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Efficient Computation of Sensitivity Coefficients ofNode Voltages and Line Currents in Unbalanced

Radial Electrical Distribution NetworksKonstantina Christakou, Member, IEEE, Jean-Yves Le Boudec, Fellow, IEEE, Mario Paolone, Senior Member,

IEEE, Dan-Cristian Tomozei, Member, IEEE

Abstract—The problem of optimal control of power distri-bution systems is becoming increasingly compelling due tothe progressive penetration of distributed energy resources inthis specific layer of the electrical infrastructure. Distributionsystems are, indeed, experiencing significant changes in terms ofoperation philosophies that are often based on optimal controlstrategies relying on the computation of linearized dependenciesbetween controlled (e.g. voltages, frequency in case of islandingoperation) and control variables (e.g. power injections, trans-formers tap positions). As the implementation of these strategiesin real-time controllers imposes stringent time constraints, thederivation of analytical dependency between controlled andcontrol variables becomes a non-trivial task to be solved. Withreference to optimal voltage and power flow controls, this paperaims at providing an analytical derivation of node voltage andline current flows as a function of the nodal power injectionsand transformers tap-changers positions. Compared to otherapproaches presented in the literature, the one proposed hereis based on the use of the [Y] compound matrix of a genericmulti-phase radial unbalanced network. In order to estimate thecomputational benefits of the proposed approach, the relevantimprovements are also quantified versus traditional methods.The validation of the proposed method is carried out by usingboth IEEE 13 and 34 node test feeders. The paper finally showsthe use of the proposed method for the problem of optimal voltagecontrol applied to the IEEE 34 node test feeder.

Index Terms—Voltage/current sensitivity coefficients, unba-lanced electrical distribution networks, power systems optimaloperation, smart grids.

I. INTRODUCTION

O ptimal controls of power systems are often based on thesolution of linear problems that link control variables

to controlled quantities by means of sensitivity coefficients.Typical optimization problems refer to scheduling of gene-rators, voltage control, losses reduction, etc. So far, thesecategories of problems have been commonly investigated inthe domain of high voltage transmission networks. However,during the past years, the increased penetration of distributedenergy resources (DERs) in power distribution systems hasraised the importance of developing optimal control strategiesspecifically applied to the operation of these networks (e.g. [1],[2], [3], [4], [5], [6]). Within this context, it is worth notingthat the solution of optimal problems becomes of interest only

Konstantina Christakou, Mario Paolone, Jean-Yves Le Boudec and Dan-Cristian Tomozei (email: konstantina.christakou@epfl.ch, jean-yves.leboudec@epfl.ch, mario.paolone@epfl.ch, dan-cristian.tomozei @epfl.ch) are with theEcole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland.

if it meets the stringent time constraints required by real-time controls and imposed by the higher dynamics of thesenetworks compared to the transmission ones.

Typical examples of optimal controls that are not yet de-ployed in active distribution networks (ADNs) are voltage andpower flow controls. Usually this category of problems hasbeen addressed in the literature by means of linear-approachesapplied to the dependency between voltages and power flowsas a function of the power injections (e.g. [4], [5], [7], [8]).

The typical approach for the solution of this class of controlproblems is the use of the sensitivity coefficients through anupdated Jacobian matrix derived from the load flow problem[9], [10], [11], [12], [13]. However, from the computationalpoint of view, the main disadvantage of such a category ofmethods is that, for every change in the operation conditionsof the network, an updated Jacobian matrix needs to be built onthe basis of the network state and needs, then, to be inverted.This procedure involves non-trivial computation constraints forthe implementation in real-time centralized or decentralizedcontrollers.

For this reason, the authors of [14] have proposed thedirect computation of voltages and network losses sensitivitycoefficients, based on the Gauss-Seidel formulation of the loadflow problem, by making use of the [Z] matrix of a balancednetwork. Also, in [7] it has been proposed the use of the [Z]matrix along with the constant-current model for loads andgenerators. In [8] the sensitivity coefficients are proposed tobe calculated starting from the network branch currents. Theapproach presented in [15], [16], [17], [18], [19] belongs to aclass of methods typically derived from circuit theory and isbased on the use of the so-called adjoint network.

In order to increase the computational efficiency of thiscategory of approaches, and to extend it to the inherent multi-phase unbalanced configuration of distribution networks, themain contribution of this paper is to provide a straightforwardanalytical derivation of node voltages and line currents sensi-tivities as a function of the power injections and transformerstap-changers positions. To this end, we propose to use theso-called [Y] compound matrix, which has the advantage ofbeing sparse.

Compared to [7] the approach here proposed takes intoaccount the whole admittance matrix of the network. Onthe other hand the analytical derivation of sensitivities in [7]was based on the approximated representation of the network

3

lines where lines shunt parameters are neglected1. The methodpresented in [8] always requires a base-case load flow solutionand it relies on the assumption that all generators are PV nodes(i.e. with fixed voltage magnitude). Also, it does not accountfor the mutual coupling between different phase conductors.

The approach that appears the more general among theabove listed is the one proposed in [14]. However, this methoddepends on a pseudo-load flow approach (i.e. it makes useof a Gauss-Seidel iterative process with a fixed number ofiterations) which influences the accuracy of the computedcoefficients. Furthermore, compared to [14] we have beenable to:• generalize the problem formulation for a generic number

of slack busses;• extend the computation of sensitivities to tap-changers

positions (i.e. changes of slack busses reference volt-ages);

• provide the proof that the analytical computation ofsensitivities admits a unique solution for the case of radialnetworks and

• take into account the inherent multiphase and unbalancednature of distribution networks.

The structure of the paper is the following: Section IIfocuses on the problem formulation by describing, in detail,the analytical procedure at the base of the proposed method. Italso includes a proof of uniqueness of the solution of the linearsystem used to calculate the sensitivity coefficients for the caseof radial networks. The same section also provides a computa-tional cost analysis of the proposed method versus traditionalapproaches. Section III validates the proposed method usingthe IEEE 13 and 34 node test feeders. Section IV shows anapplication example of sensitivity coefficients related to theoptimal voltage control in unbalanced distribution networkstaking advantage of the possibility of computing them forall the phases. Section V provides the final remarks aboutpossible applications of the proposed method.

II. PROBLEM FORMULATION

A. Classical Computation of Sensitivity Coefficients in PowerNetworks

In this paragraph we make reference to a balanced networkcomposed by K busses.

Traditionally, there are three proposed ways to calculatethe sensitivity coefficients of our interest. The first methodconsists of estimating them by a series of load flow calcula-tions each performed for a small variation of a single controlvariable (i.e. nodal power injections, Pl, Ql) [4]:2

∂|Ei|∂Pl

=∆|Ei|∆Pl

∣∣∣∣∆Pi,i 6=l=0∆Qi,i 6=l=0

∂|Iij |∂Pl

=∆|Iij |∆Pl

∣∣∣∣∆Pi,i 6=l=0∆Qi,i 6=l=0

(1)

∂|Ei|∂Ql

=∆|Ei|∆Ql

∣∣∣∣∆Pi,i 6=l=0∆Qi,i 6=l=0

∂|Iij |∂Ql

=∆|Iij |∆Ql

∣∣∣∣∆Pi,i 6=l=0∆Qi,i 6=l=0

1It is important to observe that line shunt parameters are non-negligible incase of networks characterized by the presence of coaxial cables. These typesof components are typical in the context of urban distribution networks

2In the rest of the paper complex numbers are denoted with a bar above(e.g. E) and complex conjugates with a bar below (e.g. E

¯).

where Ei is the direct sequence phase-to-ground voltage ofnode i and Iij is the direct sequence current flow betweennodes i and j (i, j ∈ {1 · · ·K}).

The second method uses the Newton Raphson formulationof the load flow calculation to directly infer the voltagesensitivity coefficients as submatrices of the inverted Jacobianmatrix (e.g. [9], [10], [11], [12], [13]):

J =

∂P

∂|E|∂P

∂θ

∂Q

∂|E|∂Q

∂θ

. (2)

It is worth observing that such a method does not allow tocompute the sensitivities against the transformers tap-changers

positions. Additionally, as known, the submatrix∂Q

∂|E|is

usually adopted to express voltage variations as a function ofreactive power injections when the ratio of longitudinal lineresistance versus reactance is negligible. It is worth notingthat such an assumption is no longer applicable to distributionsystems that require in addition to take into account activepower injections.

A third method is derived from circuit theory. In thismethod Tellegen’s theorem is applied in power networks andthe computation of sensitivities relies on the concept of theso-called adjoint networks (e.g. [15], [16], [17], [18], [19]).This approach requires a base-case load flow solution in orderto build a specific adjoint network that needs to be solved inorder to infer the desired sensitivities.

B. Analytical Derivation of Voltage and Current SensitivityCoefficients

This subsection contains the main analytical developmentof this paper related to the derivation of the voltage sensitivitycoefficients 3.

1) Voltage Sensitivity Coefficients: the analysis starts withthe voltage sensitivity coefficients. To this end, we derivemathematical expressions that link bus voltages to bus activeand reactive power injections. For this purpose, a K-bus 3-phase generic electrical network is considered. The followinganalysis treats each phase of the network separately and, thus,it can be applied to unbalanced networks.

As known, the equations that link the voltage of each phaseof the busses to the corresponding injected current are in totalM = 3K and they are given by:

[Iabc] = [Yabc] · [Eabc] (3)

where [Iabc] = [I1a , I

1b , I

1c ..., I

Ka , I

Kb , I

Kc ]T , [Eabc] =

[E1a, E

1b , E

1c ..., E

Ka , E

Kb , E

Kc ]T . We denoted by a, b, c the

three network phases. The [Yabc] matrix is formed by usingthe so-called compound admittance matrix (e.g. [20]) as

3As shown in subsection II-B2 the current sensitivities can be straightfor-wardly derived from the voltage ones.

4

follows:

[Yabc

]=

Y 11aa Y 11

ab Y 11ac · · · Y 1K

aa Y 1Kab Y 1K

ac

Y 11ba Y 11

bb Y 11bc · · · Y 1K

ba Y 1Kbb Y 1K

bc

Y 11ca Y 11

cb Y 11cc · · · Y 1K

ca Y 1Kcb Y 1K

cc

......

......

......

...Y K1aa Y K1

ab Y K1ac · · · Y KK

aa Y KKab Y KK

ac

Y K1ba Y K1

bb Y K1bc · · · Y KK

ba Y KKbb Y KK

bc

Y K1ca Y K1

cb Y K1cc · · · Y KK

ca Y KKcb Y KK

cc

.

In order to simplify the notation, in what follows we willassume the following correspondences: [Iabc] = [I1, ..., IM ]T ,[Eabc] = [E1, ..., EM ]T and

[Yabc

]=

Y11 · · · Y1M

... · · ·...

Y1M · · · YMM

.For the rest of the analysis we will consider the networkas composed by S slack busses and N busses with PQinjections, (i.e. {1, 2, · · ·M} = S ∪ N , with S ∩ N = ∅).The PQ injections are considered constant and independentof the voltage. In this respect, we are assuming that foreach separate perturbation of nodal power injections, the otherloads/generators do not change their power set points. There-fore, the computation of the sensitivities inherently accountsfor the whole response of the network in terms of variationof both active and reactive power flows. Such a consequenceallows to compute the sensitivities in the close vicinity of thenetwork state.

The link between power injections and bus voltages reads:

S¯ i

= E¯ i

∑j∈S∪N

YijEj , i ∈ N . (4)

The derived system of equations (4) holds for all the phasesof each bus of the network. Since the objective is to calculatethe partial derivatives of the voltage magnitude over the activeand reactive power injected in the other busses, we have toconsider separately the slack bus of the system. As known,the assumptions for the slack bus equations are to keep itsvoltage constant and equal to the network rated value, by alsofixing its phase equal to zero. Hence, for the three phases ofthe slack bus, it holds that:

∂Ei∂Pl

= 0 ,∀i ∈ S. (5)

At this point, by using equation (4) as a starting point onecan derive closed-form mathematical expressions to defineand quantify voltage sensitivity coefficients with respect toactive and reactive power variations in correspondence ofthe N busses of the network. To derive voltage sensitivitycoefficients, the partial derivatives of the voltages with respectto the active and reactive power Pl and Ql of a bus l ∈ Nhave to be computed. The partial derivatives with respect toactive power satisfy the following system of equations:

1{i=l} =∂E

¯ i∂Pl

∑j∈S∪N

YijEj + E¯ i∑j∈N

Yij∂Ej∂Pl

(6)

where it has been taken into account that:

∂S¯ i

∂Pl=∂{Pi − jQi}

∂Pl= 1{i=l}. (7)

The system of equations (6) is not linear over complexnumbers, but it is linear with respect to ∂Ei

∂Pl,∂E¯ i

∂Pl, therefore

it is linear over real numbers with respect to rectangularcoordinates. As we show next, it has a unique solution andcan therefore be used to compute the partial derivatives inrectangular coordinates to reduce the computational effort.

A similar system of equations holds for the sensitivitycoefficients with respect to the injected reactive power Ql.With the same reasoning, by taking into account that:

∂S¯ i

∂Ql=∂{Pi − jQi}

∂Ql= −j1{i=l} (8)

we obtain that:

−j1{i=l} =∂E

¯ i∂Ql

∑j∈S∪N

YijEj + E¯ i∑j∈N

Yij∂Ej∂Ql

. (9)

By observing the above linear systems of equations (6) and(9), we can see that the matrix that needs to be inverted inorder to solve the system is fixed independently of the powerof the l-th bus with respect to which we want to compute thepartial derivatives. The only element that changes is the lefthand side of the equations.

Once ∂Ei

∂Pl,∂E¯ i

∂Plare obtained, the partial derivatives of the

voltage magnitude can be expressed as:

∂|Ei|∂Pl

=1

|Ei|Re(E

¯ i∂Ei∂Pl

) (10)

and similar equations hold for derivatives with respect toreactive power injections.

Theorem 1: The system of equations (6), where l is fixedand the unknowns are ∂Ei

∂Pl, i ∈ N , has a unique solution for

every radial electrical network. The same holds for the systemof equations (9), where the unknowns are ∂Ei

∂Ql, i ∈ N .

Proof: Since the system is linear with respect to rectangu-lar coordinates and there are as many unknowns as equations,the theorem is equivalent to showing that the correspondinghomogeneous system of equations has only the trivial solution.The homogeneous system can be written as:

0 = ∆¯ i

∑j∈S∪N

YijEj + E¯ i∑j∈N

Yij∆j , ∀i ∈ N (11)

where ∆i are the unknown complex numbers, defined for i ∈N . We want to show that ∆i = 0 for all i ∈ N . Let usconsider two electrical networks with the same topology, i.e.same [Yabc] matrix, where the voltages are given. In the firstnetwork, the voltages are

E′i = Ei , ∀i ∈ SE′i = Ei + ∆i , ∀i ∈ N (12)

and in the second network they are

E′′i = Ei , ∀i ∈ SE′′i = Ei − ∆i , ∀i ∈ N (13)

5

Let S¯′i be the conjugate of the absorbed/injected power at the

ith bus in the first network, and S¯′′i in the second. Apply

equation (4) to bus i ∈ N in the first network:

S¯′i = E

¯′i

∑j∈S∪N

YijE′j

= (E¯ i

+ ∆¯ i

)

(∑j∈S

YijEj +∑j∈N

Yij(Ej + ∆j)

)= E

¯ i∑

j∈S∪NYijEj + ∆

¯ i∑j∈N

Yij∆j

+ ∆¯ i

∑j∈S∪N

YijEj + E¯ i∑j∈N

Yij∆j

Similarly, for the second network and for all busses i ∈ N :

S¯′′i = E

¯ i∑

j∈S∪NYijEj + ∆

¯ i∑j∈N

Yij∆j

− ∆¯ i

∑j∈S∪N

YijEj − E¯ i∑j∈N

Yij∆j

Subtract the last two equations and obtain

S¯′i − S¯

′′i = 2

(∆¯ i

∑j∈S∪N

YijEj + E¯ i∑j∈N

Yij∆j

)

By equation (11), it follows that S¯′i = S

¯′′i for all i ∈ N . Thus

the two networks have the same active and reactive powers atall non slack busses and the same voltages at all slack busses.As discussed in [21] for radial distribution networks such anassumption means that the load flow problem always has aunique solution. Therefore, it follows that the voltage profileof these networks must be exactly the same, i.e. Ei − ∆i =Ei + ∆i for all i ∈ N and thus ∆i = 0 for all i ∈ N .

2) Current Sensitivity Coefficients: From the previous ana-lysis, the sensitivity coefficients linking the power injectionsto the voltage variations are known. Thus, it is straightforwardto express the branch current sensitivities with respect to thesame power injections. Assuming to represent the lines thatcompose the network by means of π models, the current flowIij between nodes i and j can be expressed as a function of thephase-to-ground voltages of the relevant i, j nodes as follows:

Iij = Yij(Ei − Ej) (14)

where Yij is the generic element of [Yabc] matrix betweennode i and node j.

Since the voltages can be expressed as a function of thepower injections into the network busses, the partial derivativesof the current with respect to the active and reactive powerinjections in the network can be expressed as:

∂Iij∂Pl

= Yij(∂Ei∂Pl− ∂Ej∂Pl

)

∂Iij∂Ql

= Yij(∂Ei∂Ql

− ∂Ej∂Ql

)

. (15)

Applying the same reasoning as earlier, the branch currentsensitivity coefficients with respect to an active power Pl canbe computed using the following expressions:

∂|Iij |∂Pl

=1

|Iij |Re(I

¯ ij∂Iij∂Pl

). (16)

Similar expressions can be derived for the current coefficientswith respect to the reactive power in the busses as:

∂|Iij |∂Ql

=1

|Iij |Re(I

¯ ij∂Iij∂Ql

). (17)

C. Sensitivity Coefficients with respect to tap positions oftransformers

This subsection is devoted to the derivation of analyticalexpressions for the voltage sensitivity coefficients4 with re-spect to tap positions of a transformer. We assume thattransformers tap-changers are located in correspondence ofthe slack busses of the network as for distribution networksthese represent the connections to external transmission orsub-transmission networks. As a consequence, the voltagesensitivities as a function of the tap positions are equivalentto the voltage sensitivities as a function of the slack referencevoltage. We assume that the transformers voltage variationsdue to tap position changes are small enough so that thepartial derivatives considered in the following analysis aremeaningful. Furthermore, we assume that the power injectionsat the network busses are constant and independent of thevoltage.

With the same reasoning as in Sec. II-B, the analysis startsin equation (4). We write E` = |E`|ejθ` for all busses `. Fora bus i ∈ N the partial derivatives with respect to the voltagemagnitude |Ek| of a slack bus k ∈ S are considered:

−E¯ iYike

jθk = W¯ ik

∑j∈S∪N

YijEj + E¯ i∑j∈N

YijWjk, (18)

where

Wik :=∂Ei∂|Ek|

=

(1

|Ei|∂|Ei|∂|Ek|

+ j∂θi∂|Ek|

)Ei, i ∈ N .

We have taken into account that:

∂

∂|Ek|∑j∈S

YijEj = Yikejθk (19)

and

∂S¯ i

∂|Ek|= 0. (20)

The derived system of equations (18) is linear with respectto W

¯ ikand Wik, and has the same associated matrix as the

system in (6). Since the resulting homogeneous system ofequations is identical to the one in (11), by Theorem 1 it hasa unique solution.

After resolution of (18), we find that the sensitivity coef-ficients with respect to the tap position of the transformer atbus k are given by

∂|Ei|∂|Ek|

= |Ei|Re(Wik

Ei

). (21)

4Note as shown in Sec. II-B2 once the voltage sensitivities are obtainedthe ones of currents can be computed directly.

6

D. Computational Cost Analysis for Voltage Sensitivities withrespect to PQ injections

The aim of this subsection is to show the computationaladvantage of the proposed method compared to the classicalapproach with respect to the computation of voltage sensiti-vities as a function of power injections only5. Furthermore,the two methods are applied to the IEEE 13 and 34 nodetest feeders and compared in terms of CPU time necessary tocalculate the voltage sensitivity coefficients.

We are assuming that:1) there are loads/injections in all three phases of the

system and2) the phasors of phase-to-ground voltages in all the net-

work are known (e.g. coming from a state estimationprocess [22]).

In the following table, Algorithm 1 shows the steps requiredto calculate the voltage sensitivity coefficients using the tradi-tional method and Algorithm 2 shows the corresponding stepsusing the analytical method proposed here.

For the traditional method an updated Jacobian needs tobe built, and its inverse will provide the desired voltagesensitivities. For the analytical method the correspondingsteps refer to invert a square matrix of size 2N (as reportedin Section II-B1 N refers to the number of network busseswith PQ injections) and multiply the inverse matrix with onecolumn vector for each PQ bus in the network.

Algorithm 1 Computation of voltage sensitivity coefficientsusing the Jacobian method

1: build Jacobian matrix associated to the Newton Raphsonmethod

2: invert matrix J of size 2N × 2N3: extract the sub-matrices corresponding to the desired sen-

sitivity coefficients

Algorithm 2 Computation of voltage sensitivity coefficientsusing the analytical method

1: build the matrix of the linear system of equations2: invert matrix of size 2N × 2N3: do N multiplications of the inverse matrix with vectors of

size 2N × 1

In Table I the mean CPU time necessary to calculate thevoltage sensitivity coefficients is presented for the IEEE 13and 34 node test feeders respectively, when 1000 iterations ofthe method are executed. It can be observed that the analyticalapproach exhibits an improvement of performance which isof 2.34 for the IEEE 13 node test feeder and 2.52 for theIEEE 34 node test feeder. In the same table the relevant95% confidence intervals are also reported for the computationof the coefficients for the two benchmark feeders. One canobserve the advantage of the proposed analytical method asthe number of busses in the network increases. It is worthobserving that such an improvement depends not only on

5As already pointed out in Sec.II-A traditional Jacobian based sensitivitycomputations do not account indeed the variations of tap-changers.

the number of busses but also on the network topology (i.e.sparsity of the [Y] admittance matrix).

Table ICPU TIME NECESSARY FOR CALCULATING VOLTAGE SENSITIVITY

COEFFICIENTS IN THE IEEE 13 AND THE 34 NODE TEST FEEDERS WHENALL PHASES OF ALL BUSSES HAVE LOADS

Jacobian Analytical ratio13 bus feeder 28.8 ± 0.18 msec 12.5 ± 0.43 msec 2.3434 bus feeder 209.8 ± 1.30 msec 83.4 ± 0.59 msec 2.52

III. NUMERICAL VALIDATION

The numerical validation of the proposed method for thecomputation of voltage/current sensitivities is performed withtwo different approaches. In particular, as the inverse of theload flow Jacobian matrix provides the voltage sensitivities,the comparison reported below makes reference to such amethod for the voltage sensitivities only. On the contrary, asthe inverse of the load flow Jacobian matrix does not providecurrent sensitivity coefficients, their accuracy is evaluated byusing a numerical approach where the load flow problem issolved by applying small injection perturbations into a givennetwork (see Section II-A). A similar approach is deployedto validate the sensitivities with respect to tap positions of thetransformers, i.e. small perturbations of the voltage magnitudeof one phase of the slack bus and solution of the loadflow problem. Fig.1 shows the IEEE 13 nodes test feederimplemented in the EMTP-RV simulation environment ([23],[24], [25]) adopted to perform the multiphase load flow.

RL

+

PI1

_2

00

0

LF

LF

2

Sla

ck:

4.1

6kV

RM

SLL/_

0

Phase:0

RL

+

PI3_300

RL

+

PI6

_1

00

0

LF

140kW

110kVAR

Load1

LF

180kW

130kVAR

Load2

LF

140kW

110kVAR

Load3

RL

+

PI2_500

LF

88kW

80kVAR

Load4

LF

610kW

210kVAR

Load6

LF

310kW

232kVAR

Load7

LF

86kW

58kVAR

Load8

LF

137kW

88kVAR

Load9LF

37kW

30kVAR

Load10

LF

190kW

171kV

AR

Lo

ad

11

LF

190kW

145kV

AR

Lo

ad

15

RL

+

PI7

_1

00

0

LF

250kW

152kV

AR

Lo

ad

14

RL

+

PI5

_2

00

0

RL

+

PI1_300

RL

+

PI1_300

RL

+

PI1_500

RL

+

PI1_500

RL

+

PI1_500

RL

+

PI1_500

LF

Lo

ad

13

148kW

106kV

AR

LF Load5

405kW

240kVARLF Load16

405kW

240kVAR

LF Load17

405kW

240kVAR

BUS650_01

BUS632_02

ca b

BUS645_05

b

BUS634_04

b

c

aBUS646_06

b

BUS692_08

c

BUS611_11

BUS671_07

ba

c

BUS680_12

BUS652_13

a

BUS675_09a

b

c

BUS684_10

BUS633_03

Figure 1. IEEE 13 node test feeder represented in the EMTP-RV simulationenvironment.

7

−600 −400 −200 0 200 400 6000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Active power generation/absorption at phase b of bus 9 (kW)

Vol

tage

sen

sitiv

ity o

f pha

se b

of b

us 8

(p.

u.)

0

0.5

1

1.5

2

2.5

3x 10−6

Err

or b

etw

een

Jaco

bian

and

Ana

lytic

al m

etho

d

Sensitivity coefficientError

(a)

−600 −400 −200 0 200 400 600

0.22

0.24

0.26

0.28

0.3

0.32

0.34

Active power generation/absorption at phase b of bus 9 (kW)

Vol

tage

sen

sitiv

ity o

f pha

se α

of b

us 8

(p.

u.)

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−7

Err

or b

etw

een

Jaco

bian

and

Ana

lytic

al m

etho

d

ErrorSensitivity coefficient

(b)

0.8

1

1.2

1.4

1.6

1.8

2

Reactive power generation/absorption at phase b of bus 9 (kVar)

Vol

tage

sen

sitiv

ity o

f pha

se b

of b

us 8

(p.

u.)

−600 −400 −200 0 200 400 600

0

2

4

6

8

10

12x 10−6

Err

or b

etw

een

Jaco

bian

and

Ana

lytic

al m

etho

d

ErrorSensitivity coefficient

(c)

−0.42

−0.4

−0.38

−0.36

−0.34

−0.32

−0.3

Reactive power generation/absorption at phase b of bus 9 (kVar)

Vol

tage

sen

sitiv

ity o

f pha

se α

of b

us 8

(p.

u.)

−600 −400 −200 0 200 400 600−16

−14

−12

−10

−8

−6

−4

−2

0x 10

−7

Err

or b

etw

een

Jaco

bian

and

Ana

lytic

al m

etho

d

Sensitivity coefficientError

(d)

Figure 2. Voltage sensitivity coefficient of phase a and b of bus 8 withrespect to active and reactive power generation/absorption at phase b of bus9.

For the sake of brevity we limit the validation of theproposed method to a reduced number of busses exhibitingthe largest voltage sensitivity against PQ load/injections. Inparticular, we refer to the variation of voltages at bus 8 withrespect to load/injection in bus 9, i.e.

∂|Ea8 |∂P b9

,∂|Eb8|∂P b9

,∂|Ea8 |∂Qb9

,∂|Eb8|∂Qb9

In Fig.2(a) the voltage sensitivity of phase b bus 8 is shownwith respect to active power absorption and generation at phaseb of bus 9. We assume the convention that positive values ofP and Q denote power absorption, whereas negative valuescorrespond to power generation. Fig.2(b) shows for the samebusses as Fig.2(a), the same sensitivity but referring to voltageand power belonging to different phases. Additionally, Fig.2(c) and 2(d) show the voltage sensitivity of bus 8 with respectto reactive power absorption and generation at bus 9. In allthese four figures the dashed line represents the relative errorbetween the traditional approach (i.e. based on the inverseof the Jacobian matrix) and the analytical method proposedhere. As it can be observed, the overall errors are in the orderof magnitude of 10−6. In Fig.3(a) and Fig.3(b) the currentsensitivity coefficient of phase a of branch 10−13 is presentedwith respect to active and reactive power absorption/generationat phase a of bus 13. In the same figures, the dashed linesrepresent the relative error between the analytical values and

−600 −400 −200 0 200 400 600−3

−2

−1

0

1

2

Active power generation/absorption at phase α of bus 13 (kW)

Cur

rent

sen

sitiv

ity c

oeffi

cien

t of p

hase

α

of b

ranc

h 10

−13

(p.

u.)

−600 −400 −200 0 200 400 600

−10

−8

−6

−4

−2

0

2x 10−4

Err

or b

etw

een

Ana

lytic

al a

nd n

umer

ical

met

hod

ErrorSensitivity coefficient

(a)

−600 −400 −200 0 200 400 600−4

−3

−2

−1

0

1

2

Reactive power generation/absorption at phase α of bus 13 (kVar)

Cur

rent

sen

sitiv

ity c

oeffi

cien

t of p

hase

α

of b

ranc

h 10

−13

(p.

u.)

−600 −400 −200 0 200 400 600

−8

−6

−4

−2

0

2x 10−4

Err

or b

etw

een

Ana

lytic

al a

nd n

umer

ical

met

hod

Sensitivity coefficientError

(b)

Figure 3. Current sensitivity coefficients of phase a of branch 10-13 withrespect to power generation/absorption at phase a of node 13.

8

the numerical ones. Even for these coefficients extremely lowerrors are obtained.

Concerning the validation of voltage sensitivities againsttap-changer positions, we have made reference to the IEEE 13node test feeder where the slack bus and therefore the primarysubstation transformer is placed in correspondence of node 1.We assume to vary the slack bus voltage of ±6% over 72tap positions (where position ”0” refers to the network ratedvoltage). In Fig. 4 the sensitivity of voltage in phase a ofbus 7 is shown w.r.t. the tap positions in phase a, b and cof the slack. Also, in this case the difference between theanalytically inferred sensitivities and the numerical computedones is negligible (i.e. in the order of magnitude of 10−4).

Vo

ltag

e se

nsi

tivi

ty o

f p

has

e a

of

bu

s 7

Transformer's tap positions phase a

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

7.00E-04

8.00E-04

1.070

1.080

1.090

1.100

1.110

1.120

1.130

1.140

1.150

-38 -34 -30 -26 -22 -18 -14 -10 -6 -2 2 6 10 14 18 22 26 30 34 38

Sensitivity coefficient

Error

x10-4

Voltage s

ensitiv

ity o

f p

hase α

of bus 7

Transformer's tap positions phase α

(a) Voltage sensitivity coefficient of phase a of bus 7 with respect totransformer’s tap position at phase a of the slack bus.

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

1.00E-05

1.20E-05

1.40E-05

1.60E-05

1.80E-05

2.00E-05

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

-38 -34 -30 -26 -22 -18 -14 -10 -6 -2 2 6 10 14 18 22 26 30 34 38

Sensitivity coefficient

Error

0.8

0.6

0.4

0.2

x10-5

Voltage

sensitiv

ity o

f p

hase α

of bus 7

Transformer's tap positions phase b

(b) Voltage sensitivity coefficient of phase a of bus 7 with respect totransformer’s tap position at phase b of the slack bus.

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

-0.06

-0.055

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-38 -34 -30 -26 -22 -18 -14 -10 -6 -2 2 6 10 14 18 22 26 30 34 38

Sensitivity coefficient Error

0.5

Transformer's tap positions phase c

Voltage s

en

sitiv

ity o

f ph

ase α

of bus 7

x10-4

(c) Voltage sensitivity coefficient of phase a of bus 7 with respect totransformer’s tap position at phase c of the slack bus.

Figure 4. Voltage sensitivity coefficient of phase a of bus 7 with respect totransformer’s tap positions.

2

7 10

13

3 4 5

6

12 8 9

11

0.45

0.55

0.65

0.75

0.85

0.95

1.05

1.15

1.25

1500 2500 3500 4500 5500 Distance from slack (ft)

Voltage s

ensitiv

ity c

oeffic

ients

(p.u

.)

(a) Voltage sensitivity coefficients ∂|Eai |

∂Pa13

with respectto active power absorption at phase a of node 13 as afunction of the distance from the slack bus.

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2 1500 2500 3500 4500 5500

2

7 10

13

3 4 6 5

11

8

9

12

Voltage s

ensitiv

ity c

oeff

icie

nts

(p.u

.)

Distance from slack (ft)

(b) Voltage sensitivity coefficients ∂|Ebi |

∂Pa13

with respectto active power absorption at phase a of node 13 as afunction of the distance from the slack bus.

5500

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

1500 2500 3500 4500 5500

2

7 10

13

3 4

5 6

11

8

9

12

Voltage s

ensitiv

ity c

oeff

icie

nts

(p.u

.)

Distance from slack (ft)

(c) Voltage sensitivity coefficients ∂|Eai |

∂Qa13

with respectto reactive power absorption at phase a of node 13 asa function of the distance from the slack bus.

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

1500 2500 3500 4500 5500

2

7 10

13

3 4

5 6

8

9

12 11

Voltage s

ensitiv

ity c

oeff

icie

nts

(p.u

.)

Distance from slack (ft)

(d) Voltage sensitivity coefficients ∂|Ebi |

∂Qa13

with respectto reactive power absorption at phase a of node 13 asa function of the distance from the slack bus.

Figure 5. Voltage sensitivity coefficients with respect to power absorptionat phase a of bus 13 as a function of the distance from the slack bus.

9

It is worth observing that for the case of the voltagesensitivities, coefficients that refer to the voltage variationas a function of a perturbation (power injection or tap-changer position) of the same phase, show the largest couplingalthough a non-negligible cross dependency can be observedbetween different phases.

Finally, Fig.5 depicts the variation of voltage sensitivitycoefficients in all the network with respect to active andreactive power absorption at phase a of bus 13 as a functionof the distance from the slack bus in feet.

This type of representation allows to observe the over-all network behavior against specific PQ busses absorp-tions/injections. In particular, we can see that larger sensiti-vities are observed when the distance between the consideredvoltage and the slack bus increases. Furthermore, a lower, butquantified dependency between coefficients related to differentphases, can be observed. Also, as expected, reactive powerhas a larger influence on voltage variations although the activepower exhibits a non negligible influence.

From the operational point of view it is worth observingthat, figures as Fig.5, provide to network operators an im-mediate view of the response of the electrical network againstspecific loads/injections that could also be used for closed loopcontrol or contingency analysis.

RL

+

PI1

RL

+

PI2

RL

+

PI3

RL

+

PI4

RL

+

PI5

RL

+

PI6

RL

+

PI7

RL

+

PI8

RL

+

PI9

RL

+

PI10

RL

+

PI11

RL

+

PI1

2

RL

+

PI1

3

RL

+

PI14

RL

+

PI1

5

RL

+

PI1

6

RL

+

PI1

7

RL

+

PI18

RL

+

PI19

RL

+

PI20

RL

+

PI2

3

RL

+

PI2

4

RL

+

PI26

RL

+

PI28

RL

+

PI32

RL

+

PI33

LF

Load1LF

Load2

LF

Load3

LF

Load10

LF

Load11

LF

Load12

LF

Load16

LF

Load17

LF

Load23

LF

Load24

LF

Load25

LF

Load27

LF

Load31

LF

Load33

LF

Load34RL

+

PI27

LF

LF

1

Sla

ck: 24.9

kV

RM

SLL/_

0

LF

Load22

LF

Load19

RL

+

PI29

LF

Load6

LF

Load5

LF

Load26

LF

Load18

RL

+

PI3

0

LF

Load15

LF

Load14

LF

Load13

LF

Load29

LF

Load4

RL

+

PI2

5

LF

Load35

RL

+

PI3

1

LF

Load7

LF Load8

LF

Load9

RL

+

PI3

4

RL

+

PI22

LF

Load30

LFP=300kW

Q=0

DER_33

LF

Phase:0

P=600kW

Q=0

DER_24

LF

Phase:0

P=

300kW

Q=

0

DE

R_18

LF

Phase:0

P=

300kW

Q=

0

DE

R_23

LF

Load20

BUS800_01

BUS802_02

c b

BUS810_05

BUS812_06

BUS814_07

BUS850_08

BUS818_10 BUS820_11

aBUS822_12

a

BUS856_18

BUS840_34ca b

BUS852_19

BUS842_26BUS846_28

BUS848_29a

b

c

BUS862_32

BUS838_33

BUS806_03

BUS808_04

BUS816_09 b

BUS826_14

b

BUS824_13

BUS830_16

a b c

BUS828_15

ac

BUS854_17

BUS860_30

ba c

BUS832_20

c

a

b

BUS890_22

a b c

BUS836_31

BUS844_27

c

b

aBUS834_25

cba

BUS888_21

BUS858_23 BUS864_24

Figure 6. IEEE 34 node test feeder represented in the EMTP-RV simulationenvironment.

IV. APPLICATION OF THE PROPOSED PROCEDURE TO THEPROBLEM OF OPTIMAL VOLTAGE CONTROL

For the application part, the IEEE 34 test node feeder isconsidered as depicted in Fig.6. In busses 18, 23, 24 and33 we assume to have distributed energy resources that theDistribution Network Operator (DNO) can control in termsof active and reactive power. Their initial operating values,as well as their rated power outputs, are shown in Table II.Furthermore, the DNO has control on the transformer’s tappositions.

Table IIINITIAL AND MAXIMUM OPERATIONAL SET POINTS OF THE DERS AND

THE TAP-CHANGERS IN THE 34 TEST NODE FEEDER

Pinit(kW) Pmax(kW) ninit nmin nmax

DER18 210 300

0 −36 +36DER23 100 600DER24 250 600DER33 150 300

The optimal control problem is formulated as a linearone taking advantage of the voltage sensitivity coefficients.The controlled variables are the bus node voltages and thecontrol variables are the active and reactive power injectionsof the DER and the transformer’s tap positions under thecontrol of the DNO, ∆x = [∆PDER,∆QDER,∆n]. Itis important to state that, formally, this problem is a mixedinteger optimization problem due to the tap positions of thetransformers. However, for reasons of simplicity, the tappositions are considered pseudo-continuous variables whichare rounded to the nearest integer once the optimal solutionis reached. The objective of the linear optimization problemrelevant to the problem is:

min∆x‖ Ei − E ‖ (22)

The linearized relationship that links bus voltages with controlvariables is expressed in the following way (e.g. [4]):

∆|Ei| = KPi∆Pi + KQi∆Qi + Kni∆ni (23)

where KPi is the vector of sensitivity coefficients with respectto the active powers of the DERs, KQi is the vector ofsensitivity coefficients with respect to the reactive powers ofthe DERs and Kni is the vector of sensitivity coefficientswith respect to the transformer’s tap positions. The imposedconstraints on the operational points of the DERs and the tappositions are the following:

0 ≤ PDERi≤ PDERimax

(24)QDERimin

≤ QDERi≤ QDERimax

nmin ≤ n ≤ nmaxIn order to simplify the analysis, we have assumed that theDER capability curves are rectangular ones in the PQ plane.

The formulated linearized problem is solved by using thelinear least squares method. The method used to calculateanalytically the sensitivity coefficients allows us to considertwo different optimization scenarios. In the first (opt1), theoperator of the system is assumed to control the set pointsof the DERs considering that they are injecting equal powers

10

into the three phases, whereas in the second case (opt2) it isassumed to have a more sophisticated control on each of thephases independently except for the tap-changers positions.It is worth noting that this second option, although far froma realistic implementation, allows us to show the capabilityof the proposed method to deal with the inherent unbalancednature of distribution networks. Table III and Table IV showthe optimal operational set points corresponding to these cases.

Table IIIOPTIMAL OPERATIONAL SET POINTS OF THE DERS AND THE

TAP-CHANGERS IN THE 34 TEST NODE FEEDER WHEN THE SYSTEMOPERATOR HAS CONTROL ON THEIR 3-PHASE OUTPUT

Popt1 (kW) Qopt1 (kVar) nopt1DER18 300 300

-2DER23 600 600DER24 600 264.06DER33 300 -14.46

Additionally, in Fig.7 the voltage profile of the busses ofthe system is presented in the initial and the optimal cases.The solid line in the figures shows the initial voltage profile,the solid line with the markers shows the first case optimalscenario (opt1) and the dashed line represents the second casewhere the DNO has full control in each of the phases of theDERS (opt2). The offset in the graphs, observed in the slackbus, depicts the optimal tap position in each case. What canbe observed is that, when there is a possibility to control eachof the three phases of the DERs output, the optimal voltageprofile is better than the one corresponding to control of the3-phase output of the set points of the DERs.

Table IVOPTIMAL OPERATIONAL SET POINTS OF THE DERS AND THE

TAP-CHANGERS IN THE 34 TEST NODE FEEDER WHEN THE SYSTEMOPERATOR HAS CONTROL ON EACH OF THE THREE PHASES

INDEPENDENTLY

Popt2(kW) Qopt2 (kVar) nopt2DERa

18 100 100

+1

DERb18 100 -88.56

DERc18 0 83

DERa23 200 200

DERb23 200 200

DERc23 0 200

DERa24 200 102.81

DERb24 196.51 200

DERc24 111.40 200

DERa33 100 -27.88

DERb33 100 100

DERc33 98.40 100

V. CONCLUSION

In this paper we have proposed a new method for theanalytical computation of voltages and currents sensitivitycoefficients as a function of the nodal power injections. Thecontributions of the proposed method are the following: (i) itis generalized to account for a generic number of slack busses;(ii) it allows the computation of sensitivities w.r.t. tap-changerpositions (iii) it is proved to admit a unique solution for thecase of radial networks and (iv) it supports the computationof the sensitivities for a generic unbalanced electrical networkby using the [Y] compound matrix being, thus, suitable fordistribution systems.

0.94 0.945

0.95 0.955

0.96 0.965

0.97 0.975

0.98 0.985

0.99 0.995

1 1.005

1.01

0 50000 100000 150000 200000

initial voltage profile

three phase balanced injections

three phase unbalanced injections

Distance from slack bus (ft)

Vo

ltag

e (p

.u.)

Phase a

(a) IEEE 34 node test feeder - Voltage profile of phase a of the busses.

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

0 50000 100000 150000 200000

initial voltage profile

three phase balanced injections

three phase unbalanced injections

Phase b

Distance from slack bus (ft)

Vo

ltag

e (p

.u.)

(b) IEEE 34 node test feeder - Voltage profile of phase b of the busses.

0.945 0.95

0.955 0.96

0.965 0.97

0.975 0.98

0.985 0.99

0.995 1

1.005

0 50000 100000 150000 200000

initial voltage profile three phase balanced injections three phase unbalanced injections

Phase c

Distance from slack bus (ft)

Vo

ltag

e (p

.u.)

(c) IEEE 34 node test feeder - Voltage profile of phase c of the busses.

Figure 7. Initial and optimized voltage profile of the IEEE 34 node testfeeder.

Compared to the traditional use of the Jacobian load-flowmatrix, it allows us to reduce the computation time by almost afactor of three, thus enabling, in principle, its implementationin real-time optimal controllers.

The paper has also validated the proposed method bymaking reference to typical IEEE 13 and 34 nodes distributiontest feeders. The former has been used to numerically validatethe computation of the coefficients whilst the latter has beenused to show an application example related to a possibleintegration of the proposed method for the problem of optimalvoltage control in unbalanced distribution systems.

It is worth observing that the proposed analytical computa-tion of voltages and currents sensitivities enables the reductionof the computational time of several traditional power systems

11

problems involving non-negligible computational efforts, suchas real-time centralized controls, contingency analysis or op-timal planning.

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