Abstract
Volt/var optimization (VVO) is a control function that is employed in distribution systems to keep the load voltages within the standard limits, and it includes secondary objectives such as loss minimization. The power flow based VVO is the way of choice in practical applications because it can handle a variety of objective functions and provides a solution even for large-scale network instances. This paper extends the power flow based VVO to account for uncertainty in both the load values and the power generation from photovoltaic sources. The proposed method employs circular arithmetic in complex variables to compute VVO settings that guard against load uncertainty and an optimized linear decision rule that modulates the reactive power of photovoltaic inverter in function of its active power. Finally, the proposed method is tested on distribution networks with up to 3146 nodes and is shown to produce optimal solutions that are robust against power variations.
THE control of voltage and reactive power in modern distribution networks achieves the best performance when being carried out via a centralized system, as opposed to local automatic controllers. The local automatic controllers have their parameters computed using offline studies and therefore tend to be ineffective in many of the scenarios that are encountered in the practical operation. Centralized volt/var optimization (VVO) is a control function that is integrated into the system for supervisory control and data acquisition (SCADA) and the distribution management system (DMS). It makes use of a real-time network model derived from the result of the distribution system state estimator (DSSE). For the daily operation of the distribution network, the centralized VVO is commonly used in closed-loop mode; another option is the advisory mode that requires the review of the operator before implementation [
In practical applications, the centralized VVO can be rule-based or power flow based, but both start from the system state as given by the DSSE. The rule-based method employs a pre-determined set of control rules derived from offline studies, while the power flow based method computes the optimal solution of an objective function using a multi-step discrete programming search [
The periodicity of centralized VVO generally is up to 15 min. During this interval, the values of loads and power generation from photovoltaic (PV) sources can deviate from the estimates given by the DSSE and consequently employed in the VVO; such variations may give rise to significant voltage violations. A work-around is to cast the power flow model as a conic program and formulate the VVO problem using robust [
This paper extends the practical power flow based VVO to guard against uncertainty in the load and power generation from PV sources. Rather than having the demand modeled as a fixed value, it is considered to vary within a disc in the complex plane. The disc has its center at the forecasting value, and its radius is chosen to reflect the uncertainty in the actual load. The disc load model allows the multi-step discrete programming search to compute the radius of voltage variation via circular arithmetic [
The proposed VVO method is tested on radial and weakly meshed distribution networks with up to 3146 nodes. Monte Carlo analysis shows that the VVO solution computed under uncertainty is immune to power injection variations, contrary to the classical multi-step discrete programming search.
Centralized VVO can be used to optimize any mathematical function that aligns with the operation objectives of the distribution network. One possible function to be minimized involves the sum of the power loss and a penalty term for voltage magnitudes that violate their minimum/maximum limits [
(1) |
where is the active power loss of the network; is the voltage magnitude at node ; is the number of nodes; is the penalty coefficient for voltage violation; and and are the minimum and maximum voltage magnitude limits, respectively. Reference [

Fig. 1 Flowchart of DCD search for VVO.
The DCD algorithm starts by initializing the control settings of LTC transformer taps, switched capacitors, and reactive power from PV inverters. The initial values could be either the current operational ones or the nearest rounded ones computed from continuous optimization [
VVO is a problem with a non-convex search space and discrete decision variables. Therefore, the DCD search may converge only to a local optimum, close to the initial operation point, but in line with the practical necessity of reduced controller switching. The DCD search forms the basis for considering uncertainty as described below.
The previous DCD search for VVO computes the control settings for a snapshot of the system that corresponds to the estimated (constant power) complex load and the active power generation from PV inverters. Therefore, the performance of the VVO is expected to deteriorate as the power injections deviate from their estimated values over the time interval between the control set-points in the field. In practice, this may translate into voltages operating outside their bounds. This section proposes a method that hedges the VVO against power injection uncertainty. The method builds on a linear approximate relationship between the nodal voltage magnitude and the complex power injections.
Consider the network shown in
(2) |

Fig. 2 Network with complex power injections and a slack node.
where the bar sign indicates the complex conjugation.
By using Wirtinger calculus [
(3) |
where is the complex nodal voltage at the network operation point and the partial derivatives are evaluated at the same point. Exchanging terms between the summations in (3) results in:
(4) |
(5) |
To obtain the values of the partial derivatives in (5), i.e., and , at the current operation point, the nodal equations for the network are considered as specified in
(6) |
(7) |
(8) |
where is the scheduled slack node voltage; and are column vectors containing the complex power injections and voltages at nodes 2 to , respectively; and denotes the Hadamard (element-by-element) division. Taking the partial derivative of (6) with respect to (), we can obtain:
(9) |
(10) |
where and denote the Hadamard product and power, respectively. Similarly, taking the partial derivative of the conjugate of (6) with respect to (), we can obtain:
(11) |
(12) |
where is an vector of zeros except for element that has the value . Combining (9)-(12), we can obtain the following system of equations:
(13) |
where is the diagonal matrix operator that converts a vector to a square diagonal matrix with the elements of the vector on the main diagonal. The solution to (13) is the partial derivatives in the column vectors and . Cosidering (14), the complex conjugate of this solution gives the coefficients needed in (5) [
(14) |
To model the load uncertainty, it is assumed that its complex power injection can vary within an uncertainty disk in the complex plane. The center of this disk is at the forecasting load value from the DSSE, and its radius is chosen based on historical observations as a fraction of the apparent power. Therefore, the variation over the predicted value lies within a disk centered at the origin and with radius . It can be defined by the following circular complex interval [
(15) |
where the two values in are the center and radius, respectively.
(16) |
(17) |
Using circular arithmetic, is evaluated to a circular complex interval:
(18) |
(19) |
Therefore, the approximation (16) for the voltage magnitude at node decreases to a real interval:
(20) |
Using the minimum and maximum limits of the real interval (20), the VVO objective function (1) can be modified to account for load uncertainty:
(21) |
When using (21) in the DCD search in
The DCD search in
(22) |
where . To mitigate the effect of the PV active power variability on voltage excursions, the reactive power of inverters can be adjusted following an affine decision rule [
(23) |
where is the slope, which is a constant that can be optimally chosen and communicated to the inverter together with . Then, (16) can be used to estimate the voltage magnitude variation at node corresponding to a change in the complex power injection from a PV inverter at node :
(24) |
where and are the real and imaginary parts of as computed in (17), respectively. The optimal value of is calculated by minimizing the effect of on the sum of voltage deviations (24) squared over all nodes, i.e.,
(25) |
The closed-form solution is obtained by taking the derivative of (25) with respect to and setting it to be zero:
(26) |
Using (26), the reactive power contribution from the inverter at node becomes:
(27) |
If the apparent power capacity of inverters is , then the corresponding active power and reactive power (27) have to satisfy [
(28) |
(29) |
Thus, if from (27) does not satisfy (29), then it is fixed at the violated limit.
The three methods listed below are implemented for comparison.
1) Method 1 (M1): the DCD search in
2) Method 2 (M2): the DCD search in
3) Method 3 (M3): the DCD search in
The VVO implementations (M1, M2, and M3) are tested on four networks, which include modified radial version (B_R) and modified meshed version (B_M) of a Brazilian distribution network in addition to two more extensive meshed networks with 1464 and 3146 nodes, which are denoted as 1k5 and 3k, respectively.
In this ideal scenario, M1 and M2 realize the same reduction in loss value that is higher than that of M3, i.e., M3 appears at a disadvantage. The benefit of accounting for uncertainty in the VVO solutions, as in M2 and M3, is apparent whenever the injections deviate from their nominal values. A Monte Carlo simulation is carried out with 1000 trials of load and PV active power uniformly chosen from the abovementioned uncertainty sets. Half of the trials consider conforming load variations chosen from the peripheries of the uncertainty discs, while the other half do not. The results of the simulations are summarized in Tables III-V that quantify the effects of M2 and M3 on the robustness of the VVO settings.
Tables
In terms of computation time, the largest network is solved in less than 1 min using M1 and M2, and less than 13 min using M3. The computation time is recorded using a MATLAB implementation running on a MacBook Pro that has 2.9 GHz Intel Core i5 processor with a memory of 8 GB 2133 MHz.
A central aspect of M3 is the computation of the voltage radius. A Monte Carlo simulation with 10000 trials is used to validate the closed-form solution for the voltage magnitude radius , as given by (19). In each experiment, the nodal complex power injections are sampled from the boundaries of the uncertainty discs, and the current injection power flow is used to find the voltage solution of the trial and its distance from the nominal solution. The radius at each node is computed as the maximum distance observed over the 10000 trials. To quantify the accuracy of the radius computation (19), the relative error at each node is defined by (30) or (31).
(30) |
(31) |
Centralized VVO is practically solved using a multi-step discrete programming search that gives a solution corresponding to the estimated power injections, involving loads and power from PV sources. This paper proposes a methodology for extending the classical VVO solution to account for the uncertainty in the power injections during the time interval when the VVO control settings are applicable. The method is two-pronged and aims to alleviate voltage magnitude violations. It involves the use of circular arithmetic to guard against load variations and a linear decision rule that tailors the reactive power of PV inverters in function of its active power. Both components of the solution that combats uncertainty are fundamentally based on a linear approximated relationship between the nodal voltage magnitude and complex power injections. The relation is a complex variable first-order Taylor series expansion of the voltage magnitude, derived by Wirtinger calculus. The proposed method is tested on distribution networks with up to 3146 nodes and compared with the techniques that either neglect uncertainty altogether [
References
I. Roytelman and J. M. Palomo. (2016, May). Volt/var control in distribution systems. [Online]. Available: http://digital-library. theiet.org/content/books/10.1049/pbpo075e_ch8 [Baidu Scholar]
I. Roytelman, B. K. Wee, and R. L. Lugtu, “Volt/var control algorithm for modern distribution management system,” IEEE Transactions on Power Systems, vol. 10, no. 3, pp. 1454-1460, Aug. 1995. [Baidu Scholar]
W. F. Tinney, “Compensation methods for network solutions by optimally ordered triangular factorization,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-91, no. 1, pp. 123-127, Jan. 1972. [Baidu Scholar]
M. Juamperez, G. Yang, and S. B. KJÆR, “Voltage regulation in LV grids by coordinated volt-var control strategies,” Journal of Modern Power Systems and Clean Energy, vol. 2, no. 4, pp. 319-328, Dec. 2014. [Baidu Scholar]
G. Yang, F. Marra, M. Juamperez et al., “Voltage rise mitigation for solar PV integration at LV grids,” Journal of Modern Power Systems and Clean Energy, vol. 3, no. 3, pp. 411-421, Sept. 2015. [Baidu Scholar]
E. Ghiani and F. Pilo, “Smart inverter operation in distribution networks with high penetration of photovoltaic systems,” Journal of Modern Power Systems and Clean Energy, vol. 3, no. 4, pp. 504-511, Dec. 2015. [Baidu Scholar]
Y. Shi and M. Baran, “A gradient based decentralized volt/var optimization scheme for distribution systems with high DER penetration,” in Proceedings of 2019 IEEE PES GTD Grand International Conference and Exposition Asia (GTD Asia), Bangkok, Thailand, Mar. 2019, pp. 649-654. [Baidu Scholar]
IEEE PES Industry Technical Support Task Force. (2018, May). Impact of IEEE 1547 standard on smart inverters. [Online]. Available: https://resourcecenter.ieee-pes.org/publications/technical-reports/PES_TR0067_5-18.html [Baidu Scholar]
R. A. Jabr and I. Džafić, “Sensitivity-based discrete coordinate-descent for volt/var control in distribution networks,” IEEE Transactions on Power Systems, vol. 31, no. 6, pp. 4670-4678, Nov. 2016. [Baidu Scholar]
A. Borghetti, “Using mixed integer programming for the volt/var optimization in distribution feeders,” Electric Power System Research, vol. 98, pp. 39-50, May 2013. [Baidu Scholar]
A. Borghetti, F. Napolitano, and C. A. Nucci, “Volt/var optimization of unbalanced distribution feeders via mixed integer linear programming,” International Journal of Electrical Power & Energy Systems, vol. 72, pp. 40-47, Nov. 2015. [Baidu Scholar]
T. Ding, S. Liu, W. Yuan et al., “A two-stage robust reactive power optimization considering uncertain wind power integration in active distribution networks,” IEEE Transactions on Sustainable Energy, vol. 7, no. 1, pp. 301-311, Jan. 2016. [Baidu Scholar]
F. U. Nazir, B. C. Pal, and R. A. Jabr, “A two-stage chance constrained volt/var control scheme for active distribution networks with nodal power uncertainties,” IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 314-325, Jan. 2019. [Baidu Scholar]
T. Ding, Q. Yang, Y. Yang et al., “A data-driven stochastic reactive power optimization considering uncertainties in active distribution networks and decomposition method,” IEEE Transactions on Smart Grid, vol. 9, no. 5, pp. 4994-5004, Sept. 2018. [Baidu Scholar]
K. S. Ayyagari, N. Gatsis, and A. F. Taha, “Chance constrained optimization of distributed energy resources via affine policies,” in Proceedings of 2017 IEEE Global Conference on Signal and Information Processing, Montreal, Canada, Nov. 2017, pp. 1050-1054. [Baidu Scholar]
K. Baker, E. Dall’Anese, and T. Summers, “Distribution-agnostic stochastic optimal power flow for distribution grids,” in Proceedings of 2016 North American Power Symposium (NAPS), Denver, USA, Sept. 2016, pp. 1-6. [Baidu Scholar]
L. Dong, J. Li, T. Pu et al., “Distributionally robust optimization model of active distribution network considering uncertainties of source and load,” Journal of Modern Power Systems and Clean Energy, vol. 7, no. 6, pp. 1585-1595, Nov. 2019. [Baidu Scholar]
P. Chen, X. Xiao, and X. Wang, “Interval optimal power flow applied to distribution networks under uncertainty of loads and renewable resources,” Journal of Modern Power Systems and Clean Energy, vol. 7, no. 1, pp. 139-150, Jan. 2019. [Baidu Scholar]
I. Gargantini and P. Henrici, “Circular arithmetic and the determination of polynomial zeros,” Numerische Mathematik, vol. 18, no. 4, pp. 305-320, Aug. 1971. [Baidu Scholar]
K. Kreutz-Delgado. (2009, Jun.). The complex gradient operator and the CRcalculus. [Online]. Available: http://arxiv.org/abs/0906.4835 [Baidu Scholar]
I. Džafić, R. A. Jabr, and T. Hrnjić, “Hybrid state estimation in complex variables,” IEEE Transactions on Power Systems, vol. 33, no. 5, pp. 5288-5296, Sept. 2018. [Baidu Scholar]
R. A. Jabr, “High-order approximate power flow solutions and circular arithmetic applications,” IEEE Transactions on Power Systems, vol. 34, no. 6, pp. 5053-5062, Nov. 2019. [Baidu Scholar]
W. Lin, R. J. Thomas, and E. Bitar, “Real-time voltage regulation in distribution systems via decentralized PV inverter control,” in Proceedings of the 51st Hawaii International Conference on System Sciences, Hawaii, USA, Jan. 2018, pp. 2680-2689. [Baidu Scholar]
W. Lin and E. Bitar, “Decentralized stochastic control of distributed energy resources,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 888-900, Jan. 2018. [Baidu Scholar]
R. A. Jabr, “Robust volt/var control with photovoltaics,” IEEE Transactions on Power Systems, vol. 34, no. 3, pp. 2401-2408, May 2019. [Baidu Scholar]
K. Turitsyn, P. Sulc, S. Backhaus et al., “Options for control of reactive power by distributed photovoltaic generators,” Proceedings of the IEEE, vol. 99, no. 6, pp. 1063-1073, Jun. 2011. [Baidu Scholar]
R. A. Jabr. (2020, Mar.). “VVO distribution network data sets,” [Online]. Available:https://www.dropbox.com/s/pyi0tfatm8e353m/mpce_VVO.zip?dl=0 [Baidu Scholar]