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Conference Paper

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February 2019

Data-Driven Preemptive Voltage 

Monitoring and Control Using 

Probabilistic Voltage Sensitivities

Preprint

Kumarsinh Jhala,

1 Venkat Krishnan,2

Balasubramaniam Natarajan,

1 and Yingchen Zhang2

1 Kansas State University 

2 National Renewable Energy Laboratory 

To be presented at the 2019 IEEE Power and Energy 

Society General Meeting (IEEE PES GM) 

Atlanta, Georgia

August 4–8, 2019 
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Conference Paper

NREL/CP-5D00-72661

February 2019

Data-Driven Preemptive Voltage 

Monitoring and Control Using 

Probabilistic Voltage Sensitivities

Preprint

Kumarsinh Jhala,1 Venkat Krishnan,2 

Balasubramaniam Natarajan,1 and Yingchen Zhang2 

1 Kansas State University 

2 National Renewable Energy Laboratory 

Suggested Citation 

Jhala, Kumarsinh, Venkat Krishnan, Balasubramaniam Natarajan, and Yingchen Zhang. 

2019. Data-Driven Preemptive Voltage Monitoring and Control Using Probabilistic Voltage 

Sensitivities: Preprint. Golden, CO: National Renewable Energy Laboratory. NREL/CP5D00-72661. https://www.nrel.gov/docs/fy19osti/72661.pdf. 

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1

Data-Driven Preemptive Voltage Monitoring and

Control Using Probabilistic Voltage Sensitivities

Kumarsinh Jhala, Student Member, IEEE, Venkat Krishnan, Member, IEEE,

Balasubramaniam Natarajan, Senior Member, IEEE, Yingchen Zhang, Senior Member, IEEE

Abstract—Increased penetration levels of distributed variable

renewable generation can cause random voltage fluctuations

and violations at multiple nodes. Traditional methods of voltage

control typically involve reactionary responses of capacitor banks,

tap changers, and recently even smart inverters. But because

of the lack of foresight in voltage violations, these controls

are ineffective to completely mitigate the issue. Therefore, new

methods of predicting voltage violations subject to random power

injection changes in the distribution network are needed, which

can be used to guide optimal and dynamic methods of voltage

control. This work lays the foundation for such preemptive

voltage monitoring and control by proposing an analytical and

sensor data-driven voltage sensitivity analysis method. Driven by

stochastic data and forecasts, the method can be used to develop

probabilistic voltage sensitivities and consequently to predict

system nodes with high likelihood of voltage limit violations.

The effectiveness of this method is tested on IEEE 69-node

distribution system integrated with distributed solar. The results

demonstrate the proposed method’s ability to successfully predict

nodes with high probability of voltage violations for a specific

time-series simulation. The results also demonstrate the ability

to guide timely power injection control actions to mitigate future

voltage violations.

Index Terms—Distributed Generation, Distribution System,

Sensitivity, Sensor Measurement, Voltage

I. INTRODUCTION

The power system is evolving significantly with the proliferation of new smart grid technologies. Increased penetration of

renewable generation, electric vehicles and active consumers

at the grid edge create new challenges as well as opportunities.

More specifically, the distribution grid is anticipated to experience random fluctuations in voltages and even ANSI limit

K. Jhala is with the Center for Energy, Environmental, and Economic

Systems Analysis in the Energy Systems Division at Argonne National

Laboratory (e-mail: kjhala@anl.gov). This work was done while he was with

Electrical and Computer Engineering at Kansas State University.

B. Natarajan is with Electrical and Computer Engineering, Kansas State

University, Manhattan, KS-66506, USA, (e-mail: bala@k-state.edu)

V. Krishnan and Y. Zhang are with the Sensing and Predictive Analytics Group in the Power Systems Engineering Center of the National

Renewable Energy Laboratory (NREL), Golden, CO, 80401 USA, e-mail:

(venkat.krishnan@nrel.gov). K. Jhala was also a summer intern at the National

Renewable Energy Laboratory (NREL) in 2018 summer.

This work was authored in part by the National Renewable Energy

Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S.

Department of Energy (DOE) under Contract No. DE-AC36-08GO28308.

Funding provided by the U.S. Department of Energy Grid Modernization

Initiative (GMI) under the project GMLC 1.2.5 Sensing and Measurement

Strategy. The views expressed in the article do not necessarily represent the

views of the DOE or the U.S. Government. The U.S. Government retains

and the publisher, by accepting the article for publication, acknowledges that

the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide

license to publish or reproduce the published form of this work, or allow

others to do so, for U.S. Government purposes.

violations under higher penetrations of rooftop photovoltaics

(PV) [1], [2]. Traditional methods of voltage control such as

on-load tap-changing transformers, voltage regulators and capacitor banks are not designed for bi-directional current flow,

and typically are controlled to provide reactive support after

an event is detected [3]. Additionally, under increasing levels

of variable renewable, sole dependence on the installation of

more traditional voltage control devices may not be viable [4],

[5]. Therefore, extensive research have looked into utilizing

the reactive power capabilities of PV systems by integrating

them via smart inverters under either centralized [6]–[10] or

decentralized [11]–[13] control approaches. IEEE 1547 (2018)

standards and utilities have also begun to look into such smart

inverter options to mitigate voltage violations. Studies [14]

have shown Volt-Var and Volt-Watt options provide muchneeded grid support; however, these strategies also provide

reactionary support post-event, and testing studies [15] have

shown them to not eradicate voltage violations completely

given the maximum allowable limits of reactive power provisions and/or real power curtailments.

Therefore, a computationally efficient preemptive voltage

control mechanism is needed that predicts future voltage

and their uncertainty bounds and guides preventive actions

in power distribution systems interconnected with renewable

generation. This paper considers a power distribution system

with multiple active consumers with flexible load and generation participating in demand response program with thirdparty aggregators. Goal of this work is to develop a datadriven approach to predict nodes with high probability of

voltage violation and devise a preemptive control action by

modifying consumer load/generation to prevent steady-state

voltage violations.

In order to predict voltage responses, this paper assumes

the availability of voltage and power measurements from all

nodes in the distribution system, and uses them to estimate

probabilistic sensitivities of node voltages with respect to

real and reactive power injection changes at various nodes.

The probabilistic sensitivities will be used to dynamically

predict nodes with high probability of voltage violations [16],

[17] as well as mitigate them using a preemptive control

strategy that modifies nodal injections (load/generation). Major

contributions of this paper are listed below.

• Develops an analytical expression to compute the upper

bound of change in voltage resulting from change in

complex power at multiple nodes [16], which is computationally efficient and dynamic with use of sensor data.

• Provides a probabilistic voltage sensitivity analysis-based

1

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2

method that predicts probability of future voltage subject

to forecasts and detects highly vulnerable nodes (sec. III).

• A simple preemptive voltage control mechanism in IEEE

69-node system where the accuracy and effectiveness of

voltage prediction and control is tested. (section IV).

II. BACKGROUND: VOLTAGE SENSITIVITY ESTIMATION

Voltage sensitivity analysis estimates change in complex

voltage at a given node (observation node o) due to change in

complex power at another node (actor node a) in a power distribution system. Traditionally, voltage sensitivity is calculated

using Newton-Raphson load flow method (from the Jacobian

or its eigenvectors) or perturb-and-observe method [18]. Both

methods are numerical and computationally complex. This

section presents an analytical upper bound for voltage change

at an observation node (Vo) due to change in complex power

at an actor node (Sa) in a radial distribution network for

constant power model of loads, originally developed in [16].

When an actor node (a) changes its complex power from Sa

to Sa + Sa, voltage at the observation node changes from

Vo to Vo + Voa, which can be calculated from Theorem 1.

Here Voa is change in complex voltage at the observation

node o due to change in complex power at the actor node a.

Theorem 1. For a radial power distribution network, change

in voltage at an observation node due to change in power of

an actor node is upper bounded by

Voa  

SaZoa

V ⇤

a

, (1)

where V ⇤

a is complex conjugate of voltage at the actor node;

and Zoa is impedance of shared line between the observation

node o and the actor node a from the source node.

Proof. see [16]

Here, inequality sign for complex number indicates the

upper bound on real and imaginary values. Equation (1)

provides a linear upper bound on generally non-linear power

flow calculations. This notation is used throughout the paper.

Effect of multiple actor nodes on the observation node voltage

can be calculated using the following lemma.

Lemma 1. Superposition Law: If A is a set of actor nodes

in the network, effective change in complex voltage at the

observation node due to the cumulative effect of all the actor

nodes is bounded by (2).

Vo  X

a2A

SaZoa

V ⇤

a

, (2)

where A is set of all the actor nodes.

Proof. see [16]

This lemma proves that the proposed analytical method represented by (2) holds the law of superposition. The analytical

equation derived in Lemma 1 can be used to calculate the

probability distribution of voltage change at any given node

in the power distribution system. Calculating sensitivity matrix

for the IEEE 69-node test system using the classical loadflow method takes 4.52 seconds, whereas using the proposed

analytical method takes only 0.58 seconds [16]. This paper

further advances the application of these sensitivities for dynamic prediction of future power states, by integrating phasor

data from sensors and renewable forecasts. Impact of sensing

errors are not considered.

III. PREEMPTIVE ANALYSIS: PREDICTION AND CONTROL

The analytical sensitivity estimation method presented in

Section II is used as the basis for developing data-driven

probabilistic voltage sensitivity analysis for predicting the

probability of voltage violation at a given node. We make an

assumption that complex power and voltage measurements (or

estimations) are available at each node of the balanced three

phase distribution system in real time.

Random changes in power drawn/injected by active consumers with renewable generation cause random voltage fluctuations, which makes voltage at any given node in the distribution system random. This work assumes normal distribution

for power injection changes. Let V p

o be current value of

complex voltage at any observation node o, which is obtained

from measurements or state estimation, and V f

o represent

the predicted future complex voltage at node o. Due to the

variability and uncertainty introduced by renewable generation

in power system, V f

o is random and can be written as:

V f

o = V po + Vo, (3)

where Vo is random change in complex voltage at observation node due to random changes in net power injections.

Change in real and imaginary part of voltage at an observation

node due to change in complex power at an actor node can

be written as:

Voa = V r

oa + iV ioa, (4)

where

V r

oa =  1

|Va|

(Pa(Roa cos ✓a  Xoa sin ✓a)

Qa(Roa sin ✓a + Xoa cos ✓a)),

(5)

and

V i

oa =  1

|Va|

(Qa(Roa cos ✓a  Xoa sin ✓a)

+Pa(Roa sin ✓a + Xoa cos ✓a)).

(6)

From superposition law (Lemma 1), change in voltage at an

observation node due to cumulative effect of multiple actor

nodes can be written as sum of changes in voltage at the

observation node due to every actor node.

Vo = X

a

Voa = X

a

V r

oa + i

X

a

V i

oa, (7)

Behavior of nodal net-loads in a distribution network integrated with variable renewables can be modeled as random

variables. In this work, change in real and reactive power injections at a distribution system node is modeled as Gaussian random variable. Let S = [P1, ··· , Pn, Q1, ··· , Qn]

T

be a Gaussian random vector with mean μ and covariance

2

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3

matrix ⌃. Here, mean of real and reactive power reflects forecast of future net-load changes (i.e., estimated using load and

renewable generation forecasts). The covariance matrix will

be estimated using historical data, supplemented with probabilistic forecasts that provide variances of forecasts around

the mean [19]. The cross-correlations could be estimated by

generating scenarios of random forecasts for various locations,

and estimating their cross-correlations. Additionally, the effect

of spatial correlation of variable renewables is captured by

the off-diagonal elements of the covariance matrix. Detailed

modeling of variability and uncertainty in the voltage sensitivity assessment is beyond the scope of this paper, but will

be undertaken for future work.

A. Computing Voltage Sensitivity Probability Distribution

Given the normal distribution assumption for power injection changes, the resultant voltage sensitivity is also expected

to be normally distributed due to their linear relationship

modeled by equations (5) and (6) [20]. This section further

elaborates the estimation of |Vo| distributions using following steps:

1) Define ⌃, and compute vectors Cr and Ci :

⌃ =

2

6

6

6

6

6

4

2

p1 ··· cov(pn, p1) cov(q1, p1) ··· cov(qn, p1)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

cov(p1, pn) ··· 2

pn cov(q1, pn) ··· cov(qn, qn)

cov(p1, q1) ··· cov(pn, q1) 2

q1 ··· cov(qn, p1)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

cov(p1, qn) ··· cov(pn, qn) cov(q1, qn) ··· 2

qn

3

7

7

7

7

7

5

(8)

where n is number of nodes in the network. For nodes that do

not have PV, values of variance can be set to zero. In this work,

we assume that network topology is known. Value of complex

bus voltages is gathered from measurements and values of line

impedance are assumed to be known from network topology.

The vectors Cr and Ci are defined and computed as follows,

using equations (9) and (10). Values of Cr and Ci can be

computed as following.

Cr =

2

6

6

6

6

6

6

6

6

6

4

 Ro1 cos ✓1Xo1 sin ✓1

|V1|

.

.

.

 Ron cos ✓nXon sin ✓n

|Vn|

Ro1 sin ✓1+Xo1 cos ✓1

|V1|

.

.

. Ron sin ✓n+Xon cos ✓n

|Vn|

3

7

7

7

7

7

7

7

7

7

5

, (9)

Ci =

2

6

6

6

6

6

6

6

6

6

4

 Ro1 sin ✓1+Xo1 cos ✓1

|V1|

.

.

.

 Ron sin ✓n+Xon cos ✓n

|Vn|

 Ro1 cos ✓1Xo1 sin ✓1

|V1|

.

.

.

 Ron cos ✓nXon sin ✓n

|Vn|

3

7

7

7

7

7

7

7

7

7

5

. (10)

2) Compute distribution of V r

o and V io : Real and imaginary part of change in voltage at an observation node can be

written as weighted sum of elements of vector S as depicted

by equation (11) and (12). Weighted sum of Gaussian random

variables are normally distributed. Therefore, probability distribution of V r

o and V io can be derived as follows

V r

o = X

a

V r

oa = CTr S ⇠ N(CTr μ, CTr ⌃Cr) (11)

V i

o = X

a

V i

oa = CTi S ⇠ N(CTi μ, CTi ⌃Ci) (12)

3) Define bi-variate normal vector ˆVo : Covariance

between V r

o and V io can be written cov 

V r

o , V io

 =

CT

r ⌃Ci. Therefore, the change in real and imaginary part of

voltage change will be a bi-variate normal vector, i.e.

Vo ,



V r

o

V i

o



⇠ N (μ1, ⌃1) (13)

where,

μ1 =



CT

r μ

CT

i μ



, ⌃1 =



CT

r ⌃Cr CTr ⌃Ci

CT

r ⌃Ci CT

i ⌃Ci



(14)

4) Calculate probability distribution of Vf

o: Let Vfo be

vector of real and imaginary part of future voltage at node

o defined as:

Vf

o ,



V rf

o

V if

o



=



V rp

o

V ip

o



+



V r

o

V i

o



(15)

Voltage at observation node in next time slot Vf

o can be written

as Gaussian random vector as following.

Vf

o ⇠ N

✓V rf

o + CTr μ

V if

o + CT

i μ



,



CT

r ⌃Cr CTr ⌃Ci

CT

r ⌃Ci CT

i ⌃Ci

◆ (16)

The covariance matrix, voltage measurements and forecasted change in mean power injections will be updated as

new forecasts and sensor data are available. This process

will be continued for future estimations of voltage at some

regular time intervals (though in this work, for a given time

series simulation, such an update is not done for simplicity of

illustration).

B. Detecting Vulnerable Nodes and Preemptive Control

Equation (16) shows that |V f

o | is Gaussian random variable

with numerically computed mean and variance, which can be

used to find probability of voltage violation. Let Po(t) be the

probability of voltage violation at node o at time t defined as:

Po(t)=1  P(0.95 < |V f

o | < 1.05). (17)

Performing this calculation for each node in the distribution

system, we can identify nodes that have probability of voltage

violation. After identifying nodes that are highly vulnerable

to voltage violation an effective voltage control action can

be taken. To test effectiveness of the proposed method, a

voltage sensitivity-based heuristic voltage control approach

is considered, where an upper bound on maximum power

drawn/injected is enforced for nodes that are vulnerable to

voltage violation. In a case where node is vulnerable to highvoltage violation, amount of power injected by the node into

3

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4

the grid is upper bounded by x% of current power injection for

future time period. Similarly, in a case where node vulnerable

to low-voltage violation, amount of power drawn from the grid

is upper bounded by x% of current power for future time period. The amount of power injection change is estimated based

on voltage sensitivity information. Let N(t) be set of nodes

that are vulnerable to voltage violation at time t, and V t

be voltage change required to mitigate voltage violation at

node with highest/lowest voltage in case of over/under-voltage

violations in the neighborhood. Eliminating highest voltage

violation should result in elimination of voltage violation at

most of the nodes in the neighborhood. This procedure will be

undertaken iteratively until a premptive power injection control

strategy for eliminating all violations is estimated. To ensure

fairness between nodes, generation at all nodes is reduced by

same amount. Percentage load/generation curtailment can be

computed as following:

V t   X

a2N

xSaZoa

V ⇤

a

(18)

x  

V t

P

a2N

SaZoa

V ⇤

a

(19)

Here V t is selected such that the probability of voltage

violation (equation (17)) is less than the threshold used to

identify nodes that have high probability of voltage violation.

In ideal situation, an optimal control strategy will utilize this

algorithm to estimate the time for which such power injection

change has to be done. But an optimal control strategy is

beyond the scope of this paper. However to show the value of

having voltage violation foresight, we impose a simple 2-hour

time period over which the estimated power injection changes

will be implemented.

IV. SIMULATION AND RESULTS

To test proposed voltage violation prediction algorithm an

IEEE 69 bus test system is considered [16]. Synthetic sensor

data is produced using power flow solutions. A hypothetical

scenario is considered from noon to 6 p.m. with voltage and

power data available every 5-minute interval. MicroPMUs

and other distribution level sensors are capable of providing

voltage and power measurements at higher rate; however, for

simplicity of demonstration 5-minute interval is considered.

Roof top PVs in 20 nodes with varying generation capacity

are considered. Solar generation is modeled as random process

with some trend and seasonality components that reflect real

world scenario as shown in equation (20) [21]–[24].

Psolar(t) = S(t) + ns(t) (20)

Here, S(t) is PV generation mean forecast trend, and ns(t) is

zero mean Gaussian random variable that models variability

and uncertainties. Gaussian random noise in solar generation is

correlated across different nodes to reflect spatial correlation of

solar generation. In this simulation, synthetic solar generation

data is generated dynamically to reflect real world scenario.

Correlation coefficient of solar generation is chosen as 0.7.

Load at each node is synthetically produced based on typical

12.00 PM 1.00 PM 2.00 PM 3.00 PM 4.00 PM 5.00 PM 6.00 PM

Time of the Day

-400

-200

0

200

400

600

Power (kW)

Total Residential Load

Total Solar Generation

Net Power

Fig. 1: Total load and generation

load profile of residential household. Figure 1 shows load,

solar generation and net power of the distribution system. The

figure shows that load is lowest and PV generation is highest

at noon resulting in negative net power. This reverse power

flow causes high-voltage violations in parts of the distribution

system. For simplicity, this work considers a scenario where

no low-voltage violations occur. However, the proposed preemptive voltage control algorithm is general enough to apply

in case where high or low-voltage violations occur.

The aggregator receives power injection and voltage measurements at all nodes. This information along with bus

impedance data is used to compute voltage sensitivity matrix using proposed analytical method of voltage sensitivity

analysis discussed in section II in a computationally efficient

manner. The mean and variance of real and imaginary part

of voltage change at each node is estimated analytically

as shown in section III. Vectors Cr and Ci are computed

based on network topology and synthetic measurement data.

Covariance matrix ⌃ is estimated based on probabilistic variance of historical data. Based on node voltage measurements

and calculated probability distribution of voltage change, the

probability of high and low-voltage violation is computed

numerically. Nodes that have probability of voltage violation

greater than threshold are classified as highly vulnerable nodes

for voltage violation perspective. In this illustration, nodes

that have probability of voltage violation more than 50% are

considered as highly vulnerable. Number of voltage violation

in the system every time slot are shown in Figure 2 by blue

bars. Red stem plot in Figure 2 shows number of voltage

violations predicted by proposed method, based on nodes with

probability of voltage violation greater than 50%. The proposed method predicts nodes vulnerable to voltage violation

accurately, thereby giving a foresight to system operators for

optimal control.

To demonstrate preemptive control, a voltage sensitivity

based heuristic voltage control method is used. An upper

bound on power injected into the grid is enforced for next

two hours for nodes that are identified as highly vulnerable

to voltage violations. Value of upper bound is calculated

using equation (19), which is 65% of power injected into

the grid. Figure 3 shows voltage at node 65 of IEEE 69

bus test system with and without (no foresight or prediction)

preemptive voltage control. Figure 3 shows that proposed

method successfully eliminates high-voltage problems.

4

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5

12.00 PM1.00 PM 2.00 PM 3.00 PM 4.00 PM 5.00 PM 6.00 PM

Time of the Day

0

2

4

6

8

10

Number of Votlage Violation

Actual Number of Voltage Violations

Predicted Number of Voltage Violation

Fig. 2: Number of voltage violations

12.00 PM 1.00 PM 2.00 PM 3.00 PM 4.00 PM 5.00 PM 6.00 PM

Time of the Day 

.095

0.975

1.0

1.025

1.05

1.075

Voltage (p.u.)

Voltage after control

Voltage without any control

Fig. 3: Voltage at node 65 before and after control

V. CONCLUSION AND FUTURE-WORK

This paper develops a preemptive voltage control method

that mitigates voltage violation in a power distribution system

with renewable generation by taking a control action before

actual voltage violation. This paper uses an analytical method

of voltage sensitivity analysis with phasor measurements and

nodal power injection forecasts to compute probability of

future voltage violations. With this foresight, vulnerable nodes

prone to voltage violations are detected (greater than certain

probability of voltage violation), and appropriate power injection control actions can be taken. The paper illustrated a simple

preemptive control approach that is capable of mitigating

future voltage issues altogether, thereby laying the foundation for future investigations on real-time optimal control

strategies. Related future work will also include improving

voltage sensitivities prediction using empirical probabilities

(rather than gaussian distributions) and developing methods to

ensure complete system observability even under sparse sensor

proliferation scenarios.

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5

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.