1

Reverse and Forward Engineering of Local Voltage Controlin Distribution Networks

Xinyang Zhou, Masoud Farivar, Zhiyuan Liu, Lijun Chen and Steven Low

Abstract— The increasing penetration of renewable and dis-tributed energy resources in distribution networks calls for real-time and distributed voltage control. In this paper we investigatelocal Volt/VAR control with a general class of control functions,and show that the power system dynamics with non-incrementallocal voltage control can be seen as distributed algorithm forsolving a well-defined optimization problem (reverse engineer-ing). The reverse engineering further reveals a fundamentallimitation of the non-incremental voltage control: the convergencecondition is restrictive and prevents better voltage regulationat equilibrium. This motivates us to design two incrementallocal voltage control schemes based on the subgradient andpseudo-gradient algorithms respectively for solving the sameoptimization problem (forward engineering). The new controlschemes decouple the dynamical property from the equilibriumproperty, and have much less restrictive convergence conditions.This work presents another step towards developing a newfoundation – network dynamics as optimization algorithms – fordistributed realtime control and optimization of future powernetworks.

Index Terms— Distributed control and optimization, voltageregulation, network dynamics as optimization algorithms, reverseand forward engineering, power networks.

I. IntroductionTraditionally, given the predictable and relatively slow

changes in power demand, capacitor banks and load tapchangers are switched a few times per day to regulate thevoltage in distribution systems; see, e.g., [4], [5]. However,with the increasing penetration of renewable energy resourcessuch as photovoltaic (PV) and wind turbines in both residentialand commercial settings [6], [7], the intermittent and fast-changing renewable energy supply introduces rapid fluctu-ations in voltage that are beyond the capability of thosetraditional voltage regulation schemes and thus calls for newvoltage control paradigms.

A. Inverter-Based Voltage Regulation

Even though the current IEEE Standard 1547 [8] requiresdistributed generation to operate at unity power factor, in-verters can readily adjust real and reactive power outputs to

X. Zhou is with the National Renewable Energy Laboratory, Golden, CO80401, USA (Email: xinyang.zhou@nrel.gov).

Z. Liu and L. Chen are with College of Engineering and Applied Science,University of Colorado, Boulder, CO 80309, USA (Emails: zhiyuan.liu,lijun.chen@colorado.edu).

M. Farivar is with Google. (Email: mfarivar@gmail.com).S. Low is with the Division of Engineering and Applied Science,

California Institute of Technology, Pasadena, CA 91125, USA (Email:slow@caltech.edu).

Preliminary results of this paper have been presented at IEEE Conference onDecision and Control, Florence, Italy, 2013 [1], IEEE International Conferenceon Smart Grid Communications, Miami, FL, 2015 [2], and Annual AllertonConference on Communication, Control, and Computing, Allerton, IL, 2015[3].

stabilize voltages and cope with fast time-varying conditions.Indeed, the IEEE Standards group is actively exploring a newinverter-based Volt/Var control. Unlike the capacity banks ortap changers, inverters can push and pull reactive power muchfaster, in a much finer granularity and with low operationcost, enabling real-time distributed control that is needed forthe future power grid with a large number of renewable anddistributed energy resources.

Inverter-based voltage regulation has been studied exten-sively in literature. Related work roughly falls into the fol-lowing categories:

1) Centralized control: By collecting all the required in-formation and computing a global optimal power flow(OPF) problem, a central controller determines optimalset-points for the whole system; see, e.g., [9]–[13]. Cen-tralized control can incorporate general objectives andoperational constraints, but suffers from considerablecommunication overhead and long computation timeespecially when the size of the system is large. So, itusually cannot provide fast control.

2) Distributed control: For the OPF problems of certainstructures, one can design algorithms to distribute thecomputation with coordinating communication, whichis conducted either between a central controller andagents in a hierarchical way, e.g., [14]–[21], or amongneighborhoods of individual agents without any centralcontroller, e.g., [22]–[27]. Given the required commu-nication infrastructure, the scheme based on distributedOPF algorithm can provide scalable voltage control.

3) Local control: Based on only local information, localvoltage control provides fast response and, without theneed of communication, allows simple and scalableimplementation; see, e.g., [1], [28]–[34].

In this paper, we focus on analysis and design of localvoltage control. Characterization of local control, especiallysystemwide properties arising from the interaction betweenlocal controls, is challenging. In literature, the work such as[34], [35] lack analytical characterization. Other work suchas [32], [33] provide stability analysis but lack systemwideperformance characterization. There are work that providerigorous performance analysis for stability and systemwideperformance, but are subject to control functions of particulartype, e.g., linear control functions without deadband [29], [36]and quadratic control functions [28].

B. Reverse and Forward Engineering

Different from other work, in this paper we consider localvoltage control with general monotone control functions, andseek a principled way to guide systematic analysis and design

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of local voltage control with global perspective through theapproach of reverse and forward engineering. We first developmodels to understand the systemwide properties arising fromthe interaction between local controls, in particular, whetherthe power system dynamics with the existing controls canbe interpreted as distributed algorithms for solving certainoptimization problems, i.e., network dynamics as optimiza-tion algorithms. We then leverage the insights obtained formthe reverse engineering to design new local volatge controlschemes according to distributed algorithms for solving theresulting optimization problem (or its variant that incorporatesnew design objective and/or constraints).

Specifically, we first lay out a general framework for reverseengineering power system dynamics with non-incrementallocal voltage control with general control functions, and showthat it can be seen as a distributed algorithm for solving awell-defined optimization problem. We characterize the con-dition under which the dynamical system converges, which ishowever very restrictive and prevents better voltage regulationat the equilibrium (or optimum): aggressive control functionsare preferred for better voltage regulation at equilibrium, whileless aggressive ones are preferred for convergence. We aretherefore motivated to find a way to decouple the dynamicalproperty from the equilibrium property.

Notice that the optimization-based model does not onlyprovide a way to characterize the equilibrium and establishthe convergence of power system dynamics with local control(i.e., reverse engineering), but also suggests a principled wayto engineer the control to achieve the desired property (i.e.,forward engineering). In particular, new control schemes withbetter dynamical properties can be designed based on differentoptimization algorithms for solving the same optimizationproblem. Accordingly, we propose an incremental local volt-age control scheme based on the (sub)gradient algorithm forsolving the same optimization problem. This new controlscheme decouples the equilibrium property and the conver-gence property, and has much less restrictive convergencecondition. However, it converges to only within a smallneighborhood of the equilibrium. Furthermore, it requirescomputing the inverse of the control function, which may incurconsiderable computation overhead. We thus propose anotherincremental local voltage control scheme based on a pseudo-gradient algorithm that has better convergence property andsimpler implementation than the (sub)gradient control whileachieving the same equilibrium.

Similar idea of reverse and forward engineering based on theperspective of network dynamics as optimization algorithmshas been applied to distributed real-time frequency control ofthe power system, e.g., [37]–[43], as well as synchronizationof the network of coupled oscillators [44]. This paper presentsanother step towards developing a new foundation – networkdynamics as optimization algorithms – for distributed realtimecontrol and optimization of future power networks.

The rest of the paper is organized as follows. Section II de-scribes the system model and introduces the non-incrementallocal voltage control. Section III investigates the equilibriumand dynamical properties of the non-incremental local controlby reverse engineering. Section IV proposes two incremental

local voltage control schemes that decouple the equilibriumand convergence properties and have much less restrictiveconvergence conditions. Section V provides numerical exam-ples to complement the theoretical analysis, and Section VIconcludes the paper.

II. NetworkModel and Local Voltage ControlConsider a tree graph G = N ∪ 0,L that represents a

radial distribution network consisting of n+1 buses and a set Lof undirected lines between these buses. Bus 0 is the substationbus (slack bus) and is assumed to have a fixed voltage of v0 =

1 p.u. Let N := 1, . . . , n. Due to the tree topology, we alsohave the cardinality of the line set |L| = n. For each bus i ∈ N ,denote by Li ⊆ L the set of lines on the unique path from bus 0to bus i, pc

i and pgi the real power consumption and generation

respectively, and qci and qg

i the reactive power consumption andgeneration respectively. Let vi be the magnitude of the complexvoltage (phasor) at bus i. For each line (i, j) ∈ L, denote by ri j

and xi j its resistance and reactance, and Pi j and Qi j the real andreactive power from bus i to bus j. Let `i j denote the squaredmagnitude of the complex branch current (phasor) from busi to bus j. We summarize some of the notations used in thispaper in Section II-A.

A. NotationN set of buses excluding bus 0, N := 1, ..., nL set of power linesLi set of lines from bus 0 to bus ipc

i , qci real, reactive power consumption at bus i

qgi , q

gi real, reactive power generation at bus i

Pi j,Qi j real and reactive power flow from i to jri j, xi j resistance and reactance of line (i, j)vi magnitude of complex voltage at bus i`i j squared magnitude of complex current of

line (i, j)Ωi feasible power set of inverter i; Ω :=× n

i=1Ωi

[ ]Ωi projection onto set Ωi

σmax( ) maximum singular value of a matrixλmax( ) maximum eigenvalue of a matrixαi upper-bound of the (sub)derivative d fi(vi)/dvi;

A := diagα1, . . . , αn ∈ SN++

A quantity without subscript is usually a vector with appro-priate components defined earlier, e.g., v := (vi, i ∈ N), qg :=(qg

i , i ∈ N).

B. Linearized Branch Flow Model

We adopt the following branch flow model introduced in[4], [5] (DistFlow equations) to model a radial distributionsystem:

Pi j = pcj − pg

j +∑

k:( j,k)∈L

P jk + ri j`i j, (1a)

Qi j = qcj − qg

j +∑

k:( j,k)∈L

Q jk + xi j`i j, (1b)

v2j = v2

i − 2(ri jPi j + xi jQi j

)+

(r2

i j + x2i j

)`i j, (1c)

`i jv2i = P2

i j + Q2i j. (1d)

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Following [45] we assume that the terms involving `i j arezero for all (i, j) ∈ L in (1). This approximation neglects thehigher order real and reactive power loss terms. Since lossesare typically much smaller than power flows Pi j and Qi j, itonly introduces a small relative error, typically on the orderof 1%. We further assume that vi ≈ 1, ∀i so that we canset v2

j − v2i = 2(v j − vi) in equation (1c). This approximation

introduces a small relative error of at most 0.25% if there isa 5% deviation in voltage magnitude.

With the above approximations (1) is simplified to thefollowing linear model:

Pi j =∑

k∈β( j)

(pc

k − pgk

),

Qi j =∑

k∈β( j)

(qc

j − qgj

),

vi − v j = ri jPi j + xi jQi j,

where β( j) is the set of all descendants of bus j including busj itself, i.e., β( j) =

i|L j ⊆ Li

. This yields an explicit solution

for vi in terms of v0 (which is given and fixed):

v0 − vi =∑

( j,k)∈Li

r jkP jk +∑

( j,k)∈Li

x jkQ jk

=∑

( j,k)∈Li

r jk

∑h∈β(k)

(pc

h − pgh

) +∑

( j,k)∈Li

x jk

∑h∈β(k)

(qc

h − qgh

)=

∑j∈N

(pc

j − pgj

) ∑(h,k)∈Li∩L j

rhk

+∑j∈N

(qc

j − qgj

) ∑(h,k)∈Li∩L j

xhk

=

∑j∈N

Ri j

(pc

j − pgj

)+

∑j∈N

Xi j

(qc

j − qgj

),

where

Ri j :=∑

(h,k)∈Li∩L j

rhk, Xi j :=∑

(h,k)∈Li∩L j

xhk. (2)

Fig. 1. Li ∩ L j for two arbitrary buses i, j in the network and thecorresponding mutual voltage-to-power-injection sensitivity factors Ri j, Xi j.

Fig. 1 gives an illustration of Li∩L j for two arbitrary busesi and j in a radial network and the corresponding Ri j and Xi j.Since

Ri j =∂vi

∂pgj

= −∂vi

∂pcj, (3a)

Xi j =∂vi

∂qgj

= −∂vi

∂qcj, (3b)

Ri j, Xi j are also referred to as the mutual voltage-to-power-injection sensitivity factors.

Define a resistance matrix R = [Ri j]n×n and a reactancematrix X = [Xi j]n×n. Both matrices are symmetric. With the

matrices R and X the linearized branch flow model can besummarized compactly as:

v = v0 + R(pg − pc) + X(qg − qc), (4)

where v0 = [v0, . . . , v0]> is an n-dimensional vector. In thispaper we assume that v0, pg, pc, qc are given constants. Theonly variables are (column) vectors v := [v1, . . . , vn]> ofsquared voltage magnitudes and qg := [qg

1, . . . , qgn] of generated

reactive powers. Let v = v0 + R(pg − pc) − Xqc, which is aconstant vector. For notational simplicity, we will henceforthignore the superscript in qg and write q instead. Then thelinearized branch flow model reduces to the following simpleform:

v = Xq + v. (5)

We have the following result.

Lemma 1 The matrices R and X are positive definite. 2

Proof: The proof uses the fact that the resistance andreactance values of power lines in the network are all positive.Here we give a proof for the reactance matrix X, and exactlythe same argument applies to the resistance matrix R.

We prove by induction on the number k of buses in thenetwork, excluding bus 0 (the root bus). The base case ofk = 1 corresponds to a two-bus network with one line. HereX is obviously a positive scalar that is equal to the reactanceof the line connecting the two buses.

Suppose that the theorem holds for all k ≤ n. For the caseof k = n + 1 we consider two possible network topologies asshown in Figure 2:

(a) Case1: degree of bus 0 isgreater than 1

(b) Case 2: degree of bus 0 is1

Fig. 2. Two possible network structures

Case 1: bus 0 is of degree greater than 1. Split the networkinto two different trees rooted at bus 0, denoted by T1 andT2, each of which has no more than n buses excluding bus 0.Denote by X1 and X2 respectively the reactance matrices ofT1 and T2. By induction assumption X1 and X2 are positivedefinite. Note that the set Li of lines on the unique path frombus 0 to bus i must completely lie inside either T1 or T2, forall i. Therefore, by definition (2), the reactance matrix X ofthe network has the following block-diagonal form:

Xi j =

X1i j , i, j ∈ T1X2i j , i, j ∈ T20, otherwise

⇒ X =

[X1 00 X2

].

Since X1 and X2 are positive definite, so is X.Case 2: bus 0 is of degree 1. Suppose without loss of

generality that bus 0 is connected to bus 1. Denote by xthe reactance of the line connecting buses 0 and 1, and Tthe tree rooted at bus 1, excluding bus 0. Denote by Y thereactance matrix of T , and by induction assumption, Y is

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positive definite. Note that, for all buses i in the network,the set Li includes the single line that connects buses 0 and1. Therefore, by definition (2), the reactance matrix X has thefollowing form:

Xi j =

Yi j + x, i, j ∈ Tx, otherwise ⇒ X =

x . . . x...

...x . . . x

+

[0 00 Y

],

One can verify that, when Y is positive definite and x ispositive, X is positive definite. This concludes the proof.

We also refer to [29] for an alternative proof of the sameresult.

C. Inverter Model

At each bus i there is an inverter that can generate non-negative real power pi and reactive power qi that can haveeither sign. pi and qi are constrained by the apparent powercapability si of the inverter as follows:

0 ≤ pi ≤ si, 0 ≤ |qi| ≤ si, p2i + q2

i ≤ s2i . (6)

Consider power ratio cos ρi with 0 ≤ ρi ≤ π/2 such that

pi/si ≥ cos ρi. (7)

Given non-controllable pi ≤ si, the feasible (reactive) powerset Ωi for inverter i is given by:

Ωi :=qi

∣∣∣ qimin ≤ qi ≤ qi

max, (8)

where, based on (6)–(7),

qmaxi = min

pi tan ρi,

√s2

i − p2i

,

qmini = max

−pi tan ρi,−

√s2

i − p2i

.

Here, pi is further assumed to be sized appropriately to provideenough freedom in qi [34]. For buses without controllableinverters, one can set qi = qmax

i = qmini and Ωi becomes a

singleton. Define Ω :=× ni=1 Ωi for notational simplicity.

D. Local Volt/VAR Control

The goal of Volt/VAR control in a distribution network isto maintain the bus voltages v to within a tight range aroundtheir nominal values vnom

i = 1 p.u., i ∈ N by provisioningreactive power injections q. This can be modeled as a feedbackdynamical system with state

(v(t), q(t)

)at discrete time t. A

general Volt/VAR control algorithm maps the current state(v(t), q(t)

)to a new reactive power injections q(t + 1). The

new q(t + 1) updates voltage magnitudes v(t + 1) according to(5). Usually q(t + 1) is determined either completely or partlyby a certain Volt/VAR control function defined as follows:

Definition 1 A Volt/VAR control function f : Rn → Rn is acollection of local control functions fi : R→ R, each of whichmaps the current local voltage vi to a local control variableui in reactive power at bus i:

ui = fi(vi − vnomi ). ∀i ∈ N . (9)

The control functions fi are usually decreasing but notalways strictly decreasing because of a potential deadbandwhere the control signal ui is set to zero to prevent toofrequent actuation. We assume for each bus i ∈ N a symmetricdeadband (vnom

i − δi/2, vnomi + δi/2) with δi ≥ 0 around the

nominal voltage vnomi . The following assumptions are made

for fi.

Assumption 1 The control functions fi are non-increasing inR and strictly decreasing and differentiable in (−∞,−δi/2) ∪(δi/2,+∞).

Assumption 2 The derivative of the control function fi isupper-bounded, i.e., there exist αi > 0 such that | f ′i (vi)| ≤ αi forall feasible vi ∈ (−∞,−δi/2)∪ (−δi/2, δi/2)∪ (δi/2,+∞), ∀i ∈N .

Assumption 2 ensures that an infinitesimal change in voltagedoes not lead to a jump in the control variable. Define A :=diagα1, . . . , αn ∈ S

N++, and let M = σmax(AX) denote the

largest singular value of AX. We have the following result.

Lemma 2 (Lipschitz continuity) Suppose Assumptions 1–2hold. For any q, q′ ∈ Ω, we have

‖ f (v(q) − vnom) − f (v(q′) − vnom)‖2 ≤ M‖q − q′‖2. (10)

Proof: Without loss of generality, assume that vi(q) ≥vi(q′). If both vi(q) and vi(q′) are in (−∞, vnom

i − δi/2] orin [vnom

i + δi/2,+∞), by the mean value theorem we have| fi(vi(q) − vnom

i ) − fi(vi(q′) − vnomi )| ≤ αi|vi(q) − vi(q′)|. If both

are in [vnomi −δi/2, vnom

i +δi/2], 0 = | fi(vi(q)−vnomi )− fi(vi(q′)−

vnomi )| ≤ αi|vi(q) − vi(q′)|. If vi(q) ∈ [vnom

i + δi/2,+∞) andvi(q′) ∈ [vnom

i − δi/2, vnomi + δi/2], | fi(vi(q) − vnom

i ) − fi(vi(q′) −vnom

i )| = | fi(vi(q) − vnomi ) − fi(δi/2))| ≤ αi|vi(q) − (vnom

i +

δi/2)| ≤ αi|vi(q) − vi(q′)|, where the first inequality followsfrom the mean value theorem. Similarly, we can show that| fi(vi(q) − vnom

i ) − fi(vi(q′) − vnomi )| ≤ αi|vi(q) − vi(q′)| holds

under other situations too. Therefore,

‖ f (v(q) − vnom) − f (v(q′) − vnom)‖2 ≤ ‖A(v(q) − v(q′))‖2,

from which we have

‖ f (v(q) − vnom) − f (v(q′) − vnom)‖2 ≤ ‖AX(q − q′)‖2≤ M‖q − q′‖2.

See Fig. 3 (left) for an illustrative example of a piecewiselinear droop control function based on IEEE Standard 1547[8]:

fi(vi) = −αi [vi − δi/2]+ + αi [−vi − δi/2]+ (11)

with slope −αi in (−∞,−δi/2) and (δi/2,+∞). Notice that ourdesign and analysis in this paper are not limited to the linearcontrol functions.

Motivated by the IEEE Standard 1547, we consider a “non-incremental” control where the reactive power qi = ui, i ∈ N ,and obtain the following dynamical system D1 for the localVolt/VAR control: v(t) = Xq(t) + v

qi(t + 1) =[fi(vi(t) − vnom

i)]

Ωi, i ∈ N ,

(12a)(12b)

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Fig. 3. (left) The piecewise linear control function (11), (middle) its inverse function (17), and (right) the corresponding cost function (18).

where [ ]Ωi denotes the projection onto the set Ωi. A fixed point(v∗, q∗) of the above dynamical system, defined as follows,represents an equilibrium operating point of the network.

Definition 2 (v∗, q∗) is called an equilibrium point of D1, ifit satisfies

v∗ = Xq∗ + v, (13a)q∗ =

[f (v∗ − vnom)

]Ω. (13b)

In the next section, we characterize the equilibrium anddynamical properties of the system D1 by showing that it is andistributed algorithm for solving a well-defined optimizationproblem.

III. Reverse Engineering

Since fi is non-increasing, a (generalized) inverse f −1i exists.

In particular, at the origin, we assign f −1i (0) = 0 corresponding

to the deadband [−δi/2,+δi/2] of fi. This may introducediscontinuity to f −1

i at qi = 0 if the deadband δi > 0, i.e.,

f −1i (0+) ≤ −δi/2 and f −1

i (0−) ≥ δi/2, (14)

where 0+ and 0− represent approaching 0 from right and left,respectively.

Define a cost function for provisioning reactive power ateach bus i ∈ N as:

Ci(qi) := −

∫ qi

0f −1i (q) dq, (15)

which is convex since f −1i is decreasing. Then, given vi(t),

qi(t + 1) in (12b) is the unique solution to the followingoptimization problem:

qi(t + 1) = arg minqi∈Ωi

Ci(qi) + qi(vi(t) − vnom

i), (16)

i.e., (12b) and (16) are equivalent specification of qi(t + 1).Take for example the piecewise linear control function (11).

Its inverse is given by:

f −1i (qi) :=

−

qiαi

+ δi2 if qi < 0,

0 if qi = 0,−

qiαi−

δi2 if qi > 0,

(17)

and the corresponding cost function is given by:

Ci(qi) =

12αi

q2i −

δi2 qi if qi ≤ 0,

12αi

q2i + δi

2 qi if qi ≥ 0.(18)

See Fig. 3 (middle and right) for illustration.

A. Equilibrium

Consider the function F(q) : Ω→ R:

F(q) := C(q) +12

q>Xq + q>∆v, (19)

where C(q) =∑

i∈N Ci(qi) and ∆v := v − vnom, and a globaloptimization problem:

minq∈Ω

F(q). (20)

Theorem 1 Suppose Assumption 1 holds. Then D1 has aunique equilibrium point. Moreover, a point (v∗, q∗) is anequilibrium of D1 if and only if q∗ is the unique optimalsolution of (20) and v∗ = Xq∗ + v.

Proof: By Lemma 1 the matrix X is positive definite. Thisimplies that the objective function F(q) is strongly convex.Hence, the first-order optimality condition for (20) is bothnecessary and sufficient; moreover, (20) has a unique optimalsolution. We next relate it to the equilibrium point of D1.

The subdifferential of F(q) is given by:

∂F(q) = ∂C(q) + Xq + ∆v

= ∂C(q) + (Xq + v) − vnom,

where, by the definition of Ci(qi),

∂C(q) =[∂C1(q1) , . . . , ∂Cn(qn)

]>with

∂Ci(qi) =

− f −1

i (qi) if qi , 0,[−δi2 ,

δi2

]if qi = 0.

By the optimality condition, q∗ is an optimum of (20) iffthere exists a (sub)gradient ∇F(q∗) ∈ ∂F(q∗) such that

∇F(q∗)>(q − q∗) ≥ 0, ∀q ∈ Ω,

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which is equivalent to:

q∗ = [ f(Xq∗ + v − vnom)

]Ω.

It follows that a point (v∗, q∗) is an equilibrium of D1 ifand only if q∗ solves (20) and v∗ = Xq∗+ v. The existence anduniqueness of the optimal solution of (20) then implies thatof the equilibrium (v∗, q∗).

With v = Xq + v, the objective can be equivalently writtenas:

F(q, v) = C(q) +12

(v − vnom)>X−1(v − vnom) −12

∆v>X−1∆v.

(21)

Notice that the last term is a constant. Therefore, the localVolt/VAR control D1 seeks an optimal trade-off betweenminimizing the cost of reactive power provisioning C(q) andminimizing the cost of voltage deviation 1

2 (v − vnom)>X−1(v −vnom).

1) Further Characterization of Equilibrium: The first termC(q) of the objective (21) is well-defined and has the desiredadditive structure. It is however not clear what specific struc-ture the second term 1

2 (v−vnom)>X−1(v−vnom) entails. We willfurther characterize this term in this subsection.

Notice that bus 0 has a fixed voltage magnitude, whichdecouples different subtrees rooted at it. Therefore, withoutloss of generality we only consider a topology where the bus0 is of degree 1. Denote T the (sub)tree rooted at bus 1 andLT the set of links of T . Define an inverse tree T ′ that hasthe same sets of buses and lines as T but with reciprocalline reactance 1/xi j. Let L ∈ Rn×n be the weighted Laplacianmatrix of T ′ defined as follows:

Li j =

−1/xi j, (i, j) ∈ LT ,∑

(i,k)∈L 1/xik, i = j,0, otherwise.

Recall that x denotes the reactance of the line connecting buses0 and 1, we have the following result by Liu et al. [46].

Theorem 2 (from [46]) Given the tree graph G = N ∪

0,L with bus 0 being of degree 1 and its reactance matrixX defined by (2), the inverse matrix X−1 has the followingexplicit form:

X−1 = L +

1/x 0 · · · 00 0 · · · 0...

.... . .

...0 0 · · · 0

. (22)

With the above result, the cost function (21) can be rewrittenas:

F(q, v) = C(q) +12

(v1 − vnom)2

x+

∑(i, j)∈LT

(vi − v j)2

xi j

−

12

∆v>X−1∆v. (23)

whose second term (i.e., the cost of voltage deviation) consistsof two parts: the first part (v1 − vnom)2/x represents the costof voltage deviation of the bus 1 from the nominal value, andthe second part

∑(i, j)∈LT

(vi − v j)2/xi j gives the cost of voltage

deviation between the neighboring buses. This leads to a niceleader-follower structure where the first bus (the bus 1) aimsto attain the nominal voltage while each other bus tries toachieve the same voltage as that of the bus “in front of” it.

B. Dynamics

We now study the dynamic properties of the local Volt/VARcontrol D1.

Theorem 3 Suppose Assumptions 1–2 hold. If

σmax(AX) < 1, (24)

then the local Volt/VAR control D1 converges to the uniqueequilibrium point (v∗, q∗). Moreover, it converges exponentiallyfast to the equilibrium.

Proof: Write D1 equivalently as a mapping g1:

q(t + 1) = g1(q(t)) :=[f (Xq(t) + ∆v − vnom)

]Ω. (25)

By Lemma 2 and the non-expansiveness property of projectionoperator, given any feasible q, q′ we have∥∥∥g1(q) − g1(q′)

∥∥∥2 ≤ M‖q − q′‖2, (26)

where M = σmax(AX). When condition (24) holds, M < 1 andthus the mapping g1 is a contraction, implying that (v(t), q(t))converges exponentially to the unique equilibrium point underD1.

We next develop a sufficient condition for (24), which iseasier to verify in practice. Define the following matrix normsfor some W ∈ Rm×n:

‖W‖1 = max1≤ j≤n

m∑i=1

|wi j|, ‖W‖∞ = max1≤i≤m

n∑j=1

|wi j|,

‖W‖2 =√λmax(W>W) = σmax(W),

where λmax(·) denotes the largest eigenvalue of a matrix. Byutilizing the following relationship among these matrix normsbased on Holder’s inequality

‖W‖2 ≤√‖W‖1 · ‖W‖∞ , (27)

we have the following sufficient condition for convergence ofD1.

Corollary 1 Suppose Assumptions 1–2 hold. If

maxi∈N

(αi) ·maxi∈N

∑j∈N

Xi j

< 1, (28)

thenD1 converges exponentially fast to the unique equilibriumpoint (v∗, q∗).

Proof: A sufficient condition for (24) based on (27) is

‖AX‖1 < 1 and ‖AX‖∞ < 1. (29)

Given symmetric matrix X, (28) is thereafter sufficient for (29).

7

C. Limitation of the Non-Incremental Control

The local voltage control (12b) is non-incremental, as itdecides the total amount of reactive power (instead of thechange in reactive power) based on the deviation of currentvoltage from the nominal value. Intuitively, such a control maylead to over-actuation and oscillatory behavior. In order to haveconverging or stable behavior, the control function should notbe too aggressive, i.e., have small (absolute) derivative. Thiscan also be seen from Theorem 3, and in the case of the piece-wise linear control function (11), implies a small αi value.

On the other hand, seen from the equivalent objective(21), smaller cost functions Ci(qi) are preferred for bettervoltage regulation. However, a small cost function implieslarge derivative of the control function; see, e.g., the costfunction (18) that becomes smaller as αi takes larger value,as well as the numerical examples in Section V .

Hence, there is a contention or fundamental limitation forthe non-incremental control: control function with smallerderivative is preferred for convergence, while for better voltageregulation at the equilibrium control function with largerderivative is desired. This motivates us to seek new local volt-age control schemes that are not subject to such a limitation,as will be seen in the next section.

IV. Forward Engineering: Decoupling Equilibrium andDynamical Properties

The optimization-based model (20) does not only providea way to characterize the equilibrium of the local voltagecontrol (see Theorem 1), but also suggests a principled wayto engineer the control. New design goal such as fairnessand economic efficiency can be taken incorporated by engi-neering the objective function in (20), and more importantly,new control schemes with better dynamical properties canbe designed based on various optimization algorithms suchas the (sub)gradient algorithm. In this section, we applytwo iterative optimization algorithms to design local voltagecontrol schemes that can decouple the dynamical propertyfrom the equilibrium property and have less restrictive con-vergence conditions than the non-incremental local voltagecontrol studied in the previous section.

A. Local Voltage Control Based on the (Sub)gradient Algo-rithm

Given an optimization problem, we may apply differentalgorithms to solve it. A common algorithm that often admitsdistributed implementation is the (sub)gradient method [47].Applying it to the problem (20) leads to the following voltagecontrol:

qi(t + 1) =

[qi(t) − γ2

∂F(q(t))∂qi

]Ωi

, i ∈ N , (30)

where γ2 > 0 is the constant stepsize and the (sub)gradient iscalculated as follows:

∂F(q(t))∂qi

=

− f −1i (qi(t)) + vi(t) − vnom if qi(t) , 0,

vi(t) − vnom if qi(t) = 0 and−δ/2 ≤ vi(t) − vnom ≤ δi/2,

−δi/2 + vi(t) − vnom if qi(t) = 0 andvi(t) − vnom > δi/2,

δi/2 + vi(t) − vnom if qi(t) = 0 andvi(t) − vnom < −δi/2.

(31)The above control is incremental as the change in reactivepower (instead of the total reactive power) is based on thevoltage deviation from the nominal value. It is also distributed,since the decision at each bus i ∈ N depends only on its currentreactive power qi and voltage vi.

We thus obtain the following dynamical system D2:v(t) = Xq(t) + v

qi(t + 1) =

[qi(t) − γ2

∂F(q(t))∂qi

]Ωi

, i ∈ N .

(32a)

(32b)

The following result is immediate.

Theorem 4 Suppose Assumption 1 holds. Then there existsa unique equilibrium point for the dynamical system D2.Moreover, a point (v∗, q∗) is an equilibrium if and only if q∗ isthe unique optimal solution of problem (20) and v∗ = Xq∗ + v.

Since the feasible sets are bounded, we also have thebounded (sub)gradient of F(q) with some constant G > 0:

‖∇qF(q)‖2 ≤ G, ∀q ∈ Ω. (33)

Theorem 5 Suppose Assumption 1 hold. The dynamical sys-tem D2 converges as

lim supt→∞

t∑τ=1

F(q(τ)) − F(q∗)t

= γ22G2. (34)

Proof: We characterize the distance between q(t +1) andq∗ as:

‖q(t + 1) − q∗‖22≤ ‖q(t) − γ2∇qF(q(t)) − q∗‖22= ‖q(t) − q∗‖22 + γ2

2‖∇qF(q(t))‖22 − 2γ2(q(t) − q∗)>∇qF(q(t))

≤ ‖q(t) − q∗‖22 + γ22G2 − (F(q(t)) − F(q∗))

≤ ‖q(1) − q∗‖22 + tγ22G2 −

t∑τ=1

(F(q(τ)) − F(q∗)),

where the first inequality is due to non-expansiveness ofprojection operator, the second inequality is because of thedefinition of subgradient as well as the bounded gradient (33),and the last inequality is by repeating previous steps.

As ‖q(t + 1) − q∗‖22 ≥ 0, it follows that:

t∑τ=1

F(q(τ)) − F(q∗)t

≤ ‖q(1) − q∗‖22/t + γ22G2. (35)

When t → ∞, we have (34).Notice that for any control functions fi (that satisfies As-

sumptions 1–2), one can always find a small enough stepsizeγ2 such that D2 converges to a neighborhood of the (v∗, q∗) of

8

required accuracy on running average. Moreover, as shown in[2], when q∗ is not close to the non-differentiable point, D2converges exactly to the optimum. In contrast, the convergencecondition (24) for the non-incremental voltage controlD1 doesconstrain the allowable control functions fi. Therefore, D2permits better voltage regulation than D1; see the discussionat the end of Section III and the simulation results in Fig. 5

Nevertheless, the (sub)gradient nature of D2 may prevent itfrom converging to the exact optimal point. This could happenwhen the optimum is close to the non-differentiable point (q∗ =

0 in this case) where the value of subgradient (31) changesabruptly if δi , 0. Moreover, the (sub)gradient computation(31) requires computing the inverse of the control function fi,which can be computationally expensive for general controlfunctions, as well as tracking the value of vi with respectto deadband ±δi/2. These limitations motivate us to designanother incremental control scheme with better convergenceand lower implementation complexity.

B. Local Voltage Control Based on the Pseudo-Gradient Al-gorithm

The pseudo-gradient can provide a good search directionfor an optimization problem without requiring the objectivefunction to be differentiable; see, e.g., [48]. Applying it tothe problem (20) leads to the following incremental voltagecontrol at bus i ∈ N :

qi(t + 1) =[qi(t) − γ3

(qi(t) − fi(vi(t) − vnom

i ))]

Ωi, (36a)

=[(1 − γ3)qi(t) + γ3 fi(vi(t) − vnom

i )]Ωi

(36b)

where γ3 > 0 is a constant stepsize/weight and qi− fi(vi−vnomi )

is the pseudo-gradient. The above control is distributed, andis simpler to implement than the control (30).

With (36) we obtain the following dynamical system D3:v(t) = Xq(t) + v,

qi(t + 1) =[qi(t) − γ3

(qi(t) − fi

(vi(t) − vnom

i))]

Ωi,

i ∈ N .

(37a)

(37b)

Notice that D3 has the same equilibrium condition as thedynamical systems D1 and D2. The following result is im-mediate.

Theorem 6 Suppose Assumption 1 holds. There exists aunique equilibrium point for the dynamical system D3. More-over, a point (v∗, q∗) is an equilibrium if and only if q∗ is theunique optimal solution of problem (20) and v∗ = Xq∗ + v.

We now analyze the convergence of the dynamical systemD3. We first introduce the following useful results.

Denote by ∇v f the diagonal matrix with each entry(∇v f

)ii

representing the (sub)gradient defined as

(∇v f )ii

= f ′i (vi) if vi ∈ (−∞,−δi/2)∪

(−δi/2, δi/2) ∪ (δi/2,+∞)∈ [ f ′i (v−i ), f ′i (v+

i )] if vi = −δi/2∈ [ f ′i (v+

i ), f ′i (v−i )] if vi = δi/2

, (38)

which is bounded as −αi ≤ (∇v f )ii ≤ 0 based on As-sumptions 1–2. Denote by λ any eigenvalue of the matrix

∇v f X. Consider ∇v f X’s similar matrix X1/2∇v f X1/2, which issymmetric and negative semidefinite with real and nonpositiveeigenvalues. Therefore, eigenvalues of the original asymmetricmatrix ∇v f X are also real and nonpositive. Similarly, all theeigenvalues of AX are real and positive.

Theorem 7 Suppose Assumptions 1–2 hold. If the stepsize γ3satisfies the following condition:

0 < γ3 < 2/(1 + λmax(AX)

), (39)

then the dynamical system D3 converges to its unique equi-librium point.

Proof: Write D3 equivalently as a mapping g3:

q(t + 1) = g3(q(t)) :=[(1 − γ3)q(t) + γ3 f (v(q(t)))

]Ω. (40)

The Jacobian matrix of g3 without projection operator iscomputed as

∇qg3 = (1 − γ3)IN + γ3∇v f X, (41)

where IN is N × N identity matrix. On the other hand, whenprojection operator is active for some qi, the ith row of theresulting ∇qg3 is all 0. So, by Gershgorin circle theorem [49],the magnitude of ∇qg3’s eigenvalue without active projectionhas a larger bound than that with projection. Thus it issufficient to consider g3 without the projection operator in thisproof.

Denote by z the eigenvector of matrix ∇v f X correspondingto λ. By definition one has ∇v f Xz = λz. Therefore,

∇qg3z = (1 − γ3 + γ3λ)z, (42)

that is, the corresponding eigenvalue of ∇qg3 with respect toλ is 1 − γ3 + γ3λ. To ensure that g3 is stable, one must have[32], [50]

−1 < 1 − γ3 + γ3λ < 1 (43)

for any eigenvalue λ of ∇v f X. (39) is sufficient for the left-hand side of (43) and the right-hand side always holds becauseλ is nonpositive. This completes the proof.

We conclude that the voltage control D3 based on thepseudo-gradient algorithm converges to the optimum withproperly chosen stepsize, while the control D2 based on thesubgradient algorithm usually converges to within a smallneighborhood of the optimum on running average.

Remark 1 Notice that, when γ3 ≤ 1 in, the control (36) hasa nice interpretation of the new decision qi(t + 1) being a(positively-)weighted sum of the decision qi(t) at the previoustime and the local control ui(t) = fi(vi(t) − vnom

i ) in reactivepower. Similar approaches in literature are also identified asexponentially weighted moving average (EWMA) control anddelayed control, etc. However, here we do not require γ3 ≤ 1for D3 to converge, as long as the condition (39) is satisfied.

9

Network DataLine Data Line Data Line Data Load Data Load Data PV Generators

From To R X From To R X From To R X Bus Peak Bus Peak Bus CapacityBus. Bus. (Ω) (Ω) Bus. Bus. (Ω) (Ω) Bus. Bus. (Ω) (Ω) No. MVA No. MVA No. MW

1 2 0.259 0.808 8 34 0.244 0.046 18 19 0.198 0.046 11 0.67 28 0.272 3 0.031 0.092 8 36 0.107 0.031 22 26 0.046 0.015 12 0.45 29 0.2 2 13 4 0.046 0.092 8 30 0.076 0.015 22 23 0.107 0.031 13 0.89 31 0.27 26 23 13 0.092 0.031 8 9 0.031 0.031 23 24 0.107 0.031 15 0.07 33 0.45 29 1.83 14 0.214 0.046 9 10 0.015 0.015 24 25 0.061 0.015 16 0.67 34 1.34 31 2.54 17 0.336 0.061 9 37 0.153 0.046 27 28 0.046 0.015 18 0.45 35 0.13 12 34 5 0.107 0.183 10 11 0.107 0.076 28 29 0.031 0 19 1.23 36 0.675 21 0.061 0.015 10 41 0.229 0.122 30 31 0.076 0.015 20 0.45 37 0.135 6 0.015 0.031 11 42 0.031 0.015 30 32 0.076 0.046 21 0.2 39 0.456 22 0.168 0.061 11 12 0.076 0.046 30 33 0.107 0.015 23 0.13 40 0.26 7 0.031 0.046 14 16 0.046 0.015 37 38 0.061 0.015 24 0.13 41 0.457 27 0.076 0.015 14 15 0.107 0.015 38 39 0.061 0.015 25 0.2 Vbase = 12.35 KV7 8 0.015 0.015 17 18 0.122 0.092 38 40 0.061 0.015 26 0.07 S base = 1000 KVA8 35 0.046 0.015 17 20 0.214 0.046 27 0.13 Zbase = 152.52 Ω

TABLE INetwork Parameters of the SCE Circuit: Line impedances, peak spot load KVA, Capacitors and PV generation’s nameplate ratings.

Fig. 4. Circuit diagram for SCE distribution system.

V. Numerical examples

Consider a distribution feeder of South California Edison(SCE) with a high penetration of photovoltaic (PV) generation.As shown in Fig. 4, bus 1 is the substation (root bus) and fivePV generators are integrated at buses 2, 12, 26, 29, and 31. Aswe aim to study the Volt/VAR control through PV inverters,all shunt capacitors are assumed to be off. Table I containsthe network data including the line impedance, the peak MVAdemand of loads, and the capacity of the PV generators. Itis important to note that all studies are run with a full ACpower flow model with MATPOWER [51] instead of its linearapproximation. As will be seen, the results we develop for thelinearized model are corroborated numerically with the fullpower flow model.

In all numerical studies, we implement homogeneous piece-wise linear droop control functions (11) of the IEEE 1547.8Standard [8] for all PV inverters, with their deadbands from0.98 p.u. to 1.02 p.u. and slopes αi to be determined.

A. Equilibrium

As discussed in Section III-C, large (absolute) slopes ofthe control function lead to better voltage regulation at theequilibrium. To show this, we change αi from 1 to 200 andrecord the corresponding equilibrium voltages v∗. As shownin Fig. 5, v∗ gets closer to vnom as αi increases. This confirmsour previous discussion that steeper control functions are tobe implemented in order to achieve smaller voltage deviationfrom the nominal value.

1 26 40 80 120 160 200

i

0.955

0.96

0.965

0.97

0.975

0.98

0.985

Equili

brium

Voltage, p.u

.

Bus 2

Bus 12

Bus 26

Bus 29

Bus 31

Limit of D1

Fig. 5. Equilibrium voltage versus the αi value: As αi increases, theequilibrium voltage v∗i deviates less from the nominal value.

B. Dynamics

1) Convergence of Dynamical System D1: As shown inFig. 6, the dynamical system D1 displays less stable behavioras the control function become steeper with the increase ofαi value, till it ends up with oscillation when αi becomestoo large. See also the vertical dash line on Fig. 5 beyondwhich there is no convergence. As discussed in Section III-C, there is a contention between convergence and equilibriumperformance for the non-incremental voltage control (12b): Asmaller (absolute) slope is preferred for convergence, while alarger one is preferred for voltage regulation at the equilibrium.

2) Convergence of Dynamical Systems D2 and D3: Asdiscussed in Section IV, given any control function, D2 andD3 converge if small enough stepsizes are chosen, and we canthus decouple the equilibrium property from the dynamicalproperty. For instance, when αi = 27, the dynamical systemD1 does not converge; see Fig. 6(b). However, as shown inFig. 7, when the stepsizes γ2 and γ3 are properly chosen, thedynamical systems D2 and D3 converge.

3) Convergence at Non-Differentiable Point: As discussedin Section IV-A, the dynamical system D2 based on subgra-dient algorithm can only converge to within a small neigh-

10

10 20 30 40 50

iteration index

0.95

0.955

0.96

0.965

0.97

vo

lta

ge

, p

.u.

(a) i=1

10 20 30 40 50

iteration index

0.95

0.96

0.97

0.98

vo

lta

ge

, p

.u.

(b) i=9

10 20 30 40 50

iteration index

0.95

0.96

0.97

0.98

vo

lta

ge

, p

.u.

(c) i=18

10 20 30 40 50

iteration index

0.95

0.96

0.97

0.98

0.99

vo

lta

ge

, p

.u.

(d) i=27

Fig. 6. Evolution of voltage of the dynamical systemD1 with different slopesof the piecewise linear control function: Voltage does not converge when the(absolute) slope of the control function become too large (when αi > 26 inthis example).

10 20 30 40 50

iteration index

0.85

0.9

0.95

1

1.05

vo

lta

ge

, p

.u.

(a) D2:2=30

10 20 30 40 50

iteration index

0.95

0.96

0.97

0.98

0.99

vo

lta

ge

, p

.u.

(b) D2:2=23

10 20 30 40 50

iteration index

0.95

0.96

0.97

0.98

0.99

vo

lta

ge

, p

.u.

(c) D3:3=0.99

10 20 30 40 50

iteration index

0.95

0.96

0.97

0.98

0.99

vo

lta

ge

, p

.u.

(d) D3:3=0.9

Fig. 7. Evolution of voltage of the dynamical systems D2 and D3 withαi = 27: Convergence is ensured with small enough stepsizes.

borhood of the equilibrium if it is at a nonsmoooth pointof the objective function (19). We tune the parameters suchthat the equilibrium reactive power provisioned at certain bus— the bus 2 in this case — is close to zero. As shown inFig. 8, D2 eventually converges to a small region around theoptimum, even with very small stepsize. On the other hand, asshown in Fig. 9, the dynamical system D3 based on pseudo-gradient algorithm converges to the equilibrium despite thenon-smoothness of the objective function at the equilibrium.See the embedded windows in Fig. 8-9.

VI. ConclusionWe have investigated local voltage control with a general

class of control functions in distribution networks. We showthat the power system dynamics with non-incremental local

2000 4000 6000 8000 10000 12000 14000

Iteration Index

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Reactive P

ow

er

Bus 2

Bus 12

Bus 26

Bus 29

Bus 31

1.48 1.49 1.5

104

0

2

4

10-3

Fig. 8. Convergence of the dynamical system D2 to within a smallneighborhood of the equilibrium.

20 40 60 80 100 120 140

Iteration Index

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Reactive P

ow

er

Bus 2

Bus 12

Bus 26

Bus 29

Bus 31

140 145 150

0

2

4

10-3

Fig. 9. Convergence of the dynamical system D3 to the equilibrium.

voltage control can be seen as distributed algorithm for solvinga well-defined optimization problem (reverse engineering).The reverse engineering further reveals a limitation of thenon-incremental voltage control: the convergence conditionis restrictive and prevents better voltage regulation at equi-librium. This motivates us to design two incremental localvoltage control schemes based on different algorithms forsolving the same optimization problem (forward engineering).The new control schemes decouple the dynamical propertyfrom the equilibrium property, and have much less restrictiveconvergence conditions. This work presents another step to-wards developing a new foundation – network dynamics asoptimization algorithms – for distributed realtime control andoptimization of future power networks

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[37] L. Chen and S. You, “Reverse and forward engineering of frequencycontrol in power networks,” IEEE Trans. on Automatic Control, vol. 62,no. 9, pp. 4631–4638, 2017.

[38] N. Li, C. Zhao, and L. Chen, “Connecting automatic generation controland economic dispatch from an optimization view,” IEEE Trans. onControl of Network Systems, vol. 3, no. 3, pp. 254–264, 2016.

[39] C. Zhao, U. Topcu, and S. Low, “Swing dynamics as primal-dualalgorithm for optimal load control,” in Proceedings of IEEE SmartGrid-Comm, 2012.

[40] X. Zhang and A. Papachristodoulou, “A real-time control frameworkfor smart power networks with star topology,” Proceedings of AmericanControl Conference, 2013.

[41] C. Zhao, U. Topcu, N. Li, and S. Low, “Design and stability of load-side primary frequency control in power systems,” IEEE Transactionson Automatic Control, vol. 59, pp. 1177–1189, 2014.

[42] E. Mallada and S. H. Low, “Distributed frequency-preserving optimalload control,” Proceedings of the 19th IFAC World Congress, 2014.

[43] F. Dorfler, J. W. Simpson-Porco, and F. Bullo, “Breaking the hierarchy:Distributed control & economic optimality in microgrids,” IEEE Trans-actions on Control of Network Systems, vol. 3, no. 3, pp. 241–253, 2016.

[44] X. Zhou and L. Chen, “A new perspective to synchronization in networksof coupled oscillators: Reverse engineering and convex relaxation,”IFAC-PapersOnLine, vol. 48, no. 22, pp. 40–45, 2015.

[45] M. E. Baran and F. F. Wu, “Network reconfiguration in distributionsystems for loss reduction and load balancing,” IEEE Trans. on PowerDelivery, vol. 4, no. 2, pp. 1401–1407, 1989.

[46] Z. Liu, J. Shihadeh, S. You, G. Ding, X. Zhou, and L. Chen, “Signal-anticipating in local voltage control in distribution systems,” arXiv, 2018.

[47] S. Boyd and L. Vandenberghe, Convex Optimization. CambridgeUniversity Press, 2004.

[48] J. Wen, Q. Wu, D. Turner, S. Cheng, and J. Fitch, “Optimal coordinatedvoltage control for power system voltage stability,” IEEE Trans. onPower Systems, vol. 19, no. 2, pp. 1115–1122, 2004.

[49] R. S. Varga, Matrix iterative analysis. Springer Science & BusinessMedia, 2009, vol. 27.

[50] O. Galor, Discrete dynamical systems. Springer Science & BusinessMedia, 2007.

[51] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MAT-POWER: Steady-state operations, planning, and analysis tools for powersystems research and education,” IEEE Trans. on Power Systems, vol. 26,no. 1, pp. 12–19, 2011.

12

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[36] B. Zhang, A. D. Dominguez-Garcia, and D. Tse, “A local controlapproach to voltage regulation in distribution networks,” Proc. of IEEENorth American Power Symposium (NAPS), 2013, pp. 1–6, 2013.

[37] L. Chen and S. You, “Reverse and forward engineering of frequencycontrol in power networks,” IEEE Trans. on Automatic Control, vol. 62,no. 9, pp. 4631–4638, 2017.

[38] N. Li, C. Zhao, and L. Chen, “Connecting automatic generation controland economic dispatch from an optimization view,” IEEE Trans. onControl of Network Systems, vol. 3, no. 3, pp. 254–264, 2016.

[39] C. Zhao, U. Topcu, and S. Low, “Swing dynamics as primal-dualalgorithm for optimal load control,” in Proceedings of IEEE SmartGrid-Comm, 2012.

[40] X. Zhang and A. Papachristodoulou, “A real-time control frameworkfor smart power networks with star topology,” Proceedings of AmericanControl Conference, 2013.

[41] C. Zhao, U. Topcu, N. Li, and S. Low, “Design and stability of load-side primary frequency control in power systems,” IEEE Transactionson Automatic Control, vol. 59, pp. 1177–1189, 2014.

[42] E. Mallada and S. H. Low, “Distributed frequency-preserving optimalload control,” Proceedings of the 19th IFAC World Congress, 2014.

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[47] S. Boyd and L. Vandenberghe, Convex Optimization. CambridgeUniversity Press, 2004.

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[49] R. S. Varga, Matrix iterative analysis. Springer Science & BusinessMedia, 2009, vol. 27.

[50] O. Galor, Discrete dynamical systems. Springer Science & BusinessMedia, 2007.

13

[51] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MAT-POWER: Steady-state operations, planning, and analysis tools for powersystems research and education,” IEEE Trans. on Power Systems, vol. 26,no. 1, pp. 12–19, 2011.

PLACEPHOTOHERE

Xinyang Zhou received the B.S. degree in Engi-neering from Zhejiang University, China, in 2012,and the Ph.D. degree in Telecommunications fromthe University of Colorado, Boulder, CO, USA, in2018. He is currently a Postdoctoral Researcherwith the National Renewable Energy Laboratory,Golden, CO, USA. His current research interest isin distributed optimization and control for powersystems.

PLACEPHOTOHERE

Masoud Farivar is currently a research scientistat Google, Mountain View. He received his Ph.D.in Electrical Engineering from California Instituteof Technology in 2015, and a dual BSc. degree inElectrical Engineering and Computer Science fromSharif University of Technology in 2009. Masoudwas a recipient of Amgen postdoctoral fellowship,and Resnick Institute fellowship for sustainabil-ity. His research interest broadly includes MachineLearning, Optimization, and Control.

PLACEPHOTOHERE

Ziyuan Liu received a B.S. in Computer Scienceand Technology from University of Science andTechnology of China (USTC), China in 2015. Heis currently a PhD student in Computer Science atthe University of Colorado at Boulder. His currentresearch interest is in distributed optimization andcontrol for networked systems.

PLACEPHOTOHERE

Lijun Chen (M’05) is an Assistant Professor ofComputer Science at University of Colorado at Boul-der. He received a Ph.D. in Control and DynamicalSystems from California Institute of Technology in2007. He was a co-recipient of the Best Paper Awardat the IEEE International Conference on MobileAd-hoc and Sensor Systems (MASS) in 2007. Hiscurrent research interests include optimization andcontrol of networked systems, distributed optimiza-tion and control, convex relaxation and parsimonioussolutions, and game theory and its engineering ap-

plication. He is an editor for the IEEE Transactions on Communications andeditor for the IEEE Control Systems Society Conference Editorial Board. Heserves as the chair of the Special Interest Group in ICT-Enabled TransactiveEnergy and Grid Economics of IEEE ComSoc Smart Grid CommunicationsEmerging Technical Subcommittee.

PLACEPHOTOHERE

Steven H. Low (F’08) is a Professor of the Depart-ment of Computing & Mathematical Sciences andthe Department of Electrical Engineering at Caltech.Before that, he was with AT&T Bell Laboratories,Murray Hill, NJ, and the University of Melbourne,Australia. He was a co-recipient of IEEE best paperawards and is an IEEE Fellow. His research oncommunication networks has been accelerating morethan 1TB of Internet traffic every second since 2014.He was a member of the Networking and Informa-tion Technology Technical Advisory Group for the

US President’s Council of Advisors on Science and Technology (PCAST) in2006. He received his B.S. from Cornell and PhD from Berkeley, both in EE.