I Introduction
Voltage control is crucial to stable operations of power distribution systems, where it is used to maintain acceptable voltages at all buses under different operating conditions [LietAl2014]. To control voltage, reactive power is traditionally regulated through tapchanging transformers and switched capacitors [ZhangetAl2013]. With recent advances in cyberinfrastructure for communication and control, it is also possible to utilize distributed energy resources (DERs, i.e., electric vehicles [WangEtAl2016], PV panels [KanchevEtAl2011, Zhang2015]) to provide voltage regulation. There exists an extensive literature in controlling voltage in a distribution network, some of them focus on centralized control [FarivarEtAl2012pes, ValvEtAl2013], while the others address distributed algorithm [ZhuEtAl2016, LietAl2014, ZhangetAl2013, SulcEtAl2016].
In this paper, we focus on centralized control frameworks to regulate voltage through DERs, that a central controller constantly sends out regulation signals to local DERs in order to control voltage. One important issue to deploy voltage control via DERs is to deal with the uncertainty they bring to the power distribution network [TuritsynetAl2011]. Since most distribution networks still do not have realtime communication capabilities, decisions made have to be valid for a set of conditions. For example, assume that information is exchanged between a central controller and the local DERs every five minutes. Then the control signal that sets the reactive power on the DERs must be valid for the next five minutes, subject to the fast variabilities and randomness in the renewables.
One way to handle uncertainty is through stochastic programming [Dantzig2010]
. Stochastic formulation takes into account the probabilistic nature of the uncertainty. Chance constraint, which bounds the probability of a certain event, is introduced in these optimization problems. However, it is not obvious to characterize the feasible region of the chance constraint. Monte Carlo simulation is therefore adopted by many researchers, for example in
[ZhangetAl2013], to approximate the optimal solution.In this paper, we assume that the distribution of the uncertainty is known, and we adopt a stochastic approach to bound the probability that voltage stays within prescribed bounds. Unlike most of the existing literature, we propose to impose a single chance constraint on the whole system. This is different from the standard practice in literature where chance constraints are placed on every bus of the network [ZhangEtAl2011, WuEtAl2014]. Putting constraints on each single bus simplifies the problem, but suggests that the uncertainty at each bus is unrelated. However, the randomness across the buses can be well correlated in practice. In this work, we show how a single constraint can be used to capture uncertainties from all buses in the system using the linearized DistFlow model introduced in [Wu89]. Another approach is to adopt a robust optimization framework, but it is often nontrivial to set the budget of uncertainty and solving the optimization problem maybe computationally challenging [Ding13, BertsimasEtAl2013].
We show that our proposed voltage control problem is tractable without sampling techniques. We validate this statement by proving that the proposed chance constraint depicts a convex feasible region and adapts a variety of distributions. The exact optimal solution can therefore be found using standard gradient descent techniques.
In all, we make the following contributions to voltage control in distribution systems:

We consider voltage control problem with uncertainties in the system. The uncertainty is correlated across the system and is captured by a single constraint imposed onto the whole system.

The proposed framework is convex, that it only adds one additional convex constraint to the various existing voltage control problems. Therefore there is no need to adopt an approximate algorithm such as Monte Carlo sampling.
The rest of the paper is organized as follows. Section II presents the modeling of the distribution network. Section III proceeds with the formulation of the voltage control problem and demonstrate the robustness of the proposed framework with an illustrating example. Section IV states that the problem can be efficiently solved, by proving that the introduced chance constraint is convex. Section LABEL:sec:simu validates the statement by simulation results. Finally Section LABEL:sec:conc concludes the paper.
Ii Preliminary: Distribution network
In this section we present the modeling of components in a radial distribution network in power systems. For interested readers, please refer to [FarivarEtAl2012, GanetAl2012] for more details.
Iia Power flow model for radial networks
We consider a radial distribution network with buses collected in the set . We also denote a line in the network by the pair of buses it connects. Bus is the reference bus.
For each line in the network, its impedance is denoted by , where and is its resistance and reactance.
For each bus , let be the voltage magnitude at bus and be the complex power injection, i.e., the generation minus consumption. In addition, the subset denotes bus ’s neighboring buses that are further down from the feeder head. The DistFlow equations [Wu89] model the distribution network flow for every line as:
(1a)  
(1b)  
(1c) 
where are respectively the real and reactive power flow on line . We let denote the voltage at the reference bus. In addition, the term represents the loss in the power flow in line .
IiB Linear approximation of the flow model
Assuming the loss is negligible compared to line flow, a linear approximation of can be constructed. Following [Wu89], we assume that the losses is negligible. We then assume that the voltage at each bus is close to 1. This enables us to approximate by [ZhuEtAl2016]. We then obtain the linearized DistFlow model :
(2a)  
(2b)  
(2c) 
From (2), we can write the voltage magnitude in terms of reactive power injection and real power injection , and the reference voltage :
(3) 
where are matrices with and as the element in row and column, respectively. The voltage profile at bus is denoted by .
Following the findings in [LietAl2014], we give the expressions of and in terms of line resistance and reactance ^{1}^{1}1Here we do not have a factor 2 as in [LietAl2014] because we approximate by .:
(4a)  
(4b) 
where is the set of lines on the unique path from bus 0 to bus [LietAl2014]. Note that and are positive definite matrices [LietAl2014].
Iii Voltage regulation with reactive power injection
Suppose that bus is assumed to be operating at nominal voltage level, i.e., p.u. (per unit). Rewrite as the difference between the bus voltage and reference voltage , then (3) is reduced to the following form:
(5) 
As renewables introduce uncertainty in the bus voltages, the voltage profile is reformulated into the following form:
(6) 
where is the uncertainty with zero mean and covariance matrix . The covariance matrix is not necessarily a diagonal matrix since the uncertainty can be highly correlated across buses. In Section IV we illustrate a variety of distributions that can possibly follow.
In order to maintain the voltage at each bus close to the nominal level, we propose to bound the probability that the voltage profile is within some bounds:
(7) 
which is equivalent to write as:
(8) 
where and are the bounds prescribed to the random voltage profile. They indicate how far the voltage profile can be away from a nominal level. The value of indicates the probability that event occurs.
Iiia Main Optimizaton Problem
In this paper we only consider reactive power regulation and assume that 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:
(9a)  
(9b)  
(9c) 
where the box constraint on represents the limits of reactive power injection at each bus. These bounds can be interpreted as the capacity or availability of devices at each bus. In addition, note that looser bounds on and in correspond to a larger , whereas a tighter bound corresponds to a smaller . In practice, the value for is usually taken as 0.05 p.u. or 0.1 p.u. The objective function in (9a) minimize the total action of the DERs, although it can be easily replaced by any time of convex function of .
IiiB PerBus Constraints
Our approach is inherently different from the existing literature when dealing with chance constraints. In most existing literature with randomness in the distribution network, chance constraints are introduced as [ZhangEtAl2011]:
(10)  
where and extracts the th row in respective matrices. The chance constraint at bus is associated with prescribed tolerance . We assume that each bus has the same tolerance, i.e., . The optimization problem that incorporates perbus chance constraint is in the following form:
(11a)  
(11b)  
(11c) 
where .
The solution to problems described by (11) with chance constraints on each bus is discussed widely in literature, for example in [ZhangEtAl2011, WuEtAl2014]. This perbus framework is not the same as having a single constraint on the whole system, i.e., (11b) is different from (9b). Those two different frameworks do not return the same feasible set and the proposed framework, which places a single constraint on the whole system, captures the coupling between buses and is therefore more realistic and applicable. In addition, as we show in Section IV, our framework is tractable, in the sense that the chance constraint in (9b) is (after some transformations) convex.
IiiC Fourbus Toy Example
Let us take an illustrating example with a line network with four buses, shown in Fig. 1.
Suppose that the reactive power injections at bus 2 and 3 are limited to 0.2 p.u., we have a linear constraint as . Then our proposed framework is solving the following optimization problem:
(12)  
and its optimal solution is denoted by . Here we take to be .
The perbus formulation is the following optimization problem:
(13)  
and its optimal solution of (13) by . Detailed configuration of the line network is left in appendices for interested readers. Unlike the problem in (12), setting a “right” in (13) is not straightforward. Suppose we want to achieve the same level confidence as (12) where the system operates within the prescribed bounds with probability at least , then what is the right to take?
As suggested by previous studies [ZhangetAl2013], a natural candidate for is to set it equal to by thinking of each bus as independent to each other. A second candidate is simply set it at , the same as .
The main results of the fourbus line network are shown in Table I, where denotes the probability and is the figure of merit we compare the solutions with. For the proposed framework , and for the perbus framework. The bound on the voltage deviation is denoted by , which means that voltage deviates no more than 10% of the nominal level. The uncertainty is assume to be multivariate Gaussian and its covariance matrix is shown in the appendices.
Framework  

,  0.88 
,  Infeasible 
,  0.78 
It turns out that assuming each bus as independent drives the optimization problem with the perbus constraint infeasible. This is due to the fact that in order to achieve as the joint probability, the per bus constraint requires a very small that is too strict. The second value of makes the problem feasible, but at the cost of lowering the joint probability to be 0.78. This result shows the difficult of using the per bus constraint, where it is difficult to set the correct tolerance level in order to be not overly pessimistic or optimistic. Of course, one could vary and resolve (11b), but the procedure is cumbersome (especially for large networks) and the best does not guarantee that the overall probability of violation is the lowest.
In the following section, we elaborate on the framework based on (9) and show that it can be casted into a convex optimization problem.
Iv Solving the optimization problem
The proposed optimization problem in (9) has a linear objective with a box constraint and a chance constraint. In this section, we present our main statement in Theorem 1 that the proposed problem is convex.
Theorem 1.
If the uncertainty has logconcave probabilistic distribution, then the optimization problem in (9) is convex.
Theorem 1 states that even with a chance constraint whose feasible region is not obvious at first sight, it does not complicate the voltage control problem. On the contrary, the chance constraint preserves convexity of the optimization problem and is tractable.
In Theorem 1, we require that the uncertainty has a PDF that belongs to a certain class of functions, i.e., log concave functions. We now introduce the definition of logconcavity.
Definition 1.
A nonnegative function is logarithmically concave (or logconcave for short) if its domain is a convex set, and if it satisfies the inequality:
(14)  
for all and . In short, if is concave, then is logconcave.
Log concave functions enjoy many properties that lead to convexity. For example, the level set of a logconcave function is convex, which means that if is log concave, then its level set is a convex set [SaumardEtAl2014]. Therefore, the chance constraint can be easily carried over to other existing deterministic optimization frameworks and preserves the convexity of the problem.
What is more, to maximize over a log concave function, it is sufficient to take the logarithmic of the original function and apply gradient descent techniques. The local optimum is guaranteed to be the global optimum. This suggests that we can easily maximize over without using approximate algorithms such as sampling.
Therefore, to prove Theorem 1, is log concave and we introduce two lemmas that discuss logconcavity of certain functions. Lemma 1 states that the accumulated mass of a log concave probabilistic function over a convex set is log concave. Lemma 2
states that applying linear transformation to the variable in a logconcave function yields another logconcave function, given that the linear transformation has full row rank.
Lemma 1.
Denote . If the distribution is log concave in , then is log concave in .
Lemma 1 only requires the log concave distribution of to ensure that the function
is log concave. A lot of commonly known distributions fall within the logconcave probabilistic distributions, for example, Gaussian distribution and Weibull distribution which is usually adopted to generate intermittent wind energy. Detailed proof is shown in appendices.
Lemma 1 states that the objective function is log concave in . We further show that where is log concave in . This is shown in Lemma 2 and the proof is in the appendices.
Lemma 2.
Assume that is a log concave function, and that with , , . If has rank , then is also a log concave function.
Proof of Theorem 1.
We reformulate into:
(15a)  
(15b)  
(15c) 
where and .
We can scale row of the inequality inside (15) to make the expression easier to work with. Defining as and scale the inequality in (15c) by . Then (15) is equivalent to :
(16)  
where and and . The last equality is due to the symmetry of with respect to .
Denote the probability density function (PDF) of multivariate Gaussian variable as
, is finalized as the integral of the PDF of a multivariate Gaussian variable over an interval :(17)  
More generally, denotes the probability mass in from where has density function . From Lemma 1, we know that is log concave, when has logconcave distribution.
Since , where is full rank and is a constant, Lemma 2 implies is log concave function.
∎
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