7) (8) is the number of allowed operation zones. and are the lower and upper limits of the th allowed operation zone of unit . Equations (7) and (8) enforce only one of subcomponents to be nonzero while enforce other the subcomponents to be zero. Instead of using the above model for POZs, one may consider the following mixed integer constraints for dealing with POZs: where
where , and are the fuel cost coefficients of the th unit. is the number of dispatchable power generation units and is the power output of the th unit at time . is the total number of hours in the planning horizon. The constraints of the DED problem are summarized in the following:
(9)
(10)
(2)
(3) (4) (5) and are Hourly power balance is expressed using (2). total load demand and total transmission loss of the system at time , respectively. System loss is calculated using -matrix coefficients in (3). Real power generation limits are enforced and ramp-down limits using (4). The unit ramp-up are enforced using (5), respectively. Manuscript received January 08, 2013; revised April 16, 2013 and June 14, 2013; accepted July 01, 2013. Date of publication November 08, 2013; date of current version February 14, 2014. Paper no. PESL-00011-2013. A. Rabiee is with the Department of Electrical Engineering, Faculty of Engineering, University of Zanjan, Zanjan, Iran (e-mail:
[email protected]). B. Mohammadi-Ivatloo is with the Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran (e-mail:
[email protected]). M. Moradi-Dalvand was with the Department of Electrical Engineering, Power and Water University of Technology, Tehran, Iran (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2013.2288028
(11) is a binary variable, which corresponds to the th where allowed operation zone of unit at time . This model leads to a mixed integer NLP (MINLP) problem. II. IMPLEMENTATION OF OCD ON DED PROBLEM The DED problem is very large-dimension NLP (or MINLP) problem in practical large-scale power systems. Hence, for realtime implementation of the DED problem, the solution-time (or CPU-time) is very critical. If NLP model is employed for the DED problem, it could be decomposed to several simpler subproblems with lower dimensions (e.g., 24 subproblems for a DED with 24 hours time-span). Hence, the required CPU-time will be reduced significantly. In this letter, optimality condition decomposition (OCD) [2] is utilized for this aim. OCD approach relaxes complicating constraints of original large-scale NLP problem [2]. The resulting subproblems can be solved in parallel, in an iterative manner by the OCD approach. In the DED problem, ramp-rate constraints, i.e., (5), are the complicating constraints. By relaxing these constraints, the th re) of the DED in the laxed subproblem (RSP), (i.e., for th iteration of the OCD is as follows:
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(12)
RABIEE et al.: FAST DYNAMIC ECONOMIC POWER DISPATCH PROBLEMS SOLUTION VIA OPTIMALITY CONDITION DECOMPOSITION
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where
(13) The objective function of the th subproblem in iteration , i.e., (12), should be minimized subject to the following constraints: Fig. 1. Convergence characteristics of the OCD algorithm.
(14) (15) and are Lagrange multipliers correwhere sponding to complicating constrains (5) of the th subproblem and are replaced with at iteration . In (15), and in the corretheir current values, i.e., sponding equations. The parameters with a bar above them denote the values of the corresponding variables at the last ). DED problem can be solved by iteration (i.e., iteration using OCD algorithm in three steps as in the following. Step-0: Initialization In this step, all variables and Lagrange multipliers of complicating constrains (5) are initialized. In this letter, the initial values for variables are chosen by solving the DED without considering transmission losses, POZs, and ramp rate limits. This initial problem is basically simple and convex ED problem. Step-1: Independently solving of the RSPs In this phase, the RSPs are solved independently, and the optimal values for all variables are obtained, along with the Lagrange multipliers of complicating constraints (5). Step-2: Stopping criterion The algorithm stops if the variables or the overall objective function (1) value do not change significantly in two consecutive iterations [2]. Otherwise, it continues from Step-2. It is worth to mention that if the RSPs solved sequentially in time and not in parallel, the most recent obtained values for variables could be used, to accelerate the convergence of the algorithm. In other words, if at iteration , the RSP for period is solved before the RSP of period , therefore it is more appropriate to consider most recent values for (14). That is, in should be used instead of . (14), III. CASE STUDY AND NUMERICAL RESULTS The proposed method is coded in GAMS [3] environment, on a Pentium 4, dual core, 1.83-GHz personal computer with 2 GB of RAM under Windows XP. The NLP and MINLP models are solved by CONOPT and SBB solvers, respectively. The proposed real-time DED model is examined on 54-unit test system. The data of this system are adopted from [4]. The convergence characteristics of the OCD algorithm are represented in Fig. 1. The OCD is converged quickly after 4 iterations. Also, total cost and CPU-times obtained by the OCD (with and without PCA), MINLP, and NLP approaches are presented in Table I for the load demand given in [5]. It is worth to mention that the default settings of SBB solver are used to obtain the results in MINLP model (as given in Table I). By increasing the time limit (i.e., ’reslim’ option in
TABLE I COMPARISON OF OPTIMIZATION RESULTS FOR 54-UNIT TEST SYSTEM
GAMS) to 20 000 s, and by setting relative gap to zero in the options of SBB (i.e., by setting ‘optcr=0’), the optimum cost will be , with the corresponding CPU-time of 19 631.125 s. Although the obtained cost is (i.e., 0.00327%) less than the cost obtained by OCD, but the corresponding CPU-time is much more in comparison with the very low CPU-time of OCD. The negligible reduction of cost at the expense of drastically increase of CPU-time may not be desirable from the real-time operation perspective. It is noteworthy that in real-time applications, the optimal generation schedule is needed for the next few hours, subject to the forecasted load profile in the intervals equal to a fraction of an hour (e.g., 10-min intervals). The results presented in Table I substantiate that the proposed approach is well capable to attain the optimal solution of DED in a very short time, in comparison to the existing methods. Hence, the proposed approach is efficient for solution of DED in real-time environment. IV. CONCLUSION In this letter, a new fast solution approach, based on OCD, is proposed to solve DED problem in real-time applications. The numerical results obtained by different approaches (i.e., OCD, MINLP, and NLP) demonstrate that the proposed algorithm is well capable to give better results (more optimal generation schedules) in lower solution time. Besides, the proposed algorithm facilitates parallel computation ability and hence, it is applicable for real-time operation of practical power systems. REFERENCES [1] R. Jabr, “Solution to economic dispatching with disjoint feasible regions via semidefinite programming,” IEEE Trans. Power Syst., vol. 27, no. 1, pp. 572–573, Feb/ 2012. [2] A. J. Conejo, E. Castillo, R. Mnguez, and R. Garc-A-Bertrand, “Decomposition in nonlinear programming,” in Decomposition Techniques in Mathematical Programming. Berlin, Germany: Springer, 2006, pp. 187–242. [3] A. M. A. Brooke, D. Kendrick, and R. Roman, GAMS: A User’s Guide. Washington, DC, USA: GAMS, 1998. [4] [Online]. Available: motor.ece.iit.edu/data/SCUC_118test.xls (accessed Feb. 2011). [5] B. Mohammadi-Ivatloo, A. Rabiee, A. Soroudi, and M. Ehsan, “Imperialist competitive algorithm for solving non-convex dynamic economic power dispatch,” Energy, vol. 44, pp. 228–240, 2012.
