Volt/Var Optimization with Minimum EquipmentOperation under High PV Penetration

Ibrahim AlsalehElectrical Engineering Department

University of South Florida,Tampa, FL 33620, USA

Email: ialsaleh@mail.usf.edu

Lingling FanElectrical Engineering Department

University of South Florida,Tampa, FL 33620, USA

Email: linglingfan@usf.edu

Hossein Ghassempour AghamolkiEaton Corporate Research & Technology

Eden Prairie, MN 55344Email: hossein1@mail.usf.edu

Abstract—High penetration of photo-voltaic panels (PVs) in thedistribution networks causes volatile net load profiles. In turn,voltage control devices (switched capacitors, on-load tap changers(OLTCs) and voltage regulators) work more frequently to keepvoltage on distribution feeders within limits. This paper employsthe most recently developed mixed-integer programming (MIP)modeling methods to model discrete control devices includingOLTCs and switching capacitors for volt/var optimization to ex-plicitly count mechanical switching actions. Further, coordinationcapability is explored to demonstrate that the use of PV VARs caneffectively reduce operations of OLTCs and switched capacitorswhile obtaining a satisfactory voltage profile. Case studies areperformed on the IEEE 33-node distribution feeder with 125%PV penetration.

I. INTRODUCTION

Distributed generators (DGs) based on renewable energyhave been increasingly integrated into the distribution net-works. The excessive and intermittent nature of their powerproduction adversely impacts the operation of the electricalsystem, causing large and sudden fluctuations of voltages. Thisproblem is traditionally dealt with by employing switchablevoltage regulation equipment comprising on-load tap changers(OLTCs), voltage regulators (VRs) and switched capacitorbanks (SCBs) for voltage support and reactive power com-pensation. Although they are designed for a large number ofannual operations, the repetitive switching to cope with thesharp voltage sags and swells will increase maintenance costsand expedite the wear and tear of these devices [1], [2].

The volt/var optimization (VVO) with traditional equipmentsets the optimal tap position and/or capacitive reactive powerto keep voltage levels within the ±5% limits specified byANSI C84.1. Recently, standards such as IEEE 1547 allowedDG inverters to function with off-unity power factor, andthus participate in voltage regulation through reactive powergeneration/absorption. Smart technologies with two-way com-munication channels such as Advanced Metering Infrastructure(AMI) are deployed for capturing the distribution dynamicchanges and enabling the operator at the Distribution Man-agement System (DMS) to perform VVO and send controlcommands in a centralized manner [3].

The optimization problem is routinely solved by minimizingthe real power losses. However, this increases the voltage levelto the upper bound, unless being coordinated with other objec-

tives such as Conservation of Voltage Reduction (CVR) [4], orminimization of voltage deviations from the nominal value [5].

𝑞𝑚𝑎𝑥𝑔

𝑞𝑚𝑖𝑛𝑔

-𝑞𝑚𝑎𝑥𝑔

-𝑞𝑚𝑖𝑛𝑔

𝑝𝑚𝑎𝑥𝑔

𝑷

𝑸

PF limit

Fig. 1: Operating points of anoversized PV inverter.

Various methods havebeen developed on howoff-unity inverters shouldsupport the voltage[6]. One method is toscale the reactive powersupply/absorption linearlywith respect to deviationsof the local voltage [7],[8]. Another method is toset a strictly-fixed powerfactor on the inverter,where a fixed percentageof reactive power can besupplied/absorbed. Fig. 1shows the operating pointsof the inverter adopted in this paper, which is an oversizedinverter (110% of PV’s rating |S|max) with a variableleading/lagging power factor. The reactive power, generatedor absorbed during peak, is 46% of |S|max.

This paper investigates the OLTC and SCB operationsduring high penetration of PV. A centralized VVO isformulated to switch conventional devices in order to preventsevere voltage fluctuations. Some voltage-dependent loads areconfigured, whose voltages should be kept at the lower-halfas per CVR practices. For this, an additional objective isconsidered to further minimize the corresponding voltage.This would prompt the OLTC, SCBs and inverters to combatthe variable penetration by changing tap positions andgenerating/absorbing VARs to satisfy the objective. However,poor coordination among the various devices results inincreased operations of the OLTC and CBs.

Our contribution is a multi-period and multi-objective VVOthat improves the voltage profile with a coordinated operationof conventional devices and PV VARs. Two notable techniquesare adopted in our modeling: the SOCP relaxation techniquefor the branch flow model (BFM) which solves the problemto a global optimum [9], and the mixed-integer programming(MIP) technique to mimic the exact switching behavior ofan OLTC and SCBs in response to voltage deviations from

𝑃𝑖𝑗 ,𝑄𝑖𝑗

𝑟𝑖𝑗 + 𝑗 𝑥𝑖𝑗

𝑉1 𝑉𝑖

𝟏 𝒋

𝑝1𝑔

, 𝑞1𝑔

𝑝𝑖𝑔

, 𝑞𝑖𝑔 𝑝𝑗

𝑔, 𝑞𝑗

𝑔

𝑝1𝐿 , 𝑞1

𝐿 𝑝𝑖𝐿 , 𝑞𝑖

𝐿 𝑝𝑗𝐿 , 𝑞𝑗

𝐿

𝑉01

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1 : 𝑡01

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𝟎 𝒊

𝑉𝑗

𝑟01 + 𝑗 𝑥01

Fig. 2: A one-line diagram of a balanced radial distributionfeeder and system variables.desired.

The rest of the paper is organized as follows. Section IIformulates the VVO problem. The equality and inequalityconstraints related to the BFM and system models as wellas the objective function are presented successively. SectionIII conducts case studies to demonstrate the operation ofOLTC and CB with and without the coordination. Section IVconcludes the paper.

II. CENTRALIZED VVO FORMULATION

The nonlinear branch flow model describing the balancedpower flow in a radial network was first proposed in [10], andwas later convexified in [9]. Only recently has the MIP-basedOLTC model been embedded in the convex relaxation of thebranch flow model [11]–[13].

Fig. 2 shows a radial distribution feeder represented by thegraph G(N , E), where N and E correspond to the sets ofnodes and branches; respectively. A fictitious node is assumedat the transformer primary to describe the power flow into thebranch (i, j), and set relations among variables.A. Constraints

1) BFM: Equations in (1) are expressed for (i, j)

Pij =∑

k:(j,k)∈E

Pjk + rij`ij + pLj − pgj (1a)

Qij =∑

k:(j,k)∈E

Qjk + xij`ij + qLj − qgj − q

cj (1b)

vj = vi − 2(rijPij + xijQij) + (r2ij + x2

ij)`ij (1c)

`ij = (P 2ij +Q2

ij)/vit (1d)

where Pij , Qij are the real and reactive power flow from Busi to Bus j. rij and xij are the series resistance and reactanceof the line between Buses i and j. Superscript L notates loadand superscript g notates distributed generation. Superscriptc notates capacitors. Note that vi and `ij are the surrogatevariables of V 2

i and I2ij , whereas the nonequality constraint in

(1d) can be relaxed using the second-order cone programming(SOCP) relaxation. The constraint is

`ij ≥ (P 2ij +Q2

ij)/vi (2a)∥∥∥[2Pij 2Qij `ij − vi]T∥∥∥

2≤ `ij + vi (2b)

For the problem to be implementable, constraint needs tosatisfy an equality in order to comply with the physicalinterpretation. The accuracy of the relaxation is examined inSection III for the problem solution.

2) OLTC Model: An OLTC switches taps in order to adjustthe substation voltage. That is, the secondary-side voltage isincreased/decreased by changing the turns ratio so as to affectthe nodal voltages and power flows of the entire distributionsystem. Assuming (i, j) ∈ Etap and a set of discrete elementsX := {0, 1, . . . , xmax}, the OLTC model is

tij =(tmin + ∆tijx

)(3)

0 ≤ x ≤ xmax ∆tij = (tmaxij − tmin

ij )/xmax (4)

where tij is the OLTC ratio, tmaxij and tmin

ij are maximum andminimum turns ratios, ∆tij is the change per tap, and x ∈ X isthe tap position, which takes different discrete values to changethe ratio. Since the ratio also takes |X | of values, the switchingprocess can be exactly represented using binary variables.

Tij =

X∑x=0

(tminij + ∆tijx)2ux

X∑x=0

ux = 1 (5)

vij = Tijvi (6)

where Tij is composed of squared ratios to be compatible withthe voltage variable representation, and binary variables, ux,whose sum to one forces a selection of one ratio. For example,if the tap position is x = 5, then u5 becomes one to representthe corresponding ratio, while the rest of binary variables arezeros. Heed that (6) is convex if the primary-side voltage isknown. For OLTCs, the primary side is the substation as inFig. 2. The tap position can be recovered from the solution as

x = (√v∗ij/v

∗i − t

minij )/∆tij

3) Switched CB Model: A set of switchable capacitors canbe installed at the jth node, where each capacitor is switchedon to increase the voltage at the node of installation andadjacent nodes. Assuming j ∈ Ncap, an integer variable, Cj ,is defined to enforce the switching operation.

0 6 Cj 6 Nc (7)

qcj = QcCj

Nc(8)

where qcj is the SCBs’ variable included in (1b), and Qc

and Nc are the rating and number of the total SCB units;respectively.

4) PV Inverter Model: In order to represent the operatingpoints shown in Fig. 1, the reactive-power constraint is ex-pressed as

|qgi | 6√

(sgi )2 − (pgi )

2 (9)

Overcapacity of the inverter and thus reactive power gen-eration/absorption during peak PV generation are ensured ifthe nameplate MVA, namely sgi , is larger than peak PV activepower, pgmax.

5) Voltage Limits: Except for the substation node (v0 =V 2

0 = 1), ±5% of the nominal voltage are enforced as boundson each node.

V 2i min 6 vi 6 V 2

i max (10)

6) CVR Limits: The CVR originates from the fact thatvoltage-dependent loads, i.e. constant impedance or currentloads, consume more energy when the voltage is above nomi-nal, increasing annual energy costs. Therefore, CVR practicesaim at reducing voltage magnitudes to the lower-half of theallowable limit.

The constraints in (11) are used to keep the voltage of theith node between minimum and maximum thresholds.

zi > 0, zi > vi − (V thri min)2, zi > −vi + (V thr

i max)2 (11)

This is advantageous as a tighter threshold, i.e. V thrimax = 1,

can be assigned for nodes at which voltage-dependent loadsare installed, i ∈ Ncvr, and zi is minimized with large costcoefficients to create a trade-off with the loss-minimizationobjective. Moreover, this objective can be generalized for allnodes with lower costs and wider limits, say ±3%, to flattenthe voltage. The desired lower threshold is −3% so as toavoid excessive voltage drop at the point of interconnectionand maintain the safety of the equipment behind the meter.

B. Overall VVO Problem

The overall optimization problem over T horizons is for-mulated as follows.

min f =T∑t

λloss

∑(i,j)∈E

rij`t,ij + λcvr

∑i∈Ncvr

zt,i

+ λflat

∑i∈N−Ncvr

zt,i + λcap

∑i∈Ncap

|Ct,i − Ct−1,i|

+λtap

∑(i,j)∈Etap

|Tt,ij − Tt−1,ij |

s.t. (1), (2), (5)− (11)

(12)

III. CASE STUDIES AND NUMERICAL EXAMPLES

The case studies highlight the following:

1) the impacts of cloudy day and clear day on the frequencyof an OLTC’s and SCBs’ operations.

2) the effectiveness of the centralized VVO to mitigate theequipment operations and adhere to CVR limits by virtueof the inverter’s inherent VAR capability.

PV3

PV1PV2

0 1 2 3 4 5 6 7 8 9 10 11 12 13

17 16 15 14

25 26 27 28 29 30 31 3218 19 20 21

22 23 24OLTC

SCB2

SCB1

Fig. 3: Modified IEEE 33-node feeder.

00:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00

Time (hour)

0

1

2

3

4

5

6

Act

ive

Pow

er (

MW

)

LoadingClear-day PVCloudy-day PV

Fig. 4: Loading, clear-day PV and cloudy-day PV profiles.

A. Modified IEEE 33-node feeder

Fig. 3 illustrates the IEEE 33-node feeder, which wasmodified to include an OLTC, SCBs and voltage-dependentloads. The original peak load is 4.55 MVA with power factorof 0.82. Loads at each node and line parameters are obtainedfrom [14]. Three resistive loads are modeled, each with 100kW, at nodes 11, 23 and 26. The OLTC is installed on thesubstation branch, with a turns ratio varying from 0.95 to1.05. The tap position is constrained by xmax = 32, whichis a typical limit of a practical tap changer’s winding. Also,two SCBs, each with a total of 360 kVAR and three switchableunits (Nc = 3), are installed at the remote node 14, and theheavy reactive power loaded bus 29, whose adjacent buses (28and 30) consume 30% of the load.

Fig. 4 shows a loading curve obtained from PJM, anddepicted by the total active power load. Also, three 2-MW PVplants are installed at nodes 17, 24 and 32. Each PV inverterhas 110% apparent-power capacity of the peak active power.Fig. 4 shows two PV power profiles by the total MW at 5-minute resolution. The profiles mimic a real solar panel’s datacollected at the University of South Florida on May 15, andAugust 15 of 2013.

The optimization problem was formulated and solved by theCVX toolbox [15] with MATLAB R2014a, and in conjunctionwith the GUROBI solver [16].

B. Case Studies and Results

The optimization problem in (12) is solved every 15 min-utes, and multiple scenarios are carried out interchangeably.Equipment-operation penalties are fine-tuned starting withsmall values to achieve the best coordination with PV VARs[2]. For simplicity, loss reduction and CVR objectives are setwith unity penalties, while flatness is found to take effect with0.3.

TABLE I: Cost Coefficients

Objective Symbol Range Cost ($)Loss Reduction λloss - 1

CVR λcvr 0.97-1.00 pu 1Flat Profile λfalt 0.97-1.03 pu 0.3

Tap Operations λtap 0-32 taps 3SCB Operations λcap 0-3 units each 0.1

00:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00

Time (hour)

4

8

12

16

20

24

28

32T

ap P

ositi

on (

)

No PVClear-day PVCloudy-day PV

x

(a)

00:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00

Time (hour)

0

1

2

3

SC

B1

( C

t,i )

No PV Clear-day PV Cloudy-day PV

(b)

Fig. 5: (a) Tap positions. (b) Number of switched CBs.

TABLE II: Operation Counts at unity PF of PVs

Equipment No PV Clear-day PV Cloud-day PVTaps 16 36 43

SCB1 - 3 11SCB2 - 5 8

Case I: At unity power factor of the PV inverters, thethree scenarios are compared in terms of equipment operations.Since the inverter’s VAR is absent for this case, the tap-capcosts in the VVO are zeroed out (λtap = λcap = 0) so as tolet the OLTC and SCBs satisfy the operation constraints. Atno PV, the tap actions are moderate and following the load,while SCBs kept supplying full VARs. However, during bothclear-day and cloudy-day PV penetrations, the tap-cap actionsdramatically increased in frequency to cope with the dynamicnet load. PV power at a cloudy-day in particular results insignificant increase of switching actions, and is consequentlydeemed the worst-case scenario.

Case II: Keeping tap-cap costs zeroed out, the scenario withcloudy-day and unity-PF PVs is solved with and without coststhat penalize CVR and flat-profile objectives to explore thecapability of traditional equipment to abide by the thresholdsof these objectives. The main feeder is selected to examinevoltage profiles, which has an OLTC, a resistive load at node11, switched CBs at node 14, and a PV at node 17. Fig. 6ashows that without the penalties, the loss-reduction objectiveoperates taps and SCB mostly at their maximum bound, thusincreasing voltage variations at the downstream nodes andviolating CVR. The taps and SCB1 are suddenly reduced

(a)

(b)

Fig. 6: Main feeder voltages (a) without and (b) with CVRand flat-profile penalties.

when the voltage at node 17 tends to exceed 1.05 pu. On theother hand, with the penalties, Fig. 6b shows that voltages areregulated closely within the desired limits specified in TableI, and with tap-cap actions shown in Table II During eveninghours 16:00-21:00, the flatness penalty, λflat, takes less effecton the OLTC secondary voltage as more taps are switchedto counteract the heavy loading. It can be concluded thattraditional equipment can effectively regulate voltages withinthe specified thresholds. This however comes at the cost ofincreased equipment operations.

Case III: With off-unity power factor PV inverters, thepossibility of coordination between traditional equipment andinverters towards reducing tap-cap operations and improvingvoltage profiles is explored utilizing the multi-objective func-tion in (12) and considering all costs in Table I.

Being the worst-case scenario in terms of equipment oper-ation and voltage variation, the VVO is solved for the cloudy-day PV. The results in Fig. 7 focuses on the period when PVpower is most variable.

Fig. 7 shows that when the VVO is solved without theswitching penalties, the inverters are not urged to gener-ate/absorb enough VARs so as to reduce the tap-cap actions,since voltages are within the limits as in Fig. 7d. As aresult, the switching not only maintains a similar behavior,but also increased for the OLTC and SCB1 as in TableIII. Exceptionally, SCB2 remains unswitched without theswitching penalty. In contrast, with the switching penaltiesin Table I, the PV VARs coordinates well with the OLTCtaps, while keeping SCB1 unswitched as in the baseline case.The coordination can be observed at instances when PV VARsapproach zero, OLTC taps switch with smaller steps than those

9:00 12:00 15:00

Time (hour)

4

8

12

16

20

24

28

32T

ap P

ositi

on (

)

without PV VARswith PV VARswith PV VARsand switching Penaltiesx

(a)

9:00 12:00 15:00

Time (hour)

0

1

2

3

SC

B1

( C

t,i )

without PV VARs with PV VARs with PV VARs and switching penalties

(b)

9:00 12:00 15:00

Time (hour)

-1.5

-1

-0.5

0

0.5

1

1.5

PV

Rea

ctiv

e P

ower

(M

VA

R)

PV1with switching penelties

PV2 PV3

PV1without switching penelties

PV2 PV3

(c)

(d)

(e)

Fig. 7: (a) Tap positions. (b) Number of switched CBs. (c)VARs from each inverter with and without switching penalties.(d) VAR-compensated main feeder voltages without switchingpenalties, and (e) with switching penalties.

TABLE III: Cloudy-day Operations at off-unity PF of PVs

Equipment With PV VARs With PV VARs & switching penaltiesTaps 47 20

SCB1 12 -SCB2 - -

0 5 10 15 20 25 30

Lines

0

0.2

0.4

0.6

0.8

1

1.2

Exa

ctne

ss

10 -7

Fig. 8: Exactness of the centralized VVO solution.

simulated without switching penalties and/or VARs.. Also, thereactive/capacitive PV VARs boost to counteract the peaks andvalleys of PV active power. The resulting voltage profiles arefurther improved as in Fig. 7e. Table III lists the switchingcounts for each device.

C. Exactness of SOCP

The implementability of the presented centralized VVO isverified via examining the exactness of the SOCP relaxation.The SOCP relaxation is said to be exact if the solution admitsthe power flow characteristics. That is, the inequality constraintof the squared current in (2) satisfies a sufficiently small error.Therefore, the exactness for the solution of all currents andover all time horizons is examined by computing the error in(13). The results are shown in Fig. 8.

Exactness =∑t∈T

∑(i,j)∈E

|`t,ij − (P 2t,ij +Q2

t,ij)/vt,i| (13)

Since the overall errors are in the order of 1 × 10−7, thesolution is exact and the presented VVO is implementable.

IV. CONCLUSION

This paper formulates a centralized VVO that aims atreducing equipment operations and keeping overall voltageprofiles within the satisfactory limit at high penetration ofPV. CVR practices are also taken into account. Case stud-ies of three scenarios were conducted to demonstrate theeffectiveness of the VVO. At unity power factor of the PVinverters , the results have shown that the traditional equipmentapproximately fulfilled the specified limits for voltage. Thiswas at the expense of increased and repetitive operations oftaps and switched capacitors. By minimizing the switchingcost of OLTC taps and switched capacitors, off-unity in-verters operate as the fundamental voltage regulators mainly

via boosting reactive-power supply/absorption throughout thevariable PV penetration. Therefore, coordination between thetraditional equipment and PV inverters can effectively reduceequipment operations, make optimal utilization of DGs, andflatten voltage profiles even at mid-way nodes.

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