Hybrid Voltage Control in Distribution Networks Under Limited Communication Rates
Abstract
Voltage regulation in distribution networks is challenged by increasing penetration of distributed energy resources (DERs). Thanks to advancement in power electronics, these DERs can be leveraged to regulate the grid voltage by quickly changing the reactive power outputs. This paper develops a hybrid voltage control (HVC) strategy that can seamlessly integrate both local and distributed designs to coordinate the networkwide reactive power resources from DERs. By minimizing a special voltage mismatch objective, we achieve the proposed HVC architecture using partial primaldual (PPD) gradient updates that allow for a distributed and online implementation. The proposed HVC design improves over existing distributed approaches by integrating with local voltage feedback. As a result, it can dynamically adapt to varying system operating conditions while being fully cognizant to the instantaneous availability of communication links. Under the worstcase scenario of a total link outage, the proposed design naturally boils down to a surrogate local control implementation. Numerical tests on realistic feeder cases have been to corroborate our analytical results and demonstrate the algorithmic performance.
I Introduction
Recent proliferation of distributed energy resources (DERs) such as solar generation and storage devices can potentially cause some rapid voltage fluctuations in distribution networks. A promising technology to tackle this challenge is via advanced inverter control design. The fastacting power electronicinterfaced DERs can support the distribution system voltage regulation objective by controlling their reactive power (VAR) outputs.
The voltage control problem can be viewed as a special case of the optimal power flow (OPF) one that minimizes the voltage mismatch. This OPFbased approach requires the availability of full networkwide information at a centralized location [1]. It is also possible to develop local control strategies that only use local voltage magnitude information [2, 3]. Due to lack of information exchange, local designs could suffer from instability issues and suboptimality [4]. For reduced communication complexity yet globally optimal performance, several distributed optimization based techniques using information exchanges among neighboring buses have been proposed in [5, 6, 7, 8, 9]; see [10] for a review of recent distributed and decentralized methods. Albeit a distributed control framework can more effectively coordinate networkwide VAR resources, its performance highly depends on communication capabilities of distribution networks, in particular the rates of information exchange as pointed out by [10]. While the effects of bandwidth limits and quantized messages have been investigated in [11] for distributed voltage control, more focus has been on the analysis of lowrate communications, or equivalently, asynchronous control updates. For example, [12] has designed an asynchronous decentralized algorithm where each DER controller flexibly incorporates the fast incoming local information with the lowrate control signal sent by a centralized aggregator. It has been shown in [9] that distributed voltage control updates can be also executed in an asynchronous fashion. Nonetheless, existing distributed control approaches would fail to work for the worstcase scenario of a total communication outage, under which they need to freeze all updates and cannot incorporate any local voltage information. As distribution systems continue to witness lowrate communication networks, it is imperative to integrate both local and distributed control frameworks to achieve the dual objectives in terms of adaptiveness to various communication rates and globally optimal voltage regulation performance.
The present paper develops a hybrid voltage control (HVC) strategy that can dynamically adapt to varying system operating conditions while being fully cognizant to the instantaneous availability of communication links. To cope with practical communication limitations, the proposed HVC scheme consists of both distributed and local control architectures and does not require a centralized authority. We formulate the networkwide VAR optimization problem by linearizing the power flow model for analysis purposes only. The resultant quadratic programming problem is solved by a partial primaldual gradient (PPD) algorithm in discretetime domain, which is a variant of the basic primaldual (sub)gradient method, see e.g., [13]. We provide the analysis in stepsize choices that can guarantee convergence. We further use the PPDbased solver to design online HVC strategy where each bus can integrate both the local voltage measurement and the communication information shared by neighboring buses. Although a linearized model has been adopted to the algorithmic development and analysis, performance of the proposed HVC design has been verified using the full ac power flow model for unbalanced and lossy distribution networks.
Compared to existing approaches on voltage control design, the main contributions of our HVC design are threefold. First, it explicitly accounts for the VAR limits by using the projection operator. Due to the discontinuity of projection mapping, general Krasovskii’s methods [14] for analyzing the stability of primaldual gradient flow method would not hold. To tackle this problem, we have expressed the operation as a subgradient step featured by an indicator function in order to establish the stability of the PPDbased HVC design in Sec. IIIA. Second, the HVC design only requires each bus to measure its local voltage magnitude and communicate it to neighboring buses. The sensing requirement and communication overhead are minimal compared to most (de)centralized strategies. Last but not least, our hybrid design can integrate both the neighboring bus voltage information and local voltage measurements regardless of the communication link conditions. This way, the HVC design is cognizant to the instantaneous availability of communication links while effectively tracking the globally optimal VAR setting. Interestingly, although the HVC updates have been developed using the distributed PPDbased solver, it would boil down to a surrogate local voltage control update during a total communication outage. Under this worstcase scenario, satisfactory performance can still be achieved as it responds to local voltage variation.
This paper is organized as follows. Section II presents the power flow model for distribution networks and formulates the specially designed voltage optimization problem. The PPDbased HVC algorithm is developed in Section III, along with its online implementation and communicationcognizant design. Numerical test results using realistic feeders and realtimeseries data are offered in Section IV. Conclusions are presented in Section V.
Ii System Modeling and Problem Statement
We consider a distribution network given by a graph with the set of buses and the set of line segments . Per bus , let denote its voltage magnitude, and represent the active (reactive) power injection, respectively. All network quantities are in per unit (p.u.). A constant reference bus voltage is assumed for the point of common coupling. For each line , we denote and as its resistance and reactance in addition to and as the active and reactive power flow from to , respectively. The sotermed LinDistFlow model has been developed in [15] to linearize the power flow model, assuming negligible line losses and almost flat voltage. Its accuracy can be numerically corroborated by several recent work [16, 17, 4, 9]. Per bus , we consider the controllable VAR as our control input where is the VAR consumption. It is shown that one can relate the voltage to the controllable VAR as the following [4]:
(1) 
where contains bus and all of its neighboring buses; the quantity captures the system operating conditions, i.e., the effect of on when . By concatenating all scalar variables into vectors and replacing by for notational convenience, (1) can be represented in a compact form . Matrix is the Bbus matrix used in the dc power flow model (see e.g., [18, Sec. 6.16]). We reserve for our future use. By definition, matrix is a reduced, weighted graph Laplacian matrix (full rank) and thus has a unique sparsity pattern based on the network topology. Similarly, a linearized model using the graphbased matrices also holds for multiphase unbalanced networks [19]. For simplicity, we will present the proposed algorithm using (1).
We aim to optimize the network contributed from local inverters to achieve a desired voltage profile such that . This is similar to the goal of secondary voltage control design in transmission and microgrid networks to effectively coordinate the network VAR resources and enhance the voltage stability [20, 21]. Such preferred profile can be adjusted depending on certain operational specifications for the network, e.g., conservation voltage reduction implementations would be most effective if the voltage profile is flat () [22]. We first formulate the problem under a static setup where the operating condition vector is constant to develop the HVC scheme. As detailed soon in Sec. IIIB, we will extend it to an online design by dynamically updating using neighboring voltage magnitude measurements. To this end, the voltage control objective of minimizing is introduced to improve the system voltage level by coordinating networkwide VAR resources and hence providing the globally optimal VAR setting. Minimizing subject to (1) could be solved collaboratively by all VAR resources through communication between buses using distributed optimization techniques [6, 7, 8]. Nonetheless, these distributed control schemes would fail to work under a total communication outage. To make the control design more robust, we introduce another weighted objective function where we define the weighted norm for any vector . Interestingly, it turns out that minimizing subject to (1) can be tackled through a totally local control architecture, i.e., voltage droop control scheme in [4]. Albeit this weighted objective inevitably results in a suboptimal VAR setting under limited VAR resources, it turns out this objective can allow for communicationfree updates using only local voltage measurements. We form the total objective using both and , such that the resultant design would enjoy the dual benefits of being globally optimal and communicationcognizant. Combining the attractive features of both distributed and local control frameworks, we cast the HVC problem as
(2a)  
(2b)  
(2c) 
where the function based on the physical system couplings^{1}^{1}1Note that here we assume each bus has the knowledge of its own operating conditions .. Hence, both and are the decision variables in the problem (2). As detailed soon in Sec. III, inclusion of into the formulation would lead to a special feature where the local voltage measurement is part of the instantaneous gradient direction with respect to , which allows for separable problem structure and provides the VAR control law. Hence, we can deal with this part of objective without communication. Note that problem (2) is still convex because is positive definite [4]. As is a weighted graph Laplacian matrix, it has a unique structure according to the network topology, i.e., . Accordingly, coupling involving only the neighborhood in (2b) is instrumental for the HVC design in Sec. III. We also introduce the importance factor to account for the importance of the local design. To enhance the performance under the extreme total communication outage scenario, should be chosen as large as possible to facilitate the communicationfree feature of . Meanwhile, if the communication link quality is very high, one can decrease the value toward to better achieve the unweighted voltage mismatch minimization criterion. Last, (2c) is the aggregation of box constraints . This restriction either comes from the inverter apparent power limit or depends on certain inverter power factor limit. Considering the objective in (2a), one can see that the voltage mismatch error is a tradeoff between the distributed (communicationinvolved) and local (communicationfree) control objectives. Suppose that every bus has unlimited VAR capability (i.e., no presence of (2c)), (2) with arbitrary positive (including infinity) would give the same optimal solution of . This implies that the optimal solution of (2) has the potential to closely approximate the globally optimal VAR setting obtained by minimizing under abundant VAR resources.
Iii Hybrid Voltage Control
This section presents our proposed hybrid voltage control (HVC) framework. Combining the distributed and local control features, we aim to solve (2) by adopting a typed primaldual gradient algorithm. Many variants of such algorithm have been studied in both continuoustime domain (see e.g., [23, 24]) and discretetime domain (see e.g., [13, 25]). Based on the PPD, we design the HVC scheme that needs only to measure and incorporate the dynamic voltage magnitude. As detailed soon, the proposed feedback approach is very different from existing distributed control schemes in power systems since most of them are developed as a static optimization problem and overlook the communication imperfectness and online implementations. Most importantly, our HVC scheme would boil down to a surrogate local voltage control problem under the worstcase scenario of a total communication failure. Thus, it enjoys a satisfactory performance by having the cognizance to varying communication scenarios.
Introducing the Lagrangian multiplier for the equality constraints in (2b), we obtain the following Lagrangian function for the given static optimization problem (2):
(3) 
where we define and represents the inner product. The goal becomes seeking the saddle point of (3) in a physically implementable way. A popular and efficient method to numerically find the saddle point is to instead construct an augmented Lagrangian and perform alternating minimization over primal variables followed by a (sub)gradient ascent update over dual variables (see e.g., [26] for the introduction of the ADMM algorithm). Unfortunately, such algorithm would either involve a centralized computing or require multihop communication graph. Accordingly, this implementation does not work for the problem of interest here. Hence, we instead focus on the Lagrangian function (3) and adopt the partial primaldual (PPD) algorithm, which admits to a sparse communication network and turns out to cope up with total communication link failures. The "partial" update property will become clear soon, with the algorithmic design detailed here.
Using the superscript to denote the iteration number, we initialize and to be the latest VAR output setting. The PPD updates at the st iteration per bus consists of the following three steps:

Update : For given , is updated by solving
(4) which is an unconstrained convex quadratic programming. Thus, a closedform solution exists. Since the objective in (4) is decoupled in , the update per bus is obtained accordingly as
(5) This is one of the main differences to the classical gradient flow algorithm which would have used (the discretization of) . Hence, the term ”partial” is adopted here in the name of our algorithm. Similar strategy of “partial gradient update” has also been explored in [24] for a continuoustime algorithm whereas we focus on a discretetime update. Note that the variable estimates the network voltage magnitude based on the power flow model (1).

Update : Defining stepsize , we perform a gradientprojectionbased update on , as given by
(6) where the projection operator bounds any input to be within . As always holds, we use the latest iteration of (6) to be the network VAR control signal. Under this setting, the gradient direction in (6) for any pair becomes
(7) where is the instantaneous network voltage profile at time following from (1). Note that can be obtain locally by the voltage measurement unit . With , the gradient function (7) can be partially computed using the voltage measurement . By weighting the objective norm of with the positive definite matrix , its gradient direction decouples from the full vector and can be computed separately with a pair of the voltage magnitude measurement and at each bus. Thanks to the separability of box constraints, under arbitrary initialization , the update (6) for each bus becomes
(8) where denotes the projection at bus to . Albeit is constant in this static setup, it would vary according to the inverter limits and instantaneous active power generations under online implementations. Due to the physical power flow couplings, the local voltage measurementbased gradient direction contains the most uptodate networkwide information. As detailed soon, this unique feature combined with local dual variable results a robust, communicationcognizant HVC design.

Update : For a given stepsize , each multiplier is linearly updated per bus j using the iterative residual of the equality constraints which is equivalent to the gradient . Thus, we have
(9)
The PPDbased iterations in (S1S3) constitute the basis for our proposed hybrid voltage control design.
Iiia Convergence Analysis
Without the projection operator in the update (in the case ), the convergence analysis of the PPD algorithm would boil down to the stability issue of a discretetime linear system, whose analysis could be completed in a few lines of words. We would like to treat the system as a linear dynamical system (since linear system is usually considered as easy to handle), allowing us to provide a comprehensive performance analysis. However, with the presence of the projection operator, it becomes a socalled saturated linear system and its stability could be hard to determine [27]. To tackle this problem, we resort to a convex analysis approach.
The PPD algorithm actually seeks the saddle point of the Lagrangian function (3). To make our analysis more unified, let us recast the Lagrangian function (3) into a general form
(10) 
where and are both assumed to be strongly convex and have Lipschitz gradients. In addition, the indicator function
equivalently accounts for the constraint . This way, (3) becomes a special case which uses and . By definition, is a strongly convex function with Lipschitz gradient, i.e., it holds that
with some positive constant , and
with some positive constant . Here, is the standard inner product. Similarly, is a strongly convex function and is a convex function.
The saddle point containing the optimal solution of (2) satisfies the following KKT conditions:
(11a)  
(11b)  
(11c) 
where is the subdifferential set of at . A subgradient of at any point , denoted as , is defined as an element of the subdifferential set , i.e., . Thus the above subdifferential inclusion condition (11) can be reinterpreted as that there exists such that
(12a)  
(12b)  
(12c) 
In the sequel, we will always use this “reinterpreted” subgradient form to conduct the derivation since it is more straightforward. Such use of subgradient and notation have appeared in many recent references including [28, 29]. When an equation involves a subgradient, which element in the corresponding subdifferential it uses is always clear from the context. For example, the subgradient used in (12b), , equals to where
Note that is welldefined when is strongly convex or is compact, which is exactly the case in our paper. Under the aforementioned notations, our algorithm is equivalent to
(13) 
The recursive relationship of the sequence in (13) is further equivalent to^{2}^{2}2By using the notion of subgradient and indicator function, one is able to show that a gradient projection step can be understood as a subgradient step.
(14a)  
(14b)  
(14c) 
The goal of convergence analysis is to show that the firstorder residuals , , and will all go to zero when goes to infinity. Due to the fact that the optimal solution exists and is unique, vanishing firstorder residuals automatically imply the convergence of to . Keeping this in mind, in the sequel, we will provide a theorem regarding convergence under suitable stepsizes.
Theorem 1.
Let and be the smallest and largest singular values of the reduced graph Laplacian matrix , respectively. If the nonnegative stepsizes and are chosen such that
(15) 
then the sequence generated by the PPD updates (S1S3) converges to the optimizer .
Proof:
We first focus on showing that the successive difference + is an infinitely summable sequence and thus converges to zero. Next, we prove that such vanishing successive difference implies vanishing firstorder residuals. Therefore, it completes the proof.
The first step would be connecting the th iterates to the corresponding optimal quantities. To this end, substituting (14c) into (14a) and (14b), respectively, for and then subtracting it from the KKT equalities (12), we have
(16a)  
(16b)  
(16c) 
Additionally, by the strong convexity and gradient Lipschitz continuity of , we have
(17) 
Note that the first inequality of (17) is standard and obtained directly from the strong convexity and gradient Lipschitz continuity of . We omit the proof as it is derived in Theorem 2.1.11 of [30]. Reorganizing (17) gives
(18) 
Substituting (16b) into (18) for leads to
(19) 
Next, by the strong convexity of the function , we have
(20) 
Substituting (16a) into (20) for leads to
(21) 
where the weighted norm for any vector . Summing up (19) and (21) results in
(22) 
Applying the basic inequality
which holds for any and any real vectors and of the same dimension and utilizing (16c), the righthandside of (22) can be relaxed as
(23) 
where is any arbitrary constant in the interval . It then follows from (22) and (23) that
(24) 
where measures the largest singular value of a matrix. In order to conclude that is infinitely summable, based on (24), the following conditions must be met (this is done by comparing the coefficients of terms in (24) and requiring nonpositive coefficients on the righthandside of (24) after telescope cancellation):
(25) 
which is equivalent to requiring
(26) 
To maximize the possible range of , we choose . Accordingly, we obtain the stepsize rule
(27) 
Under the choice of (27), we can find the infinite summability of from (24). Hence, and . Noticing the relation (14), we thus have
(28a)  
(28b)  
(28c) 
Comparing (28) and (12), satisfies the KKT conditions (12). Therefore, we conclude converges to .
The explicit forms of and are and . The Hessian of is while that of is an identity matrix. Let us denote and as the smallest and largest singular values of the reduced Laplacian matrix , respectively. Noticing that , we have and . Obviously in this case. Finally, substituting the values of , , and into (27) leads to the specific bounds stated in the theorem, i.e., (15).
It is clear that and . Increasing the network size which leads to the growing of may reduce the bound on . Nonetheless, the dual stepsize does not significantly affect the practical convergence speed empirically. In contrast, we usually find the primal stepsize playing a more crucial role in adjusting the convergence speed. In addition, the quantity “sort of”^{3}^{3}3We comment it as “sort of” because conventionally, the algebraic connectivity of the graph is defined as the second smallest eigenvalue of the standard graph Laplacian. There is some relationship between the algebraic connectivity and , but it is beyond the scope of our discussion. entails the connectivity of the network since it is the smallest eigenvalue of the reduced graph Laplacian. But as long as the network is connected, this quantity is lower bounded away from zero. Empirically, a larger stepsize bound implies the possibility of a faster convergence. Accordingly, a better connected physical network would result in a faster convergence, which aligns with our common intuition.
Remark 1.
(Relaxations and generalizations) Our convergence analysis is conducted under the assumption of functions and both being strongly convex. This assumption can be relaxed to the socalled restricted strong convexity (Huber loss satisfies such assumption) [31]. In this case, the first inequality in (17) needs to be replaced by a looser one (worse coefficients) and a narrower stable stepsize region will be derived.
In addition, it is possible to totally remove the (restricted) strong convexity assumption on with a simple modification over our current scheme. Specifically, the (4) needs to be tweaked by appending a proximal term, as given by
Informally speaking, adding such proximal term enhances the stability of the algorithm when dealing with nonstrongly convex functions. The final stepsize rule for and would foreseeably not be affected by the strong convexity constant , but rather depend on . Consequently, needs to be adjusted together with and in the PPD algorithm to satisfy some specific rule. However, adding a proximal term could presumably slow down the convergence speed.
IiiB Online Feedback Design
Thus far, we assume the availability of static per bus . However, it would change in accordance with the system operating condition. To account for system dynamics based on (1), we need to compute and update which requires networkwide complex power injections. To this end, twoway communications between each bus and a centralized computer are necessary. Generally, this is not feasible due to limiting communication in distribution network in addition to fast system dynamics. Thanks to the sparsity of , its entry is zero for any pair of buses whose corresponding buses are not connected by a line segment. Accordingly, we propose to obtain a timevarying through neighboring voltage measurement exchanges. Each bus measures its voltage magnitude after (S2) and broadcast to its neighboring buses. Knowing incident line reactance values at each bus, we update as
(29) 
which is computed locally and adopted in (S3). Note that each bus would only need to store the neighboring line reactance values to update this feedback signal. Therefore, the memory requirement is minimal. The attractive features of the proposed feedback design are threefold. First, we can obtain locally at each bus by adopting bustobus communication architecture. This is nicely designed to our HVC scheme as the distributed feature is maintained. Second, the instantaneous voltage measurements contain the latest system information, and hence effectively approximates the dynamically varying operating conditions. Last but not least, the voltage feedback control design improves the robustness to mismatch and imperfection in system modeling (see e.g., [32, Sec. 8.9]) since the voltage measurements could potentially capture the underlying nonlinearity in the power networks.
Remark 2.
(Cyber Network Topology) The nodetonode architecture of the HVC design can be generalized to instead coordinate clusters of buses as long as the cyber network is connected; see e.g., [7]. This way, it is not necessary to have DERs to be connected to each other by a line segment. Additionally, even if the distribution network is not a complete entity (i.e., a DER is not necessarily attached to every bus), we may eliminate all the buses with no DERs to create an equivalent network by adopting the Kron Reduction method [33]. Thus, this reduced network consists of only buses with DERs installed. By adopting the voltagebased feedback signal in (29), the corresponding reduced Bbus matrix would explicitly account for all system characteristics as to the original network. Accordingly, the performance of our HVC design can still be guaranteed under a generalized distribution network where DERs are not attached to each and every bus.
IiiC Limited Communication Rates
The performance of the proposed HVC design relies on the quality of bustobus communication links, which we have assumed to be perfect throughout the algorithmic design. However, random link failures and messaging delays are common because of either network congestion, or poor signaltonoise ratios in some wireless environments for a contemporary digital communication system. It is imperative to examine how the PPDbased HVC scheme works under imperfect communication, which leads to the following two different scenarios. One is often referred to as asynchronous networking that is consisting of both link failures and messaging delays [34, 35]. Meanwhile, the other only considers link failures. The later one can also be referred to as timevarying network [36, 37]. Albeit the second scenario seems to be a special case of the first one, by playing a simple trick of embedding a time stamp in the message between each pair of buses, delays can also be treated as link failures^{4}^{4}4We assume that each bus has a clock that is aligned. As the information exchanges and dual updates could be performed at a relatively low speed, a slight mismatch between clocks does not break the viability of this approach.. Informally speaking, as long as the delay is bounded, and the timevarying communication network is connected^{5}^{5}5It is a connectivity description of graphs under timevarying scenarios. Readers are referred to Assumption 2 at page 7 of reference [37] for detailed definition., one should be able to choose small enough stepsizes to stabilize the proposed algorithm. There have been some analysis for algorithms under these conditions in the literature (see, e.g., [38, 39]). Rigorous proof of convergence properties under these conditions is out of the scope of this paper and will be a future direction. We have tested and validated the proposed design using realistic distribution networks in Sec. IV.
To tackle the challenge posed by imperfect communication networks, we leverage the work in [38, 39, 40] regarding the "freezing" strategy for distributed optimization problems. Conceptually, every PPD variable remains unchanged until new information becomes available from neighboring buses. The asynchronous version of a related distributed primaldual algorithm in [39] has been proven to be convergent under random activation of agents, i.e., link is available only when and are both randomly activated. It is assumed that the activation of each bus follows a Bernoulli distribution, independent across time. Nonetheless, under this strategy, the aforementioned PPDbased HVC updates would completely halt under the case of a total link failure in the communication network. The novelty of our work lies in the extension of the HVC scheme by modifying update steps (S1)(S3) to have: a) satisfactory performance under partial link failures, b) capability to continue providing VAR regulation under a total link failure scenario. To this end, we freeze the variables associated with the inactive bus while always updating the VAR control signal according to (8) by adapting local gradient information from . Since the voltage measurements always contain the most updated information of the network, the local voltage control design objective is advocated to continue providing the VAR support. As a result, under a total communication failure, our HVC framework boils down to a surrogate local controller design based on the current value of . This is similar to the microgrid secondary frequency/voltage control design where can be treated as an offset signal to a local droop controller [41, 21]. We denote as the subset activated nodes at iteration . Our proposed asynchronous (A)HVC algorithm is tabulated in Algorithm 1.
Iv Numerical Tests
The numerical tests presented in this section demonstrate the effectiveness of the proposed communicationcognizant HVC design for practical distribution feeders. We investigate the performance of our scheme under the settings of both static and dynamically timevarying network operating conditions. The dynamic tests are performed using the IEEE 123bus test case [42]. The desired voltage magnitude is chosen to be p.u. at every bus . Furthermore, each bus is assumed to have a certain number of PV panels installed, and thus it is able to control/provide VAR via advanced inverter design. Albeit the HVC design is based on the linearized model (1), we test and validate the performance using the full ac power flow model. All numerical tests are performed using MathWorks^{®} MATLAB 2014a software and the OpenDSS for solving the actual power flow. Accordingly, the bus voltage magnitude , instead of the one obtained from (1), is used for VAR control outputs in (8) and the following numerical tests.
Iva Static System Operating Conditions
A singlephase radial power distribution feeder that consists of 21 buses with at the head of the feeder is first adopted to test the algorithm under the static setup. The impedance of each line segment is set to be . Per bus , we fix the loading while choosing the inverter rating to be kVA where is zeromean Gaussian having variance 13.33, thus modeling the variation in inverter sizing by . Accordingly, the VAR constraints in (2c) would become active at some locations. We test the HVC algorithm with various choices of importance factor . To demonstrate the tradeoff between distributed and local control designs, we plot in Fig. 1 the optimal voltage mismatch norm error for each value with all other settings the same. Notice that increasing value in term of adding more weight on the local control objective results in a larger voltage mismatch error. This corroborates with our earlier claim that local control schemes attain a suboptimal VAR setting under limited VAR resources. Given a system model, one may study this tradeoff offline to tune the importance factor accordingly.
Based on the convergence properties in Theorem 1 with , we have . Fig. 2 plots the iterative voltage mismatch norm error in logscale for various stepsize choices assuming a perfect communication. To violate the stepssize constraints, we let and in two different scenarios, respectively. It clearly shows that the HVC design fails to converge under these cases. To stabilize our design, we bound the stepsize values to be within their limits as depicted in Fig. 2. Note that the effect of stepsize choices shows a tradeoff between the stability and convergence rate. The larger are, the faster the updates converges. Nonetheless, this could potentially lead to oscillations in the error performance, exhibiting instability under fast dynamics. To tackle this problem, once the full feeder information becomes available, our convergence properties in Theorem 1 are very useful in terms of selecting proper stepsize choices. Otherwise, it is also possible to adjust the stepsize onthefly by decreasing the values based on its local voltage oscillation intensity. To sum up, under appropriate stepsize choices, Fig. 2 validates the effectiveness of our scheme, in terms of achieving the optimal VAR setting while requiring no centralized coordination.
Moreover, to showcase the robustness of our AHVC scheme under imperfect communication links, we model the activation of every bus as a Bernoulli distribution with the same probability to each other. Since the distributed part of the HVC are most likely to be affected by random communication link failures, we let to investigate the performance of AHVC design. To this end, we have while fixing other settings to be the same as the earlier test. Fig. 3 plots the iterate voltage mismatch error in logscale under various bus activation rate ranging from , where the case of corresponds to the perfect communication scenario (synchronous case). It clearly depicts that our design enjoys a satisfactory performance guarantee under random link failures for regulating the network voltage. Informally speaking, a lower bus activation rate would lead to a slower convergence speed, with a no link failure scenario exhibiting the fastest convergence. This test verifies that our proposed AHVC design is robust against imperfect communication and thus able to cope with cyber resource constraints.
IvB Dynamic System Operating Conditions
To corroborate our HVC scheme for online implementation, we have tested the proposed algorithm on the IEEE 123bus test case [42]. Dynamic system operating conditions are generated using real load profiles from an online data repository [43], as shown in Fig. 4. The minutesampled active and reactive power consumption data along with solar profile were collected on Friday, June 20, 2010 for residential loads. Each home shares similar load and generation patterns, which are diversified by small random additive noises. For each load node of the 123bus test case, it is attached to a certain number of residential homes with solar generation rated at 3.5kW peak capacity. Additionally, for each minute time slot, physical VAR limits are updated according to their inverter ratings (i.e., 3.5kVA per inverter) and instantaneous active power from solar generations. Under these settings, it turns out that VAR resources of inverters are insufficient to achieve perfectly flat voltage at all times. This implies the VAR constraints in problem (2) are active.
Fig. 5 plots the daily voltage mismatch for the aphase of the 123bus, where the other two phases exhibit a similar comparisons. Three different control strategies including no VAR support, distributed design, and our HVC scheme are plotted. For the benchmark case of no VAR support, there are some under and overvoltage issues due to load and solar variations. To improve the voltage quality, the proposed Algorithm 1 is tested for online implementation where every PPDbased iteration of (S1)(S3) updates every 2 seconds assuming a fixed loading during corresponding oneminute interval. Additionally, we validate the robustness of the HVC design to the worstcase scenario of a total communication link outage from hour 16:00 to 24:00. Clearly, during the noon hours, that all three scenarios have similar voltage mismatch error because of limited VAR capability. For all other hours, especially during the evening hours in the zoomin view, we see that our HVC design effectively minimizes network voltage mismatch error, maintaining a nearly flat voltage profile. The attractive feature of our communicationcognizant HVC design enjoys a satisfactory performance even under the total link failure scenario, whereas the distributedbased control updates come to a complete halt, i.e. setting , resulting in a higher voltage mismatch error around hour 16:0018:00 compared to the ones in the benchmark and HVC cases. To sum up, the proposed HVC design can efficiently regulate the voltage level by coordinating networkwide VAR resources. Meanwhile, its cognizant feature to the instantaneous availability of communication links is also attractive, considering the limited deployment of cyber infrastructure in distribution networks. Therefore, the proposed HVC design would facilitate the future engagements in inverterbased VAR resources to improve voltage support by accounting for practical constraints in both physical and cyber layers.
V Conclusions
This paper has developed the communicationcognizant hybrid voltage control (HVC) scheme to coordinate networkwide VAR support in power distribution networks. Considering limited VAR resources, we have cast the specially designed voltage control problem combining both attractive features of distributed and local control architectures. The PPDbased algorithm is evoked and only requires voltage measurement exchanges among neighboring buses with local computations. Moreover, we have provided the convergence properties of the aforementioned algorithm for proper stepsize choices. To cope with cyber resource constraints and lack of reliable communication links, we have further extended the HVC design to have robustness against random communication link failures and, in particular, communicationcognizant feature to account for the worstcase scenario of a total communication outage. We have extensively validated the effectiveness of the HVC design using realistic distribution networks under both static and dynamic testing environments.
For future work, we plan to study the convergence properties of the proposed AHVC design while offering the performance analysis of the online implementation under stochastic settings. We are also interested in other cybersecurity aspects of inverter control designs.
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