Voltage stress minimization by optimal reactive power control
Abstract
A standard operational requirement in power systems is that the voltage magnitudes lie within prespecified bounds. Conventional engineering wisdom suggests that such a tightlyregulated profile, imposed for system design purposes and good operation of the network, should also guarantee a secure system, operating far from static bifurcation instabilities such as voltage collapse. In general however, these two objectives are distinct and must be separately enforced. We formulate an optimization problem which maximizes the distance to voltage collapse through injections of reactive power, subject to power flow and operational voltage constraints. By exploiting a linear approximation of the power flow equations we arrive at a convex reformulation which can be efficiently solved for the optimal injections. We also address the planning problem of allocating the resources by recasting our problem in a sparsitypromoting framework that allows us to choose a desired tradeoff between optimality of injections and the number of required actuators. Finally, we present a distributed algorithm to solve the optimization problem, showing that it can be implemented online as a feedback controller. We illustrate the performance of our results with the IEEE30 bus network.
Index terms— power networks, voltage support, reactive power compensation, resource allocation, distributed control.
I Introduction
Traditionally, the main purpose of voltage support is to maintain voltage magnitudes tightly within predetermined security constraints (e.g., within of some nominal level). Conventional wisdom suggests that such a tightly regulated voltage profile, imposed for system design reasons and good operation of the power network, should also guarantee a secure system, operating far from static bifurcation instabilities such as voltage collapse. Techniques for voltage support include shunt and static VAR compensation [1], series compensation [2], offnominal transformer tap ratios [3], synchronous condensers [4], and inverters operating away from unity power factor [5]. See [6] for a survey on the topic.
A distinct voltage control problem, which represents a key direction in power system stability analysis, has been the development of indices quantifying a power network’s proximity to voltage collapse. A broad overview of this large subfield can be found in [7, 8, 9]. The most reliable existing approaches are largely based on numerical methods and lack detailed theoretical support. They often require either continuation power flow [10] to identify the insolvability boundary, or repeated computation of loading margins over various directions in parameterspace [11].
As stressed, voltage support and distance to collapse are often analyzed separately in power systems although they are intrinsically related through the well known principle of reactive power injection. Combining the two problems represents the first contribution of the paper which is threefold. Indeed the ultimate goal in voltage support problems is the security task to confine the voltage magnitudes within predetermined bounds, as suggested by conventional engineering wisdom. Here, we follow an alternative approach: we define a particular measure for the network stress, i.e., the stress experienced by the network induced by the load profile. In particular, we begin our analysis from the recent article [12] where a sufficient and tight condition was presented for solvability of decoupled reactive power flow. This condition, rigorously proved only for the reactive decoupled case, quantifies the proximity to voltage collapse by determining a nodal measure of network stress. Based on this condition we pursue a novel systemlevel formulation of optimal voltage support encoded as an optimization problem with stressminimization, i.e., maximization of the distance to voltage collapse, as objective and subject to voltage security constraints. This approach allows us to match a local security requirement as well as a systemlevel stressminimization objective encoding the distance to collapse. By exploiting an opportune linearized reformulation, our optimization formulation becomes convex and can be efficiently solved for the optimal injections. As second contribution, we also address the planning problem of allocating the available resources by regularizing our optimization problem with a convex proxy of the cardinality function. This sparsitypromoting formulation allows us to choose a desired tradeoff between performance and a costeffective solution. Finally, we present a distributed algorithm for the stress minimization problem which is amenable to realtime implementation as a distributed feedback controller.
Compared to other approaches to voltage support problems our results do not rely on the assumption of a radial (i.e., acyclic) power grid topology [5]. This makes our approach appealing for power transmission networks. Different from the reactive power compensation literature [13] and from the voltage support literature [5], we seek stress minimization rather than optimal power flow (minimizing, e.g., losses) or voltage security tasks. Moreover, our formulation can nicely incorporate controller placement tasks.
The remainder of this paper is organized as follows. In Section II we introduce the required power system model. In Section III we present the first two main contributions of the paper: (i) we review the typical objectives for voltage regulation problems, propose a novel measure for the network stress, and formulate our optimization problem. We then present and solve the convex reformulation of the problem. (ii) We analyze a sparsitypromoting cost to address the planning problem. In Section IV we present the third contribution consisting in a distributed strategy to perform realtime stress minimization. Finally, Section V offers conclusions and future directions.
Ii Preliminaries
Iia Power Network, Generator and Load Models
A high voltage power network can be modeled as a connected, undirected and complexweighted graph where represents the set of nodes (or buses), () is the set of edges (or branches) connecting the nodes, that is the set of unordered pairs , such that and are connected to each other. Under synchronous steadystate operating conditions, all the electric quantities are sinusoidal signals at the same frequency. At every bus we have the following phasor quantities:

nodal voltage: ;

current injection: ;

power injection: ;.
where and denotes the complex conjugate operator. By collecting all quantities into vectors , Kirchhoff’s and Ohm’s laws lead to
(1) 
where denotes the imaginary unit. The symmetric and sparse susceptance matrix encodes the topology of the underlying electric network weighted by the line susceptances. Following standard assumptions we neglect line losses in highvoltage transmission networks [14, 15]. For recent studies on lossy distribution networks, see [5, 13]. From (1) we can write the Power Flow Equations (PFEs) as
(2) 
where denotes the diagonal matrix with diagonal entries . Expanding (2), for each the real and imaginary parts must satisfy
(3a)  
(3b) 
The PFEs (3a)–(3b) relate the voltage variables to the power variables , while the behavior of each bus is specified by the particular model assumed to describe it. In this paper, we partition the set of buses into two subsets, namely () which identifies powerregulated or load buses, and () which identifies voltageregulated buses.^{1}^{1}1We use the subscript for voltageregulated buses because in transmission grids these are typically generator buses. In particular, we assume the following:

Load bus model: loads are modeled as buses [15, 17]. In our setup, this model refers also to sources interfaced with power electronics and voltage support equipment such as synchronous condensers. While our results extend to ZIP load models [14], for simplicity of presentation we restrict ourselves to constant power loads ; constant impedance loads can be incorporated into the matrix as diagonal elements.
After relabeling the buses to place loads before generators, the matrix can be partitioned in the blockmatrix form
(4) 
Assumption 1 (Properties of )

is a Metzler matrix whose eigenvalues are characterized by a negative real part;^{2}^{2}2In other words, is an matrix.

the graph associated to the matrix (i.e., the graph induced by the load buses ) is connected.
Assumption 1 (i) is typically verified in practice [18], and always satisfied in the absence of phaseshifting transformers, linecharging and shunt capacitors. Regarding the shunt capacitors, they are allowed to be different from zero, however Assumption 1 (i) limits their sizes. Assumption 1 (ii) can be made without loss of generality, since connected components of the induced graph will be electrically isolated from one another by voltageregulated generator buses.
IiB Decoupled Reactive Power Flow & Critical Load Matrix
Under normal operating conditions, the highvoltage operating point is characterized by small voltages angle differences [15] which are treated as parameters [18] or considered as negligible [19]. We formalized this statement with the following
Assumption 2 (Decoupling Assumption)
In steadystate operating conditions, for , the voltage angle differences are constant and such that
Note that, under Assumption 2, from the form of Eq.(3b), it is possible to define an effective susceptance matrix by embedding the power angle terms into the original line susceptances. Under the decoupling Assumption 2 the Reactive Power Flow Equations (RPFEs) (3b) simplifies in vector notation to
(5) 
We now take into account the models and the partition introduced in Sections IIA and, accordingly we partition the vectors of voltage magnitudes and reactive power injections as , . Combining the power flow (5), the loads model and the partitioning (4), the power balance at each load can be written as
(6) 
We define the opencircuit voltages as
(7) 
which are well defined under Assumption 1. Physically, the opencircuit voltages (7) are the voltages one would measure at the load buses for zero reactive power demands . With this notation, the RPFEs (6) can be written as
(8) 
Once (8) is solved for an operating point , the reactive power injections at generators buses are uniquely determined by substituting the operating point into the final equations in (5). We define one more useful quantity.
Definition 3 (Critical Load Matrix)
Given the matrix and the opencircuit profile as defined in (7), the critical load matrix is defined as
(9) 
The critical load matrix concisely combines the network structure, generator voltages, shunts, and the relative locations of generation and load. In particular, it will help us to formulate the optimal voltage support problem to follow. Finally, it is convenient to rewrite (8) in a normalized set of variables. Using the opencircuit voltages defined in (7), we denote the vector of normalized voltages as
(10) 
Note that if the opencircuit profile is flat ( for some ), then is simply the standard vector of per unit voltages. In general, however, due to inhomogeneous generators voltage set points and the presence of shunt compensation, is not flat and the scalings in (10) are nonuniform. Substituting into the RPFEs (8) and using (9), (8) takes the simple form
(11) 
Iii Formulation of Optimal Voltage Support Problem
In this section we present our novel problem formulation. First, we present a common operational requirement highlighting its possible inadequacy to capture the safe operation of the grid. Then, we present a novel metric to measure the stress induced on the network by the load demand, representing our objective function. Finally, we present a linearization which leads to a convex reformulation of the optimization problem.
Iiia Security Constraints
A common operational requirement is that the load buses voltage magnitudes must lie within a predefined percentage deviation, typically , from a reference voltage. This tight clustering of voltages is due to the following reasons:

loads and some system components are designed to operate with a voltage in a narrow region around the network base voltage;

a flat voltage profile minimizes current flows and, consequently, minimizes resistive power losses;

a flat profile usually reduces the sensitivity of the voltage profile with respect to load changes (see Example 5);
We formalize this requirement by defining the secure set.
Definition 4 (Secure set)
Given a reference voltage , a percentage deviation and as in (7), the secure set is defined as
(12) 
Hence, if is a solution to (11), then all voltages lie within percent of the nominal voltage . While this represents a baseline operational requirement, under some circumstances it may not be sufficient to ensure safe grid operation. We present a simple example highlighting this fact.
Example 5 (Security requirement inadequacy)
Consider the simple twobuses case study consisting of a load connected to a source at voltage , as illustrated in Figure (a)a. For the case where , Figure (b)b plots the nose curve, i.e., the locus of solutions to (8) (blue solid blue) as is varied from to . Note that for a chosen , there may be two, one, or zero feasible solutions of (8). The secure set is shown as a shaded area between two dashed black lines. Also shown are the loading limits which ensure the highvoltage solution to lie in the secure set (dashed orange), and the tangent line to the nose curve at the midpoint between the dashed orange lines (dashed magenta). This tangent line captures the sensitivity of the load voltage to changes in reactive power demand. From Figure (b)b, note that if is too large, the operating point does not lie within . A standard policy is then to support the voltage level by adjusting the shunt compensation, i.e., by increasing . When , Figure (b)b demonstrates that the security requirement guarantees a “safe” distance to collapse, represented by the nose of the blue curve. Moreover, the sensitivity of the voltage to changes in load is small, meaning that relatively large changes in loading do not translate into large voltage changes. Conversely, in Figure (c)c , and the security requirement is “dangerously” close to the nose of the curve. Finally, the sensitivity line is steeper meaning that small changes in the load cause relatively big changes in the voltage. This affects the robustness of the network to small load changes.
The previous analysis highlights that the security requirement alone could be insufficient. Note that, to operate the grid in the point farthest from voltage collapse and to ultimately maximize the stability and robustness margins, a simple intuition is that of minimizing the distance of the operating point from the opencircuit solution , represented by the leftmost point on the blue curve — constrained to the fact that the operating point must belong to . As final remark, note that in general does not belong to . Thus in general, distancetocollapse minimization and voltage compensation do not coincide.
IiiB Network Stress Measure and Stress Minimization Problem
Based on the insights given by Example 5, we define the following measure quantifying the distance to collapse.
Definition 6 (Network Stress Measure)
Consider the RPFEs (11) in the normalized voltages . The network stress measure induced by the load is defined as
(13) 
Definition 6 is based on the intuition that the opencircuit profile is the network’s natural operating point in absence of loading, i.e., under “no stress”. Conversely, when the network works close to the nose tip, i.e., the farthest point from then, this is a “highstress” scenario. In this sense, the stress function (13) quantifies the loading on the network conveniently expressed in the normalized profile .
In the following, we assume that a certain number of load buses can be equipped with additional controlled devices, e.g., synchronous condensers [4]. We assume these devices can provide a controllable amount of reactive power support, and in the following we model them as controllable sources of reactive power , subject to upper and lower operational bounds. Specifically, the RPFEs (11) are modified as
(14) 
where is such that and are vectors representing the injection capacity constraints. If load bus is not equipped with a compensator, we set .
We now formulate our optimization problem of interest, which we refer to as the Stress Minimization problem.
Problem 7 (Stress Minimization)
Given , as in (9) and the capacity limits , , find and such that
The main idea behind Problem 7 is that minimizing keeps the operating point away from the tip of the nose curve, i.e., the collapse point. The standard security requirement is imposed as a hard constraint.
Since is related to through the quadratic equality constraints (14), Problem 7 is nonlinear and nonconvex. In the following, we convexify this problem through the use of a power flow linearization.
IiiC Linear Approximation and Convexification
We now introduce a suitable linearization for Problem 7 which had been first presented in [20]. From (14), assuming , we expect the normalized profile (10) to be which would be the exact high voltage solution corresponding to . Linearizing the RPFEs (14) around , to first order, the solution is given by
(16) 
That is, to first order the solution of (14) is given by a uniform component plus a deviation which is linear in the reactive injections. Using (16), the cost (13) is approximated by
(17) 
Note that the approximated cost function (17) is convex in the reactive power injections . By exploiting (16) and thanks to some algebraic manipulations, it is possible to see that the security requirement holds if and only if
(18) 
Substituting for from (16), (18) is equivalent to
(19) 
where
(20a)  
(20b) 
Thus, the security constraints have been converted into linear inequality constraints on the decision variables . We now present the convexified version of Problem 7.
Problem 8 (Convex Stress Minimization)
Remark 9 (On the stress measure)
Aside from the linearizationbased derivation in this subsection, the measure (21) is inspired by recent results [12] on the solvability of the decoupled reactive power flow equations (8), where it has been shown that represents a proper distancetocollapse measure. Indeed, if the non linear (8) has a unique highvoltage solution safe from collapse.
Observe that in Problem 8 the cost (21) and the constraints (22) are convex in the decision variables. Moreover, Problem 8 can be written as a linear program and can therefore be efficiently solved via convex optimization.
Before presenting some performance and simulations of the stress minimization procedure, notice that both Problems 7 and 8 are offline centralized procedures which, as suggested by the formulation in Section IIIB, assume that either the full set of load buses or only an a priori assigned subset of them are equipped with controllable devices. The first scenario is impractical and economically unfeasible in large networks due to the large number of devices needed. The second scenario could likely lead to a suboptimal allocation of resources if no specific allocation policies are used. In the following subsection, we refine Problem 8 to simultaneously solve for the planning problem of allocating the resources along with the systemlevel stress minimization problem.
IiiD The Planning Problem: Sparse Stress Minimization
Here we propose a modification of Problem 8 to find a desired tradeoff between the number of actuators and the minimization of the stress cost. In order to accomplish this task, we propose a sparsitypromoting approach where, by tuning an additional parameter, the user is able to control the sparsity of the solution. In this way, we simultaneously solve the systemlevel stress minimization problem as well as the planning problem of allocating a finite number of resources.
The cardinality function, , is a natural choice to account for the number of devices. However, it is discontinuous and nonconvex. A convex approximation of is the reweighted norm [21]
(23) 
where . Adding equation (23) to the cost function (21), it is possible to formulate the following problem which we refer to as the Sparse Stress Minimization problem.
Problem 10 (Sparse Stress Minimization)
Consider the same setup as in the Convex Stress Minimization of Problem 8. Then the goal is to
The parameter in the cost function (10) can be used to promote sparsity of the solution , and thereby minimize the number of required actuators. Obviously for , Problem 10 reduces to Problem 8. By increasing the value of the user can force the solver to lean towards a more sparse solution. This automatically compels the solver to optimally allocate the resources in order to find the best tradeoff between sparsity and systemlevel stress minimization.
IiiE Simulation: Planning Problem and Offline Optimization
We now present a case study to show the effectiveness of planning and the offline optimization procedure proposed. The simulations refer to Problem 10 and are implemented in MATLAB and CVX [22]. The plotted voltage profiles refer to the linearized solution (16) of the decoupled RPFEs (14). The testbed consists of:

IEEE 30 bus transmission grid [23];

a reference voltage [p.u.];

a voltage deviation limit ;

capacity limits .
Figures (a)a–(b)b–(c)c illustrates the placement of actuators for increasing values of (, and , respectively). Compensators placed by the optimization problem are indicated with a triangle. The color scheme for loads and compensators is as follows:

reactive injections, i.e., positive values, are plotted in bluescale (): the lighter the blue, the bigger the injection absolute value;

reactive consumptions/absorptions, i.e., negative values, are plotted in redscale (): the darker the red, the bigger the absolute value of the consumption;

white node means zero injections/consumptions.
First of all, for the solver places compensation everywhere. Moreover it can be seen that one compensator is red colored, meaning that it effectively absorbs reactive power. This occurs due to large reactive power injections at neighboring buses, which drive up voltage values across the network – additional reactive power must be absorbed to lower specific voltages and meet the security constraints. Sparsity promotion takes place for increasing with the solver placing compensators only where the heaviest loading occurs. Figure 3 shows, as a function of , the number of devices placed (leftblue axis) and the ratio between the value of the cost (21) after a polishing step, i.e., obtained as solution of Problem 8 given the placement obtained solving Problem 10, over the value of the cost before the optimization (rightred axis), i.e.,
It can be seen how, for increasing the final value of the cost increases since a smaller number of controllable units are less able to compensate the voltage profile. However, as can be seen from the first part of the plot, by using only 11 controllers we achieve the same performance as of using 24 compensators. This not only highlights the redundancy of using 24 compensators but that the optimal placement is necessary to achieve the same level of performance. Figures 4 shows the linearized profiles of before and after the optimization for different . It can be seen that for increasing the profile is less compensated, i.e., it is farther from the profile. Finally, as already stressed, note that stress minimization and classical voltage compensation do not coincide. Indeed, , in general, does not belong to , identified by the black dashed lines.
Iv Distributed Online Stress Minimization through Feedback Control
The previous methods of Sections IIIC and IIID are suitable only for offline optimization and planning. In this section we assume the planning problem has been solved offline, and develop a dualascent algorithm for Problem 8 which may be implemented online as a distributed feedback controller. This is motivated by different reasons among which it is worth mentioning that:

utilities could prefer not to share information with a central operator because of privacy reasons;

an online implementation can be naturally exploited as a distributed feedback controller in presence of timevarying loads, to reject disturbances, to increase the system robustness, and to track the optimal solution.
In the following, we assume that “smart agents” are embedded at all the grid’s load buses. These are characterized by mild communication and computational capabilities. Moreover, they can communicate according to a communication graph which is designed to coincide with the electrical network. It is worth mentioning that, as will be clear later, even the load buses not equipped with a controllable compensator are required to share “smartness” capabilities.
In the current formulation of Problem 8, the presence of the dense matrix in both the cost and the constraints (19) compromises the possibility to solve the stress minimization problem in a distributed fashion. However, the matrix is sparse and the graph induced by its sparsity pattern coincides with the topology of the grid which connects the load buses. We take advantage of this structure to develop a distributed algorithm to solve Problem 8.
Whereas the formulation of Problem (8) expressed the stress minimization compactly in injection coordinates , now we derive the equivalent formulation in voltage coordinates to leverage on the sparsity of . We start our analysis defining the deviation variable as
(25) 
which represents the linear deviations of the voltages as defined in (16) due to the overall reactive injection. Similar to what done in Section IIIC, from the definition of set it is possible to obtain the security constraints expressed in the coordinates. These are equal to
where
(26)  
From the definition of it is clear that
(27) 
Since the matrix is characterized by a sparsity pattern equivalent to that induced by the electric graph connecting the loads, the desired control inputs can be computed by means of a local exchange of information, namely the variables among electric neighbors. Additionally, from (27) it is easy to impose the capacity constraints, i.e., . The Problem 8 is then equivalent to
Problem 11 (Online Stress Minimization)
Now, we point out three more issues related to Problem 11: (i) the norm is not everywhere differentiable and thus not suitable for a gradientbased iterative procedure; (ii) computing the cost in (11) requires knowledge of all variables; (iii) in order to compute the derivative of the maximum function embedded in , the index where the maximum is attained must be known. Next, we propose one possible solution to these issues.
Iva A Smooth Decomposable Approximation of norm
We now present a continuously differentiable approximation for the norm which combines a smooth approximation for the maximum function, the softmax [24], and a smooth approximation for the absolute value. This reads as
(29)  
The idea behind (29) is to exploit the superlinearity property of the exponential to let the maximum component of the vector dominate the other components. The exponentiation is exploited to recover differentiability of the absolute value. The approximation approximates in the following sense; the proof can be found in Appendix A.