Optimal Voltage Regulation for Unbalanced Distribution Networks Considering Distributed Energy Resources Guodong Liu∗,Oguzhan˘ Ceylany, Yan Xu∗ and Kevin Tomsovicy ∗Power and Energy Systems Group, Oak Ridge National Laboratory Oak Ridge, Tennessee 37381 Email: [email protected], [email protected] yDepartment of Electrical Engineering and Computer Science University of Tennessee, Knoxville, Tennessee 37996 Email: [email protected], [email protected] Abstract—With increasing penetration of distributed genera- potentially could compromise operating efficiency. Within tion in the distribution networks (DN), the secure and optimal this context, new control and operation strategies capable of operation of DN has become an important concern. In this coordinating different types of distributed energy resources paper, an iterative quadratic constrained quadratic programming model to minimize voltage deviations and maximize distributed (DERs) efficiently to achieve operational objectives are in energy resource (DER) active power output in a three phase particular need of development [2]. unbalanced distribution system is developed. The optimization Management and optimizing the operation of these active model is based on the linearized sensitivity coefficients between DNs require a full AC-formulation of the power flow equa- controlled variables (e.g., node voltages) and control variables tions. Various approaches have been proposed for solving the (e.g., real and reactive power injections of DERs). To avoid the oscillation of solution when it is close to the optimum, a golden distribution optimal power flow (DOPF) problem in litera- search method is introduced to control the step size. Numerical ture. Generally, these approaches can be classified into two simulations on modified IEEE 13 nodes test feeders show the categories. In the first category, the nonlinear optimization efficiency of the proposed model. Compared to the results solved problem is directly solved using nonlinear programming meth- by heuristic search (harmony algorithm), the proposed model ods such as gradient search or interior point methods [3]-[6]. converges quickly to the global optimum. Index Terms—Voltage regulation, sensitivity coefficients, lin- The solution time and convergence characteristic of nonlinear earization, distributed energy resource (DER), golden search programming in solving the DOPF problem may not satisfy method, unbalanced distribution network. the stringent time constraints required by real-time controls. More commonly, as in the second category, the nonlinear I. INTRODUCTION optimization problem is addressed by iteratively solving the linearized problem [7]-[12]. These linear programming based With increasing penetration of distributed energy resources methods are generally more efficient in term of solution time. (e.g., wind turbines, PV panels, microturbines, fuel cells, mini- Nevertheless, the linearized problem is formulated based on hydro, battery storage, and so on) in the distribution networks the calculated sensitivity coefficients. When the solution is (DN), the traditional passive DNs without any sources are close to the optimum, the linearized sensitivities do not repre- gradually transforming into active ones with both dispatchable sent the nonlinear system correctly. In this case, the solution and non-dispatchable power sources[1]. Correspondingly, the may oscillate around the optimum and fail to converge. usual “install and forget” principle becomes infeasible and Traditionally, voltage profile and power flow in distribution This work was sponsored by the Office of Electricity Delivery & Energy feeders have been locally controlled using switched devices Reliability, U.S. Department of Energy under Contract No. DE-AC05-00OR such as shunt capacitor banks and voltage regulators. These 22725 with UT-Battelle and conducted at ORNL and UT Knoxville. This work devices are expected to switch only a few times a day to also made use of Engineering Research Center Shared Facilities supported by the Engineering Research Center Program of the National Science Foundation accommodate relatively slow variations in load. Thus, they and the Department of Energy under NSF Award Number EEC-1041877 cannot effectively respond to mitigate the frequent voltage and the CURENT Industry Partnership Program. The second author would deviations caused by the rapid output fluctuations of renewable like to thank the Scientific and Technological Research Council of Turkey (TUBITAK) for its financial support. resources. Under this circumstance, inverter-interfaced DERs Guodong Liu and Yan Xu are with the Oak Ridge National Laboratory can provide dynamic voltage/VAR support to maintain system (ORNL), Oak Ridge, TN 37831 USA (email: [email protected]; [email protected]). voltage stability and improve voltage profile [13]-[16]. The Oguzhan˘ Ceylan and Kevin Tomsovic are with the Department of Electrical 2 2 2 Engineering and Computer Science, The University of Tennessee (UT), well known apparent power equation, S = P + Q , shows Knoxville, TN 37996 USA (email: [email protected]; [email protected]). that a small change in active power output of DERs can actually provide a wide range of reactive power capability. linear equation as following: This paper is a direct extension of [15], [16] with improve- 3NG ments in multiple aspects. The main contributions of this paper X P DER Q DER 4 jVjj = Kij 4Pi + Kij 4Qi (2) are as follows. i=1 1) A nonlinear one dimensional optimization problem is where KP and KQ are the sensitivity coefficients of voltage added to determine the optimal step size of the control ij ij deviations at node j to control variables variations 4P DER variables. The problem is solved by the golden search i and 4QDER. Other constraints are included as following: method. By this way, the oscillation of solution is i (k) avoided and the convergence is accelerated. 0:95 ≤ jVjj + 4 jVjj ≤ 1:05 8 j (3) 2) In sensitivity calculation [17], various types of power DER(k) DER max injections (i.e. constant PQ, constant impedance and Pi + 4Pi ≤ Pi 8 i (4) 2 2 constant current) are taken into account and the accuracy DER(k) DER DER(k) DER 2 of calculation is improved by removing the assumption Pi + 4Pi + Qi + 4Qi ≤ Si 8 i of constant PQ injections as in [15], [16]. (5) max 3) A population based heuristic search method, harmony where Pi is the maximum real power output of DER i and search, is also used to solve the problem [18]. The results Si is the capacity of DER i or the inverter connected. indicate that the proposed linearized model is indeed A QCQP algorithm is used to solve the optimization prob- converging to the global optimum. lem (1-5). Hence, the improvement direction 4x of the control variable vector x = P DER;QDER, with P DER and QDER The rest of this paper is organized as follows. In Section being the active and reactive power set-points of all DERs, are II, the quadratic constrained quadratic programming (QCQP) determined. The initial values of the DERs’ control variables formulation for the linearized optimization problem of voltage are modified by 4x and the new improvement direction is regulation is presented. The golden search method is integrated calculated based on the updated sensitivity coefficients at to accelerate the convergence. In Section III, the proposed current operating condition. This iterative procedure continues model is validated on modified IEEE 13 nodes test feeders and until the objective function or the control variables do not the results are compared to those of the harmony search based change significantly between two consecutive iterations or a optimization. Section IV summarizes the paper and presents maximum number of iterations is reached. conclusions. It should be noted that the quadratic capacity constraint (5) can also be linearized. With linear objective function, e.g., II. FORMULATION AND METHODOLOGY summation of absolute value of voltage deviations, the QCQP In this section, the QCQP formulation for the linearized can be adapted to linear programming, which has much higher optimization problem of voltage regulation is presented. Then, solution efficiency. the golden search method is introduced to avoid the oscillation- B. Golden Search Method for Avoiding Oscillations s when close to the optimum. After that, the harmony search algorithm is introduced. The QCQP problem is formulated based on the calculated sensitivity coefficients. When the operating point is far from A. QCQP Formulation for Optimal Voltage Regulation the optimum, the solution of the QCQP gives a reasonable improvement direction 4x since the sensitivity coefficients In distribution systems operation, one wants the bus volt- represent the nonlinear system within a small error range. age magnitudes as close as possible to 1 pu. Large voltage However, as the solution approaches the optimum, the nonlin- deviations may cause inefficiencies and other problems. Other earity of system increases. When the solution is close to the considerations in the objective function is maximizing active optimum, the sensitivity coefficients can change dramatically. power output of inverter based DER. The optimal voltage In fact, their signs may change at the optimum. In this regulation problem is formulated as a QCQP. The controlled case, the linearized sensitivities do not represent the nonlinear variables are the node voltages and the control variables are system correctly. As a result, the solution oscillates around the the active and reactive power injections of the DERs. The optimum and may not converge as shown in Fig. 1. objective of the optimal voltage regulation problem is: To avoid the oscillations when close to the optimum, a 3N 3N golden search method is introduced to control the step size of G B 2 X DER X (k) 4x. After the calculation of the k+1 iteration, the elements of min −4Pi + jVjj + 4 jVjj − 1 (1) 4x (k+1) i=1 j=1 improvement direction 4x are compared with that of the last iteration 4x(k). If any pair of them has a different sign, where 4x = 4P DER; 4QDER is the control variable an oscillation is detected. Then, a nonlinear one dimensional vector. NG is the number of DERs and NB is the number optimization problem is solved by the golden search method of nodes.