1

Real-Time Local Volt/VAR Control Under ExternalDisturbances with High PV Penetration

A. Singhal, Member, IEEE, V. Ajjarapu, Fellow, IEEE, J.C. Fuller, Senior Member, IEEE,and J. Hansen, Member, IEEE

Abstract—Volt/var control (VVC) of smart PV inverter isbecoming one of the most popular solutions to address thevoltage challenges associated with high PV penetration. Thiswork focuses on the slope sensitive local droop VVC recom-mended by integration standard IEEE1547, rule21 and addressestheir major challenges i.e. vulnerability to instability (or voltageoscillations), significant steady state error (SSE) and appropriateparameter selection under external disturbances. This is achievedby proposing an autonomous, local and adaptive VVC which hastwo major features i.e. a) it is able to ensure both low SSEand control stability without compromising either and; b) itdynamically adapts its parameters to ensure good performancein a wide range of external disturbances such as sudden cloudcover, cloud intermittency, substation voltage change. Moreover,the adaptive control does not depend on the feeder topologyinformation, thus also shown to be adaptive to the error in feedertopology information. The proposed control is implementationfriendly as it fits well within the integration standard frameworkand depends only on local bus information. The performanceis compared with the existing droop VVC methods on anunbalanced 3-phase IEEE 123 test system with detailed secondaryside modeling.

Index Terms—solar energy, power distribution systems, pho-tovoltaic system, smart grid, volt/var control, smart invertercontrol, real-time control, distributed control.

I. INTRODUCTION

SOLAR photovoltaic (PV) penetration is continuously ris-ing, and is expected to be tripled in the next 5 years

in the USA [1]. High PV penetration is being fueled by thefavorable policies and significant cost reductions, nonetheless,it brings its own set of technical challenges such as voltagerise and rapid voltage fluctuations due to cloud transientswhich could lead to the reduced power quality [2], [3].Under the premise that solar will keep growing as expected,several efforts are required to address the associated voltagechallenges. To mitigate these challenges, a few simple methodsare suggested such as lowering substation voltage, increasingconductor diameter and curtailing solar generation. However,these methods are not economical and not adaptive to changingoperating conditions (low solar output during peak load) [2],[4]. In traditional volt/var control (VVC), voltage regulatingdevices such as capacitors and load tap changers (LTC) are

This work was supported in part by U.S. Department of Energys SunshotInitiative Program DE-0006341.

A. Singhal and V. Ajjarapu are with Department of Electric and Com-puter Engineering, Iowa State University, Ames, IA, 50010 USA (e-mail:ankit@iastate.edu, vajjarap@iastate.edu).

J. C. Fuller, J. Hensen are with Pacific Northwest National Lab-oratory, Richland, WA, 99453 USA (e-mail: jason.fuller@pnnl.gov, ja-cob.hansen@pnnl.gov).

supposed to maintain the feeder voltage but they are notfast enough to handle transient nature of solar generation i.e.cloud cover [5]–[7]. Therefore, PV inverters have emerged aseffective volt/var controllers to handle rapid variations in themodern distribution system by providing faster and continuousVVC capability in contrast to slower and discrete response ofthe traditional VVC devices [8]–[10].

The PV inverter VVC methods primarily fall into twobroad categories: 1) optimal power flow (OPF) based (de-)centralized approaches and 2) local control approaches. Mostof the literature deals with the OPF based methods whichare solved either in a centralized manner [5], [11], [12] orin a decentralized way using distributed algorithms [6], [13]–[16]. However, the extensive communication requirementsamong the PV devices challenge the real-time implementationof these methods. Additionally, communication delays andthe large time requirement to solve most OPFs limit theirability to respond to faster disturbances at seconds time scalesuch as cloud intermittency [8], [10], [17].Though distributedalgorithms are relatively faster, most of these methods assumeconstant substation voltage and rely on full feeder topologyinformation for control parameter selection which is usuallynot fully known to the utilities or not always reliable. Theseissues make (de-)centralized VVC methods difficult to imple-ment and also vulnerable to fast external disturbances suchas cloud transients, sudden change in substation voltage andtopology changes. To avoid these challenges, we focus on thelocal VVC approaches in this work which are usually faster,implementation friendly, and can respond to sudden externaldisturbances in distribution systems.

Among local approaches, droop VVC is the most popularlocal control framework among utilities and in the existingliterature. It was first proposed by [18] which now has beenadopted by the IEEE1547 integration standard [19] and alsobeing widely used by Rule 21 in California [20]. Nonetheless,some of the attempts to develop non-droop local VVC methodsare also worth noting [10], [17], [21]. For instance, a scaled varcontrol proposed by [17] provides stability analysis of the con-trol and demonstrates an improved local VVC performance.However, these methods require full topology information forparameter selection and do not adapt themselves to changingoperating conditions and disturbances. Moreover, they are notcompatible with the IEEE1547 standard local droop VVCframework which may jeopardize their real-time implemen-tation. Therefore, to be implementation friendly, we focuson developing standard droop compatible adaptive control.However, the droop control is highly sensitive to its droop

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(slope) parameter and the existing standards do not provideguidelines on the parameter selection. Further, it has beenshown by [22], [23] that the droop control is vulnerable toinstability and voltage flicker due to improper selection of theparameters. The desired slope to ensure stability depends onthe feeder topology and operating conditions. On the otherhand, this slope selection may adversely affect the steady stateperformance of the control which leads to high steady stateerror (SSE) as indicated by [22], and the local droop VVCwith high SSE is always prone to voltage violations in externaldisturbances as confirmed by our findings in this work. Thedelayed droop control, a variation of the conventional droopcontrol, is proposed by [23] which works well and improvesthe stability performance compared to the conventional droopunder normal operating conditions; however, under externaldisturbances, it is vulnerable to instability and violationsdue to lack of real-time parameter adaption and high SSErespectively, as detailed soon in Section II. Thus, to ensureboth control stability and set-point tracking accuracy (lowSSE), we propose a fully local and real-time adaptive VVCwhere control parameters are made self-adaptive to inaccuratetopology information and commonly occurring external dis-turbances such as cloud intermittency, cloud cover, changingload profile, and substation voltage changes.

Our work extends the previous works and provides uniquecontributions in following way: 1) The proposed controlachieves both set-point tracking accuracy (low SSE) and con-trol stability simultaneously without compromising either; 2)The proposed approach makes the control parameter selectionself-adaptive to changing operating conditions and externaldisturbances; 3) It is purely local in nature with no requirementof the additional communication links and also compatiblewith the existing onboard droop controls specified in the recentstandards (IEEE1547, Rule 21); 4) The real-time adaptivenature and tight voltage control (low SSE) feature of theproposed control opens interesting opportunities for operatorsto utilize PV inverters not only to mitigate over-voltage butalso for other volt/var related applications such as CVR, lossminimization, providing var support to transmission side etc.;and 5) A detailed modeling of secondary side of an unbalanceddistribution system is used to verify the control approach withhouse-level loads and heterogeneous inverter population.

Overall, in this new environment of increasing renewables,demand response and other pro-active functionalities, theunexpected external disturbances in distribution system willbecome more common; and the standard based proposed adap-tive local VVC framework is will facilitate an easy plug-and-play implementation without reliance on much communicationnetwork, which is the main motivation behind this work.

The layout of the remainder of the paper is as follows. InSection II, stability conditions and SSE expressions of thedroop control are derived and discussed to establish the basefor the adaptive control development. Based on the analysis,the adaptive control strategy is proposed in Section III. Sim-ulation results on the test system are discussed in Section IV.Finally, concluding remarks are presented in Section V.

𝝁𝒗𝒎𝒊𝒏

𝒗𝒎𝒂𝒙

𝒒𝒎𝒂𝒙

𝒒𝒎𝒊𝒏

𝑽𝒐𝒍𝒕𝒂𝒈𝒆 (𝒑𝒖)

𝑽𝑨𝑹(𝒑𝒖)

𝑃𝑟𝑜𝑣𝑖𝑑𝑖𝑛𝑔 𝑉𝐴𝑅

𝐴𝑏𝑠𝑜𝑟𝑏𝑖𝑛𝑔 𝑉𝐴𝑅

−𝒎𝒔𝒍𝒐𝒑𝒆

𝒅

Fig. 1. Conventional droop VVC framework recommended by IEEE 1547.8

II. BACKGROUND AND PROBLEM SETUP

Consider a general N + 1 bus distribution system with onesubstation bus and N load buses with PV inverters. The powerflow equations for the system can be written as

P inv − Pd = gp(V, δ)

Qinv −Qd = gq(V, δ)(1)

Where Q = [Q2 Q3 . . . QN+1]T and P inv =[P2 P3 . . . PN+1]T are inverter reactive and real powerinjection vectors respectively at each bus. Pd and Qd aresimilar vectors of real and reactive power loads at each bus.gp and gq are well-known power flow equations with voltagemagnitude and angles as variables at all load buses [24].

The standard droop function fi(.) at ith bus is shown inFig.1. It is a piecewise linear function with a deadband d andslope m. Assuming the operating point is in non-saturationregion, the inverter var dispatch at time t can be written as afunction of previous voltage and other control parameters as

Qi,t+1 = fi(Vi,t) = −mi(Vi,t − µi ± d/2) (2)

where, Qi,t and Vi,t are the inverter var injection and thevoltage magnitude respectively. Subscripts i and t denote ith

bus and time instant t. µi is the reference voltage and mi isthe slope of the curve. We consider the same slope for boththe regions in the droop control for a given inverter as shownin Fig.1. m can be maintained at desired value by changingcontrol parameters as

mi = qi,max/(µi−d/2−vi,min) = qi,min/(µi+d/2−vi,max)

where vi,min, vi,max, qi,min, qi,max are the four control set-points. It should be noted that in existing droop methodsthese parameters are either constant or un-controlled. Whereas,in this work, these parameters are dispatched based on theproposed adaptive control strategy. As detailed soon, dynamiccontrol over these parameters leads to more reliable controlperformance compared to previous works.

A. Stability Analysis

As described in [22], [23], the local droop VVC can bemodeled as feedback dynamical system φ with N states[Q2,t Q3,t . . . QN+1,t]

T at discrete time t.

Qt+1 = φ(Qt) = f(h(Qt)) (3)

Where the vector f(.) = [f2 f3 . . . fN+1] contains localVVC functions which map the current voltage vector Vt to newinverter var injections vector Qt+1 i.e. Qi,t+1 = fi(Vi,t). The

3

new var vector Qt+1, in turn, leads to the new voltage vectorVt+1 according to power flow equations (1). The function his an implicit function vector derived from (1) i.e. hi(Qi,t) =Vi,t. It is shown in [23] that the system φ is locally stable inthe vicinity of an equilibrium point (Q̄) if all eigenvalues ofthe matrix ∂φ/∂Q have magnitude less than 1.[

∂φ

∂Q

]Q=Q̄

=

[∂f

∂V

] [∂V

∂Q

](4)

In the case of droop control, ∂f/∂V is a diagonal matrix withslope at each inverter as diagonal entries.[

∂f

∂V

]= M = −diag(mi) = −

m2 · · · 0...

. . ....

0 · · · mN+1

(5)

Lets define A = ∂V/∂Q and aij = ∂Vi/∂Qj which is avoltage sensitivity matrix with respect to var injection and canbe extracted from the power flow Jacobian matrix from (1) asshown in [23].

[∆V ] = [A] [∆Q] =

a22 · · · a2,N+1

.... . .

...aN+1,2 · · · aN+1,N+1

[∆Q] (6)

Now, the sufficient condition for the control to be stable canbe written as

ρ(MA) < 1 (7)

Where ρ is spectral radius of a matrix which is defined asthe largest absolute value of its eigenvalues. Condition (7)provides useful information for evaluating the stability ofspecific inverter slope settings. However, in order to obtaininformation for selecting the inverter slopes, we will deriveanother conservative sufficient condition for stability usingspectral radius upper bound theorem [25].

Theorem 1: Let ‖.‖ be any matrix norm on Rn×n and let ρbe the spectral radius of a matrix, then for all X ∈ Rn×n:

ρ(X) ≤ ||X|| (8)

Proposition: If sum of each row of MA is less than 1, i.e.

mi.

N∑j=1

|aij | < 1 ∀i, (9)

Then the droop control will be stable i.e. ρ(MA) < 1Proof: Using Theorem 1, if we apply ‖.‖∞ on MA,

then, ρ(MA) ≤ ‖MA‖∞ = max1≤i≤N∑N

j=1 |mi.aij |. Bycondition 9, if mi.

∑Nj=1 |aij | < 1 ∀i, then the maximum of

sum of rows will also be less than one. Thus, the upper boundon spectral radius will always be less than one.

Remark 1: The condition (9) provides useful informationto select slope for each inverter to ensure control stability,i.e. mi < mc

i , where mci is critical slope given by mc

i =(∑

j |aij |)−1. It should be noted that, usually, the entries ofthe sensitivity matrix A do not remain constant. Changes inoperating conditions (cloud cover, load changes) as well aschanges in feeder topology lead to change in values of aijand mc

i ; thus can potentially cause instability if mi are notupdated dynamically. Intuitively, entries of A can also be seen

as proportional to reactance of feeder lines [22] i.e. longerlines are more likely to have higher magnitude of aij andlower value of critical slope. Therefore, PV inverters on ruralnetwork with long lines, especially towards the feeder end,will be more sensitive to instability and their slope selectionshould be more conservative. Therefore, non-adaptive and ho-mogeneous slope selection for all inverters make system proneto control instability. Also, the un-controlled change in qmin

and qmax with change in solar generation leads to undesiredslope. For instance, in case of cloud cover, the generation dropsand the qmax limit will be increased automatically leadingto very high slope exactly when var support is not neededwhich creates stability/flicker issues. It is worth mentioningthat the attempt to lower the effective slope by adding a delayblock after droop in delayed droop [23] improves the stabilitycompared to conventional droop i.e. Qt+1 = fi(Vi,t) + τ.Qt,where τ is a delay coefficient. However, because of its non-adaptive nature and un-controlled parameters, it leads to issuesunder external disturbances and topology changes which willbe illustrated through a comparison later in the section.

B. Steady State Error (SSE) Concerns

One of the major drawbacks of the droop control is asignificant deviation from the set-point in steady state. Toderive the analytical expression for SSE, lets assume thesystem is in equilibrium point (Q,V ) at t = 0. Controlequation (2) can be written in vector form at t = 0, as

[Q] = [M ][V − µ] (10)

Now, consider an external disturbance perturb the equilibriumby causing sudden change in voltage, ∆V d, at t = 0.

[Qt=1 −Q] = [∆Q]t=1 = [M ]∆V d (11)

Now (2) can be rewritten as following for t > 0

[∆Q]t+1 = [M ][∆V ]t (12)

Where [∆Q]t+1 = [Qt+1 − Qt] and [∆V ]t = [Vt − Vt−1].Using (7) and (13), we can write

[∆V ]t+1 = [A][M ][∆V ]t (13)

[∆V ]t+1 = [A.M ]t[∆V ]t=1 (14)

Using (7), (12) and (15),

[V ]t+1 = [V ]t + [A.M ]t+1∆V d (15)

[V ]t+1 = [V ]t=0 +

t+1∑i

[A.M ]i∆V d (16)

In this case, the geometric progression series of matrices onlyconverges if the condition (7) holds true (the stable case). Dueto disturbance, initial voltage changes by ∆V d i.e. [V ]t=0 =V + ∆V d. The new equilibrium voltage can be written as

limt→∞

[V ]t+1 = V + ∆V d + [I −A.M ]−1∆V d (17)

SSE vector can be written as

SSE = limt→∞

[V ]t+1 − µ (18)

4

Solar

Load (600 kW)

(1 MW)1 2 3

Solar

Load

1 2 3Solar

Load

4switch1

Fig. 2. A small 4 bus example to demonstrate the impact of feederreconfiguration and other external disturbances

20 40 60 80 100 120(a)

1

1.04

1.08

V3 (

pu)

conventional droop

delayed droop

20 40 60 80 100 120(b)0.9

1

1.1

V3 (

pu) delayed droop

20 40 60 80 100 120(c)

control iterations (seconds)

1

1.05

1.1

V3 (

pu) Switch1 closed Switch1 open

sudden cloud cover:

generation drop

upper voltage limit

substation voltage change

SSE

Fig. 3. Impact of external disturbances on droop VVC under different slopesettings: a) impact of change in substation voltage on conventional and delayedVVC at conservative slope settings; b) Impact of sudden cloud cover and; c)topology change on delayed droop VVC at non-conservative slope setting.

Equation (18) shows that, for a given disturbance, the only wayto decrease SSE is to set higher values of slopes mi whichin turn might violate stability condition (9). In fact, in mostcases, its a compromise between achieving acceptable low SSEand control stability in droop VVC. It can also be shown thatthe delayed droop [23] does not improve SSE compared toconventional control, though it improves the stability.

Remark 2: Note that it might be possible to maintainvoltages within the ANSI range with high SSE, close toboundaries, for a given system conditions. But, any unexpectedexternal disturbance can instantly push the voltages out oflimits as illustrated later. Moreover, the tight voltage regulationcapability with low SSE is not only desired just to maintainvoltages within the ANSI range, rather it makes the systemmore flexible and provides enough room to the operator toperform other voltage-dependent applications such as CVR,loss minimization etc.; thus fully utilizing the PV inverterscapability.

C. Illustration

To corroborate the above analysis, we will illustrate theimpact of external disturbances using a small modified IEEE4 bus test system shown in Fig.2. 600 kW load and 900 kWsolar generation is added at node 3. A similar node 4 is addedvia a normally open switch to simulate the change in feedertopology. We will consider two types of initial slope settingsto convey the main outcome of the analysis i.e. conservative(m = 2) and non-conservative (m = 6). Solar generationis applied at t = 20 to observe the impact of VVC withµ = 1 at node 3 voltage profile. Fig.3 (a) demonstrates howconservative settings cause high SSE (though, within the ANSIlimit initially) for the droop controls (both conventional and

droop) which leads to over-voltage violation due to a smallchange in substation voltage from 1.03-1.05 pu at t = 60.On the other hand, using non-conservative settings to reduceSSE makes the system prone to control instability or voltageflicker as shown in Fig.3(b) and (c). Conventional droop isnot shown as it is always unstable in this case. Fig.3(b) showsthat sudden drop in solar generation due to cloud cover att = 60 increases qmax and makes the slope very high whichcauses voltage oscillations. Further, to simulate the impact oftopology change or error in topology information, switch1 isclosed at t = 60. Delayed droop, as discussed before, is stableunder normal conditions, however, change in feeder topologyleads to voltage oscillations as shown in Fig.3(c) at non-conservative settings. This example demonstrates it is difficultto achieve both low SSE and control stability under externaldisturbances with existing droop controllers. Moreover, thisproblem becomes more crucial in a large realistic system dueto thousands of independent inverter devices, dynamic natureof generation and loads and higher possibility of inaccuracyin feeder topology information and in parameter selection.

Therefore, our intention is to develop a new droop basedadaptive VVC strategy 1) to achieve both low SSE andlow voltage oscillations (stability) simultaneously; 2) to makecontrol parameters dynamically self-adaptive to external dis-turbances and inaccurate feeder information in real-time ;and 3) to keep VVC purely local and compatible to VVCframework recommended by recent utility and IEEE standards.

III. ADAPTIVE CONTROL STRATEGY

This section will introduce the proposed adaptive local VVCfunction fpi (Vi,t) which can be written as follows:

Qi,t+1 = fpi (Vi,t) = P[qpi −mpi (Vi,t − µi)] (19)

Where P is the saturation operator with (qpmin,i, qpmax,i) as

saturation var limit parameters applied at cut-off parameters(vpmax,i, v

pmin,i). qpi is error adaptive parameter and its main

function is to provide SSE correction. Desired adaptive slopemp

i can be set as,

mpi =

qpmin,i − qpi

µi + d/2− vpmax,i

=qpmax,i − q

pi

µi − d/2− vpmin,i

(20)

There are two unique features of this control. First, thefunctions of maintaining control stability and low SEE aredecoupled. Two different parameters mp and qp are used toachieve control stability and low SSE respectively with differ-ent approaches so that none of the objectives are compromised.Second, all these parameters are dynamically adapted in real-time. Superscript p denotes the adaptive nature of the controlparameters. To achieve this, a two-layer control framework isproposed as shown in Fig.4. The inner layer is a fast VVCfunction fpi (Vi,t) to track the desired set-point µ accordingto (19). The outer layer dispatches the control parameters(mp

i , qpi ) based on the proposed adaptive algorithm described

later in the section. The outer layer works on a relativelyslower time scale (to) to allow inner fast control to reachsteady state before dispatching new control parameters, thusavoiding hunting and over-corrections. Control time-line isshown in Fig.5. Control parameters are updated at every period

5

𝑄𝑖,𝑡+1 = 𝑓𝑖𝑝𝑉𝑖,𝑡 , 𝑞𝑖,𝑡𝑜

𝑝, 𝑚𝑖,𝑡𝑜

𝑝

Voltage profile of

last 𝑇 minutes

𝑉𝑖,𝑡

𝑉𝑚𝑒𝑎𝑠

Adaptive

control algorithm

Power

Circuit

Updated control

parameters 𝑚𝑖𝑝, 𝑞𝑖

𝑝

Outer loop: 𝑇 = ~ 1 minutes

Inner loop : 𝑡 = 1 second

𝑄𝑖,𝑡+1

Dis

trib

uti

on

Sys

tem

Net

wo

rk

Droop settings

𝑣𝑚𝑖𝑛𝑝

, 𝑣𝑚𝑎𝑥𝑝

,𝑚𝑝

𝑞𝑚𝑖𝑛𝑝

, 𝑞𝑚𝑎𝑥𝑝

, 𝑞𝑝

Inverter

μ 𝑉𝑖𝑄𝑖

𝑞𝑝

𝑞𝑚𝑎𝑥𝑝

𝑞𝑚𝑖𝑛𝑝

−𝑚

𝑣𝑚𝑖𝑛𝑝

𝑣𝑚𝑖𝑛𝑝

Fig. 4. Two-layer framework of the proposed adaptive control approach

Inner control loop Inner control loop

Outer adaptive control loop

𝑡𝑜 = 0

Outer adaptive control loop

𝑡𝑜 = 1

Outer adaptive control loop

𝑡𝑜 = ℎ

𝑡𝑖𝑛 = 1,2…𝑇 𝑡𝑖𝑛 = 1,2…𝑇

𝑡 = (𝑡𝑜+1) . 𝑡𝑖𝑛

Inner loops

𝑡𝑜 = 0 𝑡𝑜 = 1 Outer loops𝑡𝑜 = ℎ

𝑡𝑖𝑛 = 1,2…𝑇 𝑡𝑖𝑛 = 1,2…𝑇

𝑡 = (𝑡𝑜+1) . 𝑡𝑖𝑛

Fig. 5. Time-line of adaptive inner and outer loop control

T , control horizon of the outer loop control. Each iterationof the inner and outer loop control is denoted by tin andto respectively. The outer loop adaptive algorithm consists oftwo strategies where qpi and mp

i are dynamically adapted totake care of SSE and voltage instability/flicker respectively asdescribed below.

A. Error Adaptive Control: Strategy I

The aim of the strategy I is to minimize SSE by utilizing varresources efficiently. Therefore, set-point deviation (SSEavg)is used as the control criteria and defined for each outer loopas

SSEavg,i(t0) =

T∑tin=1

(Vt0tin,i − µi)/T (21)

SSEavg,i denotes the average set-point deviation of voltage atith inverter bus. A tolerance band for SSEavg,i can be definedas µi ± εsse, where εsse is tolerance for the deviation.

In this strategy, the adaptive term qpi (to) in (20) is updatedat each outer loop interval to, based on SSEavg,i during thelast time horizon T as

qpi (to) = qpi (to − 1)− kdi .SSEavg,i(to) (22)

It is important to note that SSEavg is used as an algebraicvalue with sign rather than its absolute magnitude. The signof the error decides whether qpi needs to be moved positiveor negative. If the voltage settles on a higher value than theset point, a negative term is added in qpi to facilitate more varabsorption to lower the voltage. Similarly, a positive term isadded in qpi to provide more var when voltage settles lowerthan the set point. A constant kdi > 0 is a correction factor. Ahigher kdi brings SSEavg within the desired range faster andvice versa.

Fig.6 depicts the adaptive control fpi (Vi,t) with differentqpi values. It should be noted that the solid curve with qpi =0 is same as the conventional droop control fi(Vi,t) in (2).Fig.6 brings out an important feature of the proposed controlfpi (Vi,t) that it can be seen as shifted and adaptive droop VVCwhich makes it compatible with integration standards.

VA

R (

pu

)

Fig. 6. Adaptive VVC with different error adaptive parameter qp

While it is possible to defer real power solar generation,in this work the consumer value is maximized by limitingvar output to leftover capacity and not deferring real powergeneration. To utilize the inverter capacity entirely, qpmax andqpmin are also updated in every outer loop as

qpmax(to)=√s2−p2

pv(to); qpmin(to)=−√s2−p2

pv(to) (23)

where, s is inverter rating and ppv(to) is the average solar PVreal power generation in the last outer loop time interval.

B. SSE correction in Adaptive Control

In this section, we will verify how the proposed control(19) helps to mitigate the SSE. Consider the system is atequilibrium point (Q,V ) at t = 0 with SSE = V −µ. Now ifthe parameter qp is changed at t = 0 by ∆qp, the new voltagedeviation (SSEadp) can readily be obtained by followingthe procedure provided in the Section II.B by replacing theconventional droop (2) with the adaptive control (19):

SSEadp = V − µ+ [I −AM ]−1A∆qp (24)

To achieve SSEadp = 0, ∆qp required will be,

∆qpreq = −(A−1 −M)SSE (25)

Equation (25) provides the analytical expression of therequired change in qp parameter to achieve zero SSE in justone iteration. However, the solution requires the information ofA matrix, SSE and slope (M) at all inverter buses which is notavailable to local bus controllers. Moreover, estimation of A iscontingent to error in centralized feeder topology informationand might not be reliable. Interestingly, our proposed localupdate strategy ∆qpi (to) = −kdi .SSEavg,i(to) is local versionof the analytical solution (25) and is able to correct SSE,though, it may take more than one iterations to achieve nearzero SSE. The important part is that it requires only localbus information, making it purely local and more feasible.Value of kdi can be decided once from the offline studies.Nevertheless, the update strategy can always be made fasterand more accurate using (25), if information at other nodes isalso available in future.

C. Adaptive Slope Control: Strategy II

The objectives of the strategy II are to ensure stability aswell as to keep voltage fluctuations within the IEEE 141standard limit [26] by adapting parameter mi. Therefore,voltage flicker (VF) is used as the control criteria whichis defined as the voltage fluctuations in loads which causeirritation to user eyes [26]. Based on [26], we define the short-term flicker calculation for each inverter bus at the beginning

6

Critical Zone Decrease 𝑚 by Δ𝑣𝑓

Subcritical Zone Decrease 𝑚 by Δ𝑣𝑓 < Δ𝑣𝑓

Safe Zone No control action

Relaxed Zone Increase 𝑚 only if 𝑆𝑆𝐸𝑎𝑣𝑔is violated

𝑉𝐹𝑙𝑖𝑚 − 𝜖𝑣𝑓

𝑉𝐹𝑙𝑖𝑚

𝑉𝐹𝑙𝑖𝑚

𝑽𝑭

Fig. 7. Control action region for adaptive outer loop control strategy II forflicker mitigation

𝒗𝒎𝒊𝒏𝟐 𝒗𝒎𝒊𝒏𝟏

𝒗𝒎𝒂𝒙𝟐𝒗𝒎𝒂𝒙𝟏

𝝁𝒗𝒎𝒊𝒏

𝒗𝒎𝒂𝒙

𝒒𝒎𝒂𝒙

𝒒𝒎𝒊𝒏

Voltage (pu)VAR(pu)

Adaptive control strategy 1: adaptive slope

𝑚2 < 𝑚0 < 𝑚1

𝑚2

𝑚0

𝑚1

Fig. 8. Adaptive strategy II: changing slope of droop curve by changing vmin

and vmax parameters to keep flicker in the limit

of each outer loop as

V F (to) =

T∑ti=1

(Vtoti − Vtoti−1)/Vt0ti

T× 100 (26)

As seen in (15), voltage fluctuations are proportional toslope and can be reduced by decreasing mi. For this purpose,the voltage flicker range is divided into four control regions asshown in Fig.7. The IEEE standard 141 flicker curve providesthe maximum fluctuation limit (V Flim) beyond which wedefine as critical flicker zone. The same standard also gives aborderline flicker limit (V Flim). The region between (V Flim)and V Flim is termed as the subcritical flicker zone. Furtherwe define a tolerance (V Flim − εvf ) and the tolerance bandis termed as the safe flicker zone. The region below safeflicker zone is defined as the relaxed flicker zone. In criticalzone, we update the parameters by a larger amount (∆vf ) toavoid control instability and to return to subcritical zone faster.In subcritical zone, the slope is decreased in a smaller step(∆vf ) to avoid over-correction which might impact SSEavg

negatively. As soon as we reach the safe zone, no controlaction is taken. This is the desired range of control parameters.Though rarely required, in the relaxed zone, slope is increasedto improve SSE only if SSE is out of range. Correction factors(∆vf ) are estimated offline in this work, though they canalways be made responsive to the online control performance,if required.

It is worth noting here that the the main feature of the pro-posed control lies in the decoupling of the two functionalitiesi.e. SSE and slope. since SSE is catered by qpi , slope canalways be in the conservative range (safe or relaxed zones)to ensure control stability. In this work, we use the earlierderived condition (9) to choose initial slopes. It is estimatedusing offline studies for the base case, however, to keep safemargin it can be further reduced by a certain factor. Fig.8.depicts the control strategy II with adaptive mi.

Thus, we get the new parameters qpi (to), qpmin,i(to),

qpmax,i(to) from strategy I and mpi (to) from strategy II. Finally,

vpmin,i(to) and vpmax,i(to) parameters are calculated using (20)and dispatched to be used in the inner loop. Overall detailedalgorithm of the adaptive control strategy is shown below.

Algorithm 1: Adaptive control scheme

1. Real-time measurement and control criterion calculation1.1. Collect Vt=to.tin∀tin = 1, 2, . . . , n1.2. Calculate SSEavg(to) and V F (to)

2. Go to adaptive strategy I: error adaptive2.1. If |SSEavg(to)| > µ+ εsse

qp(to) = qp(to − 1)− kd.SSEavg(to)

2.2. Else, qp(to) = qp(to − 1)2.3. update qmax(t0) and qmin(t0): equation (23)

3. Go to adaptive strategy II: slope adaptivemp(t0) = mp(t0 − 1) + ∆m

3.1. If V F (to) > V Flim ∆m = −∆vf

3.2. Else if V F (t0) > V Flim ∆m = −∆vf

3.3. Else if V F (t0) > (V Flim − εvf ) ∆m = 03.4. Else, check if |SSEavg| > µ+ εsse ∆m = ∆vf

4. Update final parameters vmin and vmax: equation (20)5. to = to + 1, go to step 1

IV. CASE STUDY

A. 4 Bus System IllustrationThe proposed adaptive VVC performance is compared with

delayed control for the example system described in theSection II.C in Fig.9. Since it is a small system, outer looptime horizon of 10 seconds is adequate to demonstrate adaptivenature of the control. Other system setup and parametersselection are same as described earlier. At t = 60, whenvoltage profile get a surge due change in substation voltagefrom 1.03 to 1.05, the adaptive control starts adapting itself tore-track the set-point in 2 steps. Whereas, the delayed VVC isleads to voltage violation due to high SSE and non-adaptivenature. Similarly under non-conservative settings, in (b) and(c), adaptive control is able to maintain the control stabilityunder the impact of sudden cloud cover and topology changesunlike the delayed control.

B. Large Test Case ModelingThe proposed adaptive control is tested on IEEE123 bus

test system which is an unbalanced three-phase feeder [27].To create a more realistic simulation, the test system is furtherexpanded with detailed secondary side house-load modeling at120 volts resulting in 1500 nodes as shown in Fig.10 usingGridLAB-D platform; GridLAB-D is an open-source agent-based simulation framework for smart grids developed byPacific Northwest National Lab [28]. Each residential load ismodeled in detail with ZIP loads and temperature dependentHVAC load [29] [30]. Diversity and distribution of parameterswithin the residential loads is discussed in [31]. The feeder ispopulated with 1280 residential houses with approximately 6MW peak load. Inverter ratings are considered 1.1 times thepanel ratings. Uniformly distributed solar PVs throughout thefeeder create lesser problems than the PV units distributedin one area of the feeder. Therefore, to demonstrate theeffectiveness of the control in more severe case, PV units

7

0 20 40 60 80 100 120(c)

control iterations (seconds)

1

1.05

1.1

V3 (

pu

)

Delayed (switch1 closed) Adaptive (switch1 closed)

0 20 40 60 80 100 120(b)0.9

1

1.1

V3 (

pu

)

Delayed droop Adaptive

0 20 40 60 80 100 120(a)0.98

1

1.02

1.04

1.06

1.08V

3 (

pu

)

delayed droop adaptive

µ = 1

voltage violation

cloud cover

substation voltage change

Fig. 9. Adaptive VVC performance comparison with delayed VVC underimpact of : a) substation voltage change; b) sudden cloud cover and; c)topology change

A B C

Fig. 10. IEEE 123 bus test system with detailed secondary side modeling

are distributed randomly at 500 houses only in right half ofthe feeder. Temperature and solar irradiance data for January2, 2011 is obtained from publicly available NREL data forHawaii [32]. Load and solar profiles for the day have beenshown in Fig.11.

C. Performance Metrics

We will be using three performance metrics to evaluatethe proposed control approach. First metric is mean steadystate error (MSSE) which evaluates control set-point trackingperformance. This is the absolute average (in percent) ofvoltage set point deviation at all of the houses with solar PV

Fig. 11. Total feeder load and solar PV profile for 24 hours

over the concerned time period. It is calculated as

MSSE =

n∑i=1

h∑t=1

|Vt,i − µ|h

.1

n× 100 (27)

where n is the total number of solar PV units and h is totaltime duration. Second metric is flicker count (FC) where oneflicker violation at one house is considered when V F value,as defined in (26), exceeds V Flim. The total number of suchflicker violations at all of the houses is termed as FC. Highervalue of this metric is an indication of lesser power qualityand an oscillatory voltage profile that in turn indicates thepossibility of unstable control. The third metric is voltageviolation index (V V I) is the total number of voltage violationsat all of the buses. Based on ANSI standards [33], a voltageviolation is counted if the voltage at a bus violates either 1)1.06-0.9 pu band instantaneously (range A) or 2) 1.05-0.95 puband continuously for 5 minutes (range B).

D. Results

In this section, we will demonstrate the effectiveness of theproposed control scheme in a wide range of external distur-bances and operating conditions. The adaptive control perfor-mance (blue) will be compared with existing droop controllersi.e. conventional (orange) and delayed droop (black). Voltageprofiles and parameters dispatched are shown at a randomlychosen solar PV unit at bus 92 whereas the performancemetrics are calculated for the whole system. Dashed and solidred lines denote the voltage violation limits and voltage set-point respectively. Outer loop horizon T is taken as 1 minute.

1) Dynamic Tests with Daily Load and Solar VariationA day-long load and solar profile can be seen as continuous

external disturbances in the system. Fig.12(a) shows thatduring the daytime, non-adaptive droop controls are not able totrack the set point voltage which might lead to voltage viola-tions e.g. around 12 noon when the solar generation is at peak.µ = 0.97 and homogeneous conservative settings (m = 3)are used for conventional and delayed control. Whereas theadaptive control adapts its parameters at each bus differentlyto keep a flat voltage profile throughout the day; note, thismay not be entirely desirable for the utility, or the owners,due to increased var flows, but rather indicates the flexibilityof the system for applications such as CVR, loss minimizationetc. Fig.12(b) shows the dynamic dispatch of adaptive errorparameter qp at bus 92. The performance metrics for the wholesystem are compared in I. In this case, high MSSE in delayedcontrol is because of selecting a conservative slope settingwhich can be improved by choosing higher slope, however,it will make the control highly vulnerable to sudden externaldisturbances as demonstrated in the next results. Whereas dueto its decoupled functionality, the proposed control is capableof achieving near zero MSSE even at conservative settings,thus not making system prone to instability or voltage flicker.

2) Dynamic Tests with Sudden External DisturbancesReliable performance under external disturbances is a

unique feature of the proposed control. To demonstrate this,the control performance is tested with sudden external distur-bances. A smaller window of 1-2 hours is considered whensolar is at its peak to observe the most severe impact of

8

0 3 6 9 12 15 18 21 24(b)

Time (hours)

-0.5

0

0.5

qp

Adaptive

0 3 6 9 12 15 18 21 24(a)

0.9

0.95

1

1.05

1.1

Vol

tage

(pu

)

No var controlConventional droopDelayed droopAdaptive

Fig. 12. Comparison of adaptive control performance throughout the day: a)voltage profile; b) dispatch of error adaptive parameter (qp)

TABLE IPERFORMANCE METRICS COMPARISON FOR 24-HOUR PROFILE

MetricsNo

controlConventional

droopDelayeddroop

Adaptivecontrol

MSSE 5.2% 4.3% 4.3% 0.3%VVI 5× 105 21853 16137 0FC 0 0 0 0

disturbances.a) Sudden cloud cover and cloud intermittency: Usually

cloud covers cause two types of disturbances in PV generationi.e. intermittency and sudden drop in the generation as shownin Fig.13 (a) and (b) respectively. Cloud intermittency dataof 30 seconds scale is considered. µ = 1 and m = 5are used for non-adaptive controls. Fig.14 shows how cloudintermittency causes high voltage fluctuations in conventionalcontrol which leads to violations. Delayed control reduces theflicker significantly compared to conventional (from 6919 to107), however, still results in a good number of violationsdue to high SSE as shown in Table II. Though, the effect ofintermittency is also visible in adaptive control voltage profile(Fig.14 (b)), it manages to achieve zero indices of flicker andviolations. It demonstrates the effectiveness of control in fasterdisturbances.

On the other hand, using non-conservative settings (m =10)to decrease violations can cause stability issues withsudden cloud cover as shown in Fig.15. At 11.30 AM, acloud cover results in a sudden drop in real power gen-eration (Fig.13(b)) which frees the inverter capacity. Sinceconventional and delayed controls utilize all the free capacityimmediately without monitoring, it increases the slope bya significant amount and results in voltage oscillations asshown in Fig.15(a). Whereas, the adaptive control dynamicallyregulates the settings in real-time to ensure stable voltageprofile as well as quick restore of the set-point tracking asvisible in Fig.15(b).

b) Change in substation voltage: The primary side of substa-tion voltage keeps changing due to changes in the transmissionsystems. conservative setting (m = 5) is used here for non-adaptive controls. In Fig.16, at 12 noon, the feeder experiencesa surge in primary substation voltage from 1 to 1.07 pu.Conventional control experiences high voltage oscillations.Delayed control does not experience voltage flicker but since

5 7 9 11 13 15 17 19(b)

Time (hours)

0

2

4

6

Pow

er (

MW

)

5 7 9 11 13 15 17 19(a)

Time (hours)

0

2

4

6

Pow

er (

MW

)

Fig. 13. Solar profile with a) sudden cloud cover and b) cloud intermittency

11:15 11:45 12:15 12:45(b)

Time (hours)

0.95

1

1.05

1.1

V (

pu)

Adaptive

11:15 11:45 12:15 12:45(a)

0.95

1

1.05

1.1

V (

pu)

Conventional Delayed

Fig. 14. Control performance comparison under cloud intermittency

11:00 11:15 11:30 11:45 12:00 12:15 12:30 12:45 13:00

(b)

Time (hours)

0.95

1

1.05

V (

pu

)

Adaptive

11:00 11:15 11:30 11:45 12:00 12:15 12:30 12:45 13:00

(a)

0.95

1

1.05

V (

pu

)

Delayed

Fig. 15. Control performance comparison under sudden cloud cover

it cannot reduce the SSE on its own, it waits for substationtap changer to operate to bring voltage within the limit again.Whereas adaptive control suffers from few instantaneous vi-olations but immediately starts re-tracking the set-point, thusavoiding violations for long time period.

d) Sudden load decrease: A sudden 40% load reduction isapplied at 12:00 to test the robustness of the proposed controlas shown in Fig.17. The conservative parameter (m = 5),which provided stable voltage performance under normalconditions is causing instability on a sudden load disturbanceas shown in Fig.17(b). Delayed control improves the stabil-ity but the voltage is vulnerable to overvoltage violations.

TABLE IIPERFORMANCE METRICS COMPARISON FOR INTERMITTENT

SOLAR-PROFILE FOR A TWO-HOUR WINDOW

Metrics No control Conventional Delayed Adaptive

MSSE 3.5% 2.00% 2.00% 0.40%VVI 5× 105 21853 16137 0FC 122 6919 107 0

9

11:30 12:00 12:30(b)

Time (hours)

0.9

1

1.1

V(p

u)

Delayed Adaptive

11:30 12:00 12:30(a)

0.9

1

1.1

V(p

u)

Conventional

upper

voltage limit

voltage violation

voltage violation

surge in substation

voltage

Fig. 16. Impact of change in substation voltage on control performances

11:30 11:45 12:00 12:15 12:30 12:45

0.8

1

1.2

Vol

tage

(pu

)

Conventional droop

11:30 11:45 12:00 12:15 12:30 12:45Time (hours)

0.9

1

1.1

Vol

tage

(pu

)

Delayed Adaptive

11:30 11:45 12:00 12:15 12:30 12:45

2

3

4

Pow

er (

MW

)

Feeder loadSudden LoadReduction

upper voltage limit

Fig. 17. Impact of sudden load change on controls: a) sudden load reductionapplied; b) voltage with conventional control; c) voltage with adaptive control

Whereas, the adaptive control adapts itself to the disturbanceand maintains a stable and flat voltage profile without set pointdeviation.

3) Adaptive to the error in feeder topology informationUsually in a large real-world system, fully reliable feeder

topology info is not available or there are a lot of changes inthe feeder which might not be communicated. This leads tochange in feeder topology and sensitivity matrix A, thus oldcontrol settings might create issues. The proposed control isalso adaptive to such errors or changes in feeder information.To simulate this, 25 new solar PV houses were added at theend of the original test system and the old non-conservativesettings (m = 10) were used for delayed control. Fig.18compares the voltage profile before and after feeder changefor delayed and adaptive controls. It can be seen that thevoltage profile changes from smooth (FC=0,VVI=0) to highlyfluctuating (FC=4047,VVI=1615) in the slightly expandedfeeder with the delayed control. Whereas, the adaptive controlprovides a better performance with zero flicker and violations.

V. CONCLUSION

In this study, a real-time adaptive and local VVC schemeis proposed to mitigate voltage challenges associated withhigh PV penetration under external disturbances. In specific,the proposed approach addresses two major issues of slopesensitive droop VVC methods. First, the proposed framework

11:15 11:45 12:15(a)

0.9

0.95

1

1.05

V(p

u)

Delayed Adaptive

11:15 11:45 12:15(b)

Time (hours)

0.9

0.95

1

1.05

V(p

u)

Delayed Adaptive

Fig. 18. Control performance comparison before and after feeder growth

(shifted and adaptive droop) enables VVC to achieve high set-point tracking accuracy (low SSE) and control stability (lowvoltage flicker) simultaneously without compromising eitherby decoupling the two functionalities. Second, the adaptivealgorithm enables dynamic self-adaption of control parametersin real-time which eliminates another major challenge ofselecting appropriate control settings under wide range ofoperating conditions/external disturbances such as fast cloudtransients, substation voltage change etc. All this is achievedwhile keeping the control purely local with no need of cen-tralized topology information and ensuring that the developedcontrol framework is compatible with the integration standards(IEEE1547) and utility practices (Rule 21). These featuresmake the proposed VVC feasible and implementation friendly.The satisfactory performance is demonstrated by comparingwith existing droop methods in several cases.

It is worth mentioning that the proposed local VVC frame-work is easily extendable to centralized approaches. In fact,due to its tight voltage regulation feature and adaptive na-ture under external disturbances, it facilitates the use of PVinverters for other system-wide volt/var applications such asCVR, loss minimization, increasing PV penetration capacityetc. The integration with supervisory control and the impactof the VVC on transmission system will be explored in futurestudies.

APPENDIX

Small 4-bus example system information: length of lines1-2, 2-3 and 3-4 are 2000, 4500 and 4500 feet respectively.Transformer is step-down (12.47kV/4.16 kV).

In all cases d = 0 and τ = 0.1 are considered. Notethat in conventional and delayed droop control, all settingsremain constant throughout the day except qmin and qmax

which change with change in PV generation and cloud cover.For adaptive control, all of these settings are decided by theproposed algorithm.

TABLE IIICONVENTIONAL AND DELAYED DROOP VVC SETTINGS: 4 BUS SYSTEM

droop qmin qmax vmin vmax

conservative (m = 2) -0.2 0.2 0.9 1.1non-conservative (m = 6) -0.2 0.2 0.967 1.033

10

TABLE IVCONVENTIONAL AND DELAYED DROOP VVC SETTINGS: 123 BUS SYSTEM

droop qmin qmax vmin vmax

conservative (m = 3) -0.4 0.4 0.867 1.133conservative (m = 5) -0.4 0.4 0.92 1.08

non-conservative (m = 10) -0.4 0.4 0.96 1.04

REFERENCES

[1] “SEIA ANNUAL REPORT 2016,” SOLAR ENERGY INDUSTRIESASSOCIATION, Tech. Rep., 2016. [Online]. Available: https://www.seia.org/research-resources/2016-annual-report

[2] E. J. Coster et al., “Integration Issues of Distributed Generation inDistribution Grids,” Proc. of the IEEE, vol. 99, pp. 28–39, Jan. 2011.

[3] b. Mather et al., “High-Penetration PV Integration Handbook for Dis-tribution Engineers,” NREL, Tech. Rep., Jan. 2016.

[4] C. L. Masters, “Voltage rise: the big issue when connecting embeddedgeneration to long 11 kV overhead lines,” Power Engineering Journal,vol. 16, no. 1, pp. 5–12, Feb. 2002.

[5] H. G. Yeh, D. F. Gayme, and S. H. Low, “Adaptive VAR Control forDistribution Circuits With Photovoltaic Generators,” IEEE Trans. PowerSys., vol. 27, no. 3, pp. 1656–1663, Aug. 2012.

[6] B. A. Robbins et al., “A Two-Stage Distributed Architecture for VoltageControl in Power Distribution Systems,” IEEE Trans. Power Sys.,vol. 28, no. 2, pp. 1470–1482, May 2013.

[7] M. McGranaghan et al., “Advanced grid planning and operation,” SandiaNational Lab, Tech. Rep., 2008.

[8] K. Turitsyn et al., “Options for Control of Reactive Power by DistributedPV Generators,” Proc. IEEE, vol. 99, no. 6, pp. 1063–1073, Jun. 2011.

[9] M. Bollen and A. Sannino, “Voltage control with inverter-based dis-tributed generation,” IEEE Trans. on Power Del., vol. 20, no. 1, pp.519–520, Jan. 2005.

[10] P. M. S. Carvalho et al., “Distributed Reactive Power Generation Controlfor Voltage Rise Mitigation in Distribution Networks,” IEEE Trans. onPower Sys., vol. 23, no. 2, pp. 766–772, May 2008.

[11] E. DallAnese, S. V. Dhople, and G. B. Giannakis, “Optimal Dispatch ofPhotovoltaic Inverters in Residential Distribution Systems,” IEEE Trans.Sust. Energy, vol. 5, no. 2, pp. 487–497, Apr. 2014.

[12] M. Farivar et al., “Optimal inverter VAR control in distribution systemswith high PV penetration,” in IEEE PES Gen. Meet., Jul. 2012, pp. 1–7.

[13] K. Turitsyn, P. ulc, S. Backhaus, and M. Chertkov, “Distributed controlof reactive power flow in a radial distribution circuit with high photo-voltaic penetration,” in IEEE PES General Meeting, Jul. 2010, pp. 1–6.

[14] B. Zhang et al., “An Optimal and Distributed Method for VoltageRegulation in Power Distribution Systems,” IEEE Trans. on Power Sys.,vol. 30, no. 4, pp. 1714–1726, Jul. 2015.

[15] E. DallAnese et al., “Decentralized Optimal Dispatch of PhotovoltaicInverters in Residential Distribution Systems,” IEEE Trans. on EnergyConv., vol. 29, no. 4, pp. 957–967, Dec. 2014.

[16] W. Zheng et al., “A Fully Distributed Reactive Power Optimizationand Control Method for Active Distribution Networks,” IEEE Trans.on Smart Grid, vol. 7, no. 2, pp. 1021–1033, Mar. 2016.

[17] H. Zhu and H. J. Liu, “Fast Local Voltage Control Under LimitedReactive Power: Optimality and Stability Analysis,” IEEE Transactionson Power Systems, vol. 31, no. 5, pp. 3794–3803, Sep. 2016.

[18] B. Seal and M. McGranaghan, “Standard Language Protocols for PVand Storage Grid Integration,” EPRI, Tech. Rep., May 2010.

[19] “IEEE Standard for Interconnecting Distributed Resources with ElectricPower Systems - Amendment 1,” IEEE Std 1547a, pp. 1–16, May 2014.

[20] “Rule 21 Interconnection Standard, CA.” [Online]. Available: http://www.cpuc.ca.gov/General.aspx?id=3962

[21] B. Zhang, A. D. Domnguez-Garca, and D. Tse, “A local control approachto voltage regulation in distribution networks,” in NAPS, Sep. 2013.

[22] M. Farivar, L. Chen, and S. Low, “Equilibrium and dynamics of localvoltage control in distribution systems,” in 52nd IEEE Conference onDecision and Control, Dec. 2013, pp. 4329–4334.

[23] P. Jahangiri and D. C. Aliprantis, “Distributed Volt/VAr Control by PVInverters,” IEEE Trans. Power Sys., vol. 28, no. 3, Aug. 2013.

[24] J. J. Grainger and W. D. Stevenson, Power system analysis. McGraw-Hill, Jan. 1994.

[25] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge UniversityPress, Oct. 2012.

[26] “IEEE Recommended Practice for Electric Power Distribution for In-dustrial Plants,” IEEE Std 141-1993, pp. 1–768, Apr. 1994.

[27] “IEEE 123 Bus Test System.” [Online]. Available: https://ewh.ieee.org/soc/pes/dsacom/testfeeders/

[28] D. P. Chassin et al., “GridLAB-D: An Agent-Based Simulation Frame-work for Smart Grids,” Journal of App. Math., p. e492320, Jun. 2014.

[29] K. P. Schneider et al., “Multi-State Load Models for Distribution SystemAnalysis,” IEEE Trans. on Power Sys., vol. 26, no. 4, Nov. 2011.

[30] K. P. Schneider, E. Sortomme, S. S. Venkata, M. T. Miller, andL. Ponder, “Evaluating the magnitude and duration of cold load pick-up on residential distribution using multi-state load models,” IEEETransactions on Power Systems, vol. 31, no. 5, pp. 3765–3774, Sept2016.

[31] J. Fuller et al., “Evaluation of Representative Smart grid InvestmentGrant Project technologies,” PNNL, Tech. Rep., Feb. 2012.

[32] M. Sengupta and A. Andreas, “Oahu Solar Measurement Grid:1-SecondSolar Irradiance; Oahu, Hawaii (Data),” NREL, Tech. Rep., 2010.

[33] “ANSI C84.1: American National Standard for Electric Power Systemsand EquipmentVoltage Ratings (60 Hertz),” Oct. 2016.