Voltage regulation is crucial to maintain voltage stability in distribution systems under different operating conditions . Traditionally, voltage is regulated by changing reactive power through tap-changing transformers and switched capacitors . With the advances of distributed energy resources (DERs, i.e., electric vehicles , PV panels [4, 5]) reactive power support to regulate voltage may be available. These control strategies have been analyzed in literature by either centralized schema [6, 7] or distributed algorithms [8, 1, 2, 9].
DERs bring significant uncertainty and fast variations in real power to the distribution system [10, 5, 11, 12]. Since most distribution systems do not yet have real-time communication capabilities, a decision made must be valid for a set of possible conditions. For example, in a system with solar PVs, the buses communicate with the feeder (or some other coordinators) every 5 minutes to receive a command for setting its reactive power, but the changes in solar irradiation result in sub-minute timescales changes in its active power. In this paper, we consider a centralized control framework, where periodic control signals are designed to regulate voltages for the entirety of the period in the presence of randomness.
A natural framework to handle the uncertainties introduced by the fast variation in the real power output of the DERs is chance constraints. Indeed, chance constraints bound the probability of voltage constraint violations. Chance constraints may yield difficult optimization problems, since algorithms may have to be designed on a case-by-case basis. Authors in [14, 15, 16] offer different relaxation techniques to break chance constraints into simpler ones and formulate stochastic optimizations in power systems as conic problems. However, this relaxation can be very conservative and may not apply to large-scale problems . In , the authors probe the boundary of the feasible region by the so-called “-efficient points”. However, the proposed procedure requires solving a mixed integer programming at each iteration. In , the authors adopt a gradient-like algorithm that efficiently finds a sub-optimal solution.
However, in all of these papers, the authors focus on solving the optimization problem, instead of explicitly characterizing the feasible region due to uncertainty. We note that it is actually an important topic in some other engineering areas to find a good approximation of the feasible region. In integrated circuit design, the problem of design centering [20, 21], i.e., finding good design parameters inside the feasible region, is crucial in ensuring a good yield in the presence of statistical uncertainties. There are two potential benefits of having an approximation of the feasible region: first, the grid operator can evaluate how safe or crucial the current operation since there is a visualization of the region; second, the algorithm does not need to be re-run when the objective functions or any of the additional constraints change (i.e., different configuration or requirement of the system).
In this paper, we provide an explicit ellipsoidal approximation of the feasible region bounded by the chance constraints of interest. We choose ellipsoids due to their convex geometry and tractability when incorporated in convex objective functions 
. In addition, we provide a sampling algorithm that efficiently trains the proposed model. Compared to random sampling strategy, our algorithm achieves smaller estimation error with comparable number of samples.
Our contributions to ellipsoidal approximation in chance constrained optimization problems are:
Our proposed method is data driven: we do not assume any prior knowledge on the probabilistic distribution of the uncertainty in the system.
We provide a novel view on approximating chance constrained optimizations by applying a machine learning approach.
We present an efficient training algorithm that achieves a small estimation error.
The rest of the paper is organized as follows. In Section I-A, we review the literature on ellipsoidal approximation of feasibility sets and introduce our proposed method. In Section II, we present the mathematical formulation of voltage control in power distribution systems. In Section III we find an approximation of the feasible region by a machine learning approach, and in Section IV we show how this learning model can be efficiently trained using active sampling techniques. Simulation results based on the IEEE standard bus system are shown in Section V. Conclusions are presented in Section VI.
I-a Prior results
The estimation and approximation of feasibility sets using simple geometric structures have been addressed in many applications.
propose to use supporting hyperplanes for a similar purpose. More recently, in, the authors propose to approximate the region of interest by convex piecewise linear machine. In , the authors use polytopes to estimate feasible sets of a model predictive control problem. A problem of these approaches is that linear approximation may end up with many piecewise segments, leading to exponentially many constraints.
An alternative to linear approximation is to use more versatile and smooth functions, for example a polynomial function, among which ellipsoids are commonly adopted. In , the authors propose to approximate convex sets by ellipsoids based on a modification of the volumetric cutting plane method. Authors in 
evaluate the optimality of ellipsoidal approximation by moments of the support function. This approach unfortunately requires the solution of a high dimensional integral. In, the authors use ellipsoids to approximate polytopes. Ellipsoidal approximation is also widely used in control problems to describe the attraction domain  and to approximated polyhedral sets in power systems .
However, either the metrics are hard to compute in a high dimensional setting, or the close form of the target convex set is required in previous papers on ellipsoidal approximation. Instead, we consider a data-driven approach and propose an efficient sampling algorithm that finds the boundary of an unknown feasibility region. By doing so, we do not require any specific knowledge of the geometry on the feasible region, except that it is a bounded set and it is easy to check whether a point is inside the set or not.
More specifically, we use a support vector machine (SVM) approach to identify the boundary of the feasibility set by an ellipsoid. Since we assume that checking whether a point is feasible or not is relatively inexpensive, we seek to collect favorable sample points and update the learned boundary using those queried points.
Our approach stems from the field of active learning 
. Unlike passive learning where samples are collected regardless of the machine learning model itself, in active learning, the samples are selected and queried from the oracle manually to optimally train a specific model. The goal is to query a small number of samples to achieve similar accuracy as passive learning. Active Learning is a rich field with many results obtained over the years, we recommend the interested reader to access[32, 33, 34] and the references within, to see the label (query) complexity of different sampling methods for either the consistent or the agnostic case.
Ii Problem Set-up
Ii-a Power flow model for radial networks
We first present the modeling of components in a radial distribution network in power systems. For interested readers, please refer to [35, 36] for more details. We specifically consider a linear approximation of the system, which is presented in , assuming that the losses are negligible and the voltage at each bus is close to 1 in p.u. . We consider a distribution network with buses ordered as , where bus is the feeder (reference bus). The linearized relationship between voltage and power injection of the system is given by:
where is an all-one vector, is the nominal voltage value at reference bus , represent respectively the voltage, reactive power injection and real power injection at each of the buses. The matrices and are associated with the resistance and reactance of the distribution system .
Ii-B Voltage control in power distribution systems
To facilitate analysis, rewrite as the difference between the bus voltage and the reference voltage , then (1) becomes:
Installment of DER at each of the buses produces uncertainty in power injection, resulting in uncertainty in voltages. The voltage profile is reformulated into the following form:
where is the uncertain power injection due to DER at each bus, and represents the uncertainty of the system that has covariance matrix . We assume that has a continuous density distribution. The covariance matrix is not a diagonal matrix, even when has i.i.d. distribution.
The voltages in the system should be maintained within a tight bound usually plus/minus 5% of nominal. With uncertainties introduced by the operation of the DERs, we model this constraint in a probabilistic fashion. We use the following chance constraint which bounds the probability of the voltages staying in the prescribed bounds:
which is equivalent as:
where and are the voltage bounds. The value of is a parameter that can be chosen to indicate the probability that event occurs.
In this paper we only consider reactive power regulation and assume that the active load injection is determined exogenously and the controllable variable is the reactive power injection . Denote by , for a given tolerance level , the centralized voltage regulation problem is then captured as the following:
where the cost function can be any convex cost function, for example . This cost function encourages small amount of reactive power support to maintain the acceptable voltage deviation due to uncertainty. In addition, can be subject to box constraints due to limit of regulatory power.
Computing requires computing an integral on the density of on a multi-dimensional box, which does not generally have a close form. Thus it is hard to parameterize the geometry of the constraint . However, its shape is reminiscent of a perfect ellipsoid that motivates us to approximate the region with an ellipsoid. Once the region is approximated, the optimization problem for voltage regulation becomes:
Therefore, the central question in this paper is: for a given tolerance level , how can one approximate the feasible region by an ellipsoid quickly and efficiently? In Section III, we discuss more details on how we answer this question by a SVM machine learning approach, and in Section IV we present an efficient training algorithm.
Iii The learning problem
Finding an approximate ellipsoid is non-trivial, since we do not know the distribution of the randomness in the system. Even if the distribution is known, the problem is still hard due to the multivariate integral computation. However, if we can empirically evaluate the feasibility and infeasibility of sufficiently many points in the space, eventually we obtain a reasonable estimate of the feasible region. This evaluation can be done by an oracle defined as the following:
Oracle: An oracle can efficiently check whether (empirically) given a particular . If , the oracle returns a label , otherwise a label . We say that the point is queried, when the oracle checks whether the constraint holds for a specific . The query process is illustrated in Fig. 1.
Given that such an oracle exists, we can efficiently query as many ’s as possible and hence estimate this boundary accurately. Since a perfect oracle that tests whether exactly holds requires a multi-dimensional integral, and hence it is computationally challenging, we resort to an approximate test to verify whether . The approximate oracle then tests whether:
is greater than , given that are i.i.d. samples of . In practice, can be historical observations of the randomness Here is the indicator function and (8) is an approximate evaluation of the target chance constraint (6b).
Let us define . An ellipsoid can always be represented as , where . Mathematically, we want to approximate the unknown geometry by an ellipsoid denoted by using the oracle. Let subscript denote the index of query. If we have queried points, i.e., , to estimate , we solve the following optimization problem:
where is the label and if the oracle estimates it is feasible, and otherwise. This formulation is similar to the renowned SVM, except that the loss term is not the hinge loss, i.e., , but . We adopt this loss term because if has the same sign as , then the loss term is zero.
When the boundary is truly an ellipsoid, the optimization returns an ellipsoid such that no false positive nor false negatives occurs. Otherwise, we tune the weights ’s such that the value is high for false positives in the objective in (9). This minimizes the occurrence of false positives, and the optimization problem approaches the maximum volume ellipsoid  as number of queried points goes to infinity. In this way, we obtain a feasible solution to (6) by solving (9). In the next section, we discuss how to efficiently query a small number of ’s to find such an ellipsoid with high accuracy.
Iv Querying procedure
To illustrate our proposed algorithm, in this section we assume that the ground truth boundary is an ellipsoid, and we show that only logarithmically many points are needed to guarantee a small error in parameter estimation. More specifically, in Section IV-A we present the difference between two sampling procedures for training the model in (9). In Section IV-B we propose an efficient algorithm to sample the necessary points and in Section IV-C we illustrate our main result on the performance of the proposed algorithm. We show in Section V that the proposed approach also performs well for shapes with non-ellipsoidal boundaries.
Iv-a Random sampling and active sampling
A naive way to query points in order to estimate from (9) is to query a random number of points. However, this approach suffers from two shortcomings: 1). Random sampling queries points in the whole space equally probably. However, points that are further away from the boundary do not contribute to model change as much as the points close to the boundary. 2). If the whole search space is large and the true feasible region is small, the chance that we query a feasible point is relatively low, which does not yield accurate ellipsoidal estimation.
On the other hand, if we query actively the sample points that are close enough to the boundary (and diversely located around the boundary), then with the same number of queries that random sampling uses, we should get a better understanding of what the boundary looks like. This is illustrated in Fig. 2.
Now the question becomes: how to query the points such that they are close enough to the boundary while maintaining sufficient diversity? If the points are not diverse enough, for example the queried points are concentrated inside a small region within the feasible set, we end up with introducing too much bias into the learning model. In Section IV-B, we present the algorithm to query points such that they are located near the boundary and discuss an easy-to-implement approach to ensure diversity.
Iv-B Binary search
We propose a random binary search algorithm in Algo. 1 that queries points with binary cuts along many randomly directions. Overall, the binary cut procedure efficiently locates points close to the boundary with the smallest amount of queries; diversity in queried points is maintained through querying many random directions. In Algo. 1, subscript stands for each instance of queried point. In total, Algo. 1 requires at most oracle calls.
We first explain the inner ”while” loop for each fixed direction from line 3 to line 13 in Algo. 1. The inner loop guarantees that when it terminates, we have at least one feasible point and one infeasible point along direction . Further, their distance from the true boundary is no larger than . If we can gather many pairs of such ’s and ’s which are diverse enough, we should be able to obtain a good estimation of the true ellipsoid, illustrated by Fig. 2. In Theorem 1, we state that to achieve a small error in parameter estimation, it suffices to query logarithmically such points.
To ensure maximum diversity in queried points, we generate sufficiently many random directions ’s and run Algo. 1. This is described by the outer ”while” loop, which loops over many different random directions ’s. Once the points are collected using Algo. 1, we can train the model in (9). Considerations on estimation errors are provided in Section IV-C.
Iv-C Main result
Before presenting the main result in this paper, we first introduce some assumptions and notations used in this section. Assumption 1: The ground truth boundary is an ellipsoid described by .
Notation 1: Using subscript for each query, we query points, i.e., .
Notation 2: Let us adopt notation . Let , and .
We now present our main result on sampling complexity in Theorem 1.
Proof of Theorem 1 is in Appendix -A for interested readers. From Theorem 1, we first observe that under the assumption that the true boundary is an ellipsoid, Algo. 1 achieves a logarithmic bound in estimation error on the number of queries, as compared to the standard result that random sampling achieves a linear bound [39, 40]. This improvement in sampling complexity is due to the fact that Algo. 1 actively
selects the best points to query instead of blindly querying random points. Second, to achieve the bound, the queried points have to be diverse, i.e., the smallest eigenvalue of the data covariance matrix is bounded away from zero. This enables us to fully explore the unknown feasible region, and reduces bias in the learning model.
In this section, we validate our approach using the IEEE standard test feeder. Here we use IEEE 13 bus feeder . The test feeder is shown in Fig. 3, where we assume bus 1 is the reference bus. More results based on synthetic data are left to Appendix -C for interested readers.
In this distribution test feeder, we assume that there is no distributed generation, so the active power injection on each bus is negative. The line impedance is retrieved from . In addition, we restrict the available reactive power regulation at the buses to be no more than 0.1 (p.u.). We assume that and . The randomness
presented in this system is assumed to be a multivariate normal distribution with zero mean and high degrees of correlation, i.e., bus 2, 3, 9, 10, 12, 13 have non-trivial correlations. The distribution itself is unknown by the algorithm; however, historical observations of, i.e., ’s are available.
The indicator functions in this optimization problem can be replaced by ancillary binary variables and (10) can be solved by mixed integer programming (MIP). Details of the MIP formulation are presented in . The results comparing MIP and the proposed ellipsoidal approximation are summarized in Table I.
|Running time (seconds)|
|Empirical risk level|
From Table I, we observe that using ellipsoidal approximation significantly reduces computational time. It also yields a comparable result to that from MIP, with a slightly conservative risk level. In addition, MIP is inefficient in finding a sub optimal solution to (10), yielding a relative duality gap of 58.4 % after 42 seconds, whereas our algorithm finds a sub optimal solution within just 19 seconds. This makes our algorithm more adaptable for real-time operation.
We proposed a stochastic programming framework to solve voltage regulation problems. In order to approximate the feasible region of the chance constraint, we formulate the approximation procedure using a machine learning framework. We present an efficient active sampling algorithm that only queries points close to the boundary of the chance constraint. We find that this procedure queries logarithmically many points under the assumption that the true region is an ellipsoid. We extend this result to non-ellipsoidal regions using IEEE standard test feeders. Simulation results show that our proposed algorithm has significant better performance than standard approaches. In the future, we aim to extend the result to larger power distribution systems.
-  N. Li, G. Qu, and M. Dahleh, “Real-time decentralized voltage control in distribution networks,” in Communication, Control, and Computing (Allerton), 2014 52nd Annual Allerton Conference on. IEEE, 2014, pp. 582–588.
-  B. Zhang, A. Dominguez-Garcia, and D. Tse, “A local control approach to voltage regulation in distribution networks,” in North American Power Symposium (NAPS). IEEE, 2013, pp. 1–6.
-  D. Wang, X. Guan, J. Wu, P. Li, P. Zan, and H. Xu, “Integrated energy exchange scheduling for multimicrogrid system with electric vehicles,” IEEE Transactions on Smart Grid, no. 4, pp. 1762–1774, 2013.
-  H. Kanchev, D. Lu, F. Colas, V. Lazarov, and B. Francois, “Energy management and operational planning of a microgrid with a pv-based active generator for smart grid applications,” IEEE transactions on industrial electronics, no. 10, pp. 4583–4592, 2011.
-  B. Zhang, A. Y. S. Lam, A. D. Domínguez-García, and D. Tse, “An optimal and distributed method for voltage regulation in power distribution systems,” IEEE Transactions on Power Systems, vol. 30, no. 4, pp. 1714–1726, 2015.
-  M. Farivar, R. Neal, C. Clarke, and S. Low, “Optimal inverter var control in distribution systems with high pv penetration,” in Power and Energy Society General Meeting. IEEE, 2012, pp. 1–7.
-  G. Valverde and T. Van Cutsem, “Model predictive control of voltages in active distribution networks,” IEEE Transactions on Smart Grid, no. 4, pp. 2152–2161, 2013.
-  H. Zhu and H. Liu, “Fast local voltage control under limited reactive power: Optimality and stability analysis,” IEEE Transactions on Power Systems, no. 5, pp. 3794–3803, 2012.
-  P. Sulc, S. Backhaus, and M. Chertkov, “Optimal distributed control of reactive power via the alternating direction method of multipliers,” IEEE Transactions on Energy Conversion, no. 4, pp. 968–977, 2014.
-  K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov, “Options for control of reactive power by distributed photovoltaic generators,” in Proceedings of the IEEE, vol. 99, no. 6. IEEE, 2011, pp. 1063–1073.
-  B. A. Robbins, C. N. Hadjicostis, and A. D. Domínguez-García, “A two-stage distributed architecture for voltage control in power distribution systems,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 1470–1482, May 2013.
-  B. Jin, P. Nuzzo, M. Maasoumy, Y. Zhou, and A. Sangiovanni-Vincentelli, “A contract-based framework for integrated demand response management in smart grids,” in Proceedings of the 2nd ACM International Conference on Embedded Systems for Energy-Efficient Built Environments. ACM, 2015, pp. 167–176.
-  M. Hajian, M. Glavic, W. Rosehart, and H. Zareipour, “A chance-constrained optimization approach for control of transmission voltages,” IEEE Transactions on Power Systems, vol. 27, no. 3, pp. 1568–1576, 2012.
-  M. Lubin, Y. Dvorkin, and S. Backhaus, “A robust approach to chance constrained optimal power flow with renewable generation,” IEEE Transactions on Power Systems, no. 5, pp. 3840–3849, 2016.
-  H. Zhang and P. Li, “Chance constrained programming for optimal power flow under uncertainty,” IEEE Transactions on Power Systems, no. 4, pp. 2417–2424, 2011.
-  H. Wu, M. Shahidehpour, Z. Li, and W. Tian, “Chance-constrained day-ahead scheduling in stochastic power system operation,” IEEE Transactions on Power Systems, no. 4, pp. 1583–1591, 2014.
-  K. Baker and B. Toomey, “Efficient relaxations for joint chance constrained ac optimal power flow,” Electric Power Systems Research, vol. 148, pp. 230–236, 2017.
-  G. Martinez, Y. Zhang, and G. Giannakis, “An efficient primal-dual approach to chance-constrained economic dispatch,” in IEEE North American Power Symposium, 2014, pp. 1–6.
-  P. Li, B. Jin, D. Wang, and B. Zhang, “Distribution system voltage control under uncertainties using tractable chance constraints,” ArXiv, 2017. [Online]. Available: https://arxiv.org/abs/1704.08999
-  S. M. Harwood and P. I. Barton, “How to solve a design centering problem,” Mathematical Methods of Operations Research, vol. 86, no. 1, pp. 215–254, 2017.
-  S. Director and G. Hachtel, “The simplicial approximation approach to design centering,” IEEE Transactions on Circuits and Systems, vol. 24, no. 7, pp. 363–372, 1977.
-  S. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004.
-  S. Sapatnekar, P. Vaidya, and S. Kang, “Feasible region approximation using convex polytopes,” in Circuits and Systems, 1993, ISCAS’93, 1993 IEEE International Symposium on. IEEE, pp. 1786–1789.
-  Y. Zhou, B. Jin, and C. Spanos, “Learning convex piecewise linear machine for data-driven optimal control,” in Machine Learning and Applications (ICMLA), 2015 IEEE 14th International Conference on. IEEE, 2015, pp. 966–972.
-  B. Jin, M. Maasoumy, P. Nuzzo, and A. Sangiovanni-Vincentelli, “Online computation of polytopic flexibility models for demand shifting applications,” in 2017 13th IEEE Conference on Automation Science and Engineering (CASE), Aug 2017, pp. 900–905.
-  K. Anstreicher, “Ellipsoidal approximations of convex sets based on the volumetric barrier,” Mathematics of Operations Research, vol. 24, no. 1, 1999.
-  Y. Kiselev, “Approximation of convex compact sets by ellipsoids. ellipsoids of best approximation,” Proceedings of the Steklov Institute of Mathematics, vol. 262, no. 1, 2008.
-  F. Dabbene, P. Gay, and B. Polyak, “Recursive algorithms for inner ellipsoidal approximation of convex polytopes,” Automatica, vol. 39, no. 10, 2003.
-  B. Polyak and P. Shcherbakov, “Ellipsoidal approximations to attraction domains of linear systems with bounded control,” in American Control Conference, 2009. ACC’09. IEEE, 2009, pp. 5363–5367.
-  A. Saric and A. Stankovic, “Applications of ellipsoidal approximations to polyhedral sets in power system optimization,” IEEE Transactions on Power Systems, vol. 23, no. 3, pp. 956–965, 2008.
-  S. Tong and D. Koller, “Support vector machine active learning with applications to text classification,” Journal of machine learning research, vol. 4, 2001.
-  M. Balcan, A. Beygelzimer, and J. Langford, “Agnostic active learning,” Journal of Computer and System Sciences, vol. 75, no. 1, pp. 78–89, 2009.
-  S. Dasgupta, “Coarse sample complexity bounds for active learning,” in Advances in neural information processing systems, pp. 235–242.
M. Balcan, A. Broder, and T. Zhang, “Margin based active learning,” in
International Conference on Computational Learning Theory, pp. 35–50.
-  M. Farivar and S. Low, “Branch flow model: Relaxations and convexification,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on. IEEE, 2012, pp. 3672–3679.
-  L. Gan, N. Li, U. Topcu, and S. Low, “Branch flow model for radial networks: convex relaxation,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on. IEEE, 2012.
-  M. Baran and F. Wu, “Optimal capacitor placement on radial distribution systems,” IEEE Transactions on Power Delivery, no. 1, pp. 725–734, 1989.
-  V. Kekatos, G. Wang, A. Conejo, and G. Giannakis, “Stochastic reactive power management in microgrids with renewables,” IEEE Transactions on Power Systems, no. 6, pp. 3386–3395, 2015.
-  A. Ehrenfeucht, D. Haussler, M. Kearns, and L. Valiant, “A general lower bound on the number of examples needed for learning,” Information and Computation, vol. 82, no. 3, pp. 247–261, 1989.
-  S. Hanneke, “A general lower bound on the number of examples needed for learning,” The Journal of Machine Learning Research, vol. 17, no. 1, pp. 1319–1333, 2016.
-  W. Kersting, “Radial distribution test feeders,” IEEE Transactions on Power Systems, no. 3, pp. 975–985, 1991.
-  J. Luedtke and S. Ahmed, “A sample approximation approach for optimization with probabilistic constraints,” SIAM Journal on Optimization, no. 2, pp. 674–699, 2008.