IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 31, NO. 2, MARCH 2016
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Optimal Power Flow With the Consideration of Flexible Transmission Line Impedance Tao Ding, Student Member, IEEE, Rui Bo, Senior Member, IEEE, Fangxing Li, Senior Member, IEEE, and Hongbin Sun, Senior Member, IEEE
Abstract—This letter presents an optimal power flow model with the consideration of flexible transmission line impedance. By the use of big-M based complementary constraints, the original non-convex model can be transformed into a mixed integer quadratic programming that can be solved by branch and bound method. Numerical results from several test systems show that flexible transmission lines can provide better economic dispatch and reduce the total generation cost. Moreover, some infeasible optimal power flow scenarios can become solvable with optimal adjustment of the flexible transmission line impedance.
II. MATHEMATICAL FORMULATION The general DC optimal power flow can be mathematically expressed as a quadratic program with linear constraints and a quadratic cost function in the following form: (1a)
Index Terms—Big-M, complementary constraint, flexible transmission line impedance, optimal power flow.
(1b) (1c) (1d)
I. INTRODUCTION
O
PTIMAL power flow (OPF) is widely used in economic dispatch to minimize the total generation cost, while guaranteeing the energy balance within generator physical limits and transmission line capacity limits. Traditionally, the transmission elements are treated as fixed assets in the network, except during times of forced outages or maintenance [1]. However, with installing more flexible elements, such as FACTs [2], power system operation becomes more flexible. The transmission line impedances can be changed by TCSC to increase transfer capacity nowadays. It is expected that, with the breakthroughs in the area of materials, sensors, and controls in the future, economically viable flexible devices such as piezoresistive and thermistor impedance may be developed and installed in power systems on a large scale that can enable the adjustment of transmission line impedance. By reducing impedance, transfer capacity can be improved. By increasing impedance on congested transmission lines, power will be shifted to other facilities to mitigate the congestion. In this letter, we propose an optimal power flow model with the consideration of flexible transmission line impedances (FTLIs) to achieve better operational economics. Certainly, possible power system protection problems should be taken into account in practical application [5]. Manuscript received June 30, 2014; revised October 23, 2014; accepted January 27, 2015. Date of publication March 26, 2015; date of current version February 17, 2016. This work was supported in part by NSFC (51428701, 51321005), the 973 Program of China (2013CB228206), and US NSF EEC-1041877. Paper no. PESL-00094-2014. T. Ding is with the Department of Electrical Engineering, Tsinghua University, Beijing, China, and also with The University of Tennessee, Knoxville, USA (e-mail:
[email protected]). R. Bo is with the Mid-Continent Independent Transmission System Operator (Midwest ISO), St Paul, MN 55108 USA. F. Li is with the Department of Electrical Engineering and Computer Science (EECS), The University of Tennessee, Knoxville, TN 37996 USA. H. Sun is with Department of Electrical Engineering, Tsinghua University, Beijing, China. Digital Object Identifier 10.1109/TPWRS.2015.2412682
(1e) where is the generation of the th generator and is the load demand at bus is the triplet coefficients of quadratic cost function of the th generator; denotes the set of indices of generators connected to bus ; the subscript -ij denotes the th line with its “from” bus and “to” bus ; the set of “from” buses and “to” buses are and is the voltage angle of the th bus and is the reactance of the th trans“ref” refers to reference bus; mission line; and are the minimum and maximum generation limit of the th generator; is the transmission caand are the lower and upper bound pacity of the th line; of voltage angle at bus and denote the total number of buses, generators, and transmission lines, respectively. If the reactance of each line is taken as a variable, in the range of for , it yields a new model: (2a) (2b) Note that (2) is a non-convex model, because the term becomes a bilinear function, instead of a linear function, when is taken as a variable. Here, we assume the and are positive. Fortunately, model (2) has a special structure that the bilinear terms always show up together and the only appear in (2b). Therefore, we introduce variables a new dummy vector for each line, whose physical meaning is the power flowing on each line. After , obtaining the optimal solution of (2) with variables the optimal reactance can be uniquely determined by . Therefore, replace the variable by and the constraint (2b) becomes
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(3)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 31, NO. 2, MARCH 2016
However, the sign of cannot be determined beforehand, so it is difficult to transform (3) into a linear constraint by multiplying for both sides of the inequalities. Moreover, if the denominator is zero, the numerator must be zero. According to the different sign of , (3) can be simplified as
TABLE I TRANSMISSION LINE LIMIT AND COMPUTATIONAL TIME
(4) These “if” constraints can be simplified by the use of binary variables and big-M complementary constraints [3], such that
is a big number and where 1) When
TABLE II GEN COST ($) OF OPTIMAL POWER FLOW WITH/WITHOUT FTLIS
(5) is a dummy binary variable. , we have .
Therefore,
whichever equals, 1 or 0, there must be one equation , and the other one is a that leads to redundant constraint that is always satisfied, such that . , the constraints 2) When and are complementary as only one constraint would be active, and the other one is a redundant constraint, leading to . As discussed above, (4) is equivalent to (3), so that the original model (2) can be transformed into a mixed integer quadratic programming (MIQP), which can be easily solved by branch and bound method or cutting plane method. But it has been reported in [3] that should be sufficiently large but not overly big, because an extremely large number may cause numerical instability. Moreover, when solving MIQP, large will result in large feasible region of the relaxed model, so that more iterations are needed to find the optimal solution. In our model, it can be found that the angle difference of line must be in with respect to the power system stability reis chosen to be . quirement, so Finally, the proposed economic dispatch model with flexible transmission line impedances can be cast as follows: (6a) (6b) (6c) (6d) (6e) (6f) III. NUMERICAL EXAMPLE In this section, the proposed model has been studied on six test systems available from MATPOWER [4] and implemented in MATLAB with the CPLEX 12.5. The range of impedance is assumed to be and is chosen as 10% and 20%, respectively. Note that many transmission line limits are not available in [4]. In order to study the impact of FTLIs on optimal power flow, uniform transmission line limits are assumed in Table I. In addition, two scenarios are taken into considera-
tion, where the second scenario (“S2”) has tighter transmission line limits than the first scenario (“S1”). The computation time is presented in Table I, where it needs more time for large systems. Note, if there are limited FTLIs in the system, the discrete variables in the model (6) will be greatly reduced, which will alleviate the computational time. Besides, Table II shows the generation cost and it can be observed that the cost with FTLIs is reduced comparing to that without FTLIs. As well, with the increase of (i.e., increase the available range of impedances), the total cost will be further reduced. Moreover, none of the traditional optimal power flow without FTLIs can be converged due to tight constraints for “S2”, but with the help of FTLIs, they may become feasible. For the first three systems, when %, the optimal power flow is still infeasible, but when %, it become feasible. For the %, the optimal power flow can last three systems, when be converged and the cost from % is better than %. IV. CONCLUSION This letter proposes an optimal power flow model with the consideration of flexible transmission line impedance, which can be formulated as a mixed integer quadratic programming. The results and comparisons on six test systems show that a more economic dispatch is achieved as transmission line impedance becomes flexible. Most importantly, the flexible transmission line impedance can help traditionally unsolvable optimal power flow find an optimal solution. REFERENCES [1] K. W. Hedman, R. P. O'Neill, E. B. Fisher, and S. S. Oren, “Optimal transmission switching with contingency analysis,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1577–1586, Aug. 2009. [2] S. N. Singh and A. K. David, “A new approach for placement of FACTS devices in open power markets,” IEEE Power Eng. Rev., vol. 21, no. 9, pp. 58–60, 2001. [3] T. Ding, R. Bo, W. Gu, and H. Sun, “Big-M based MIQP method for economic dispatch with disjoint prohibited zones,” IEEE Trans. Power Syst., vol. 29, no. 2, pp. 976–977, Mar. 2014. [4] R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, “MATPOWER: Steady-state operations, planning, and analysis tools for power systems research and education,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 12–19, Feb. 2011. [5] P. K. Dash, A. K. Pradhan, G. Panda, and A. C. Liew, “Adaptive relay setting for flexible AC transmission systems (FACTS),” IEEE Trans. Power Del., vol. 15, no. 1, pp. 38–43, Jan. 2000.
