Small Signal Stability Analysis of Distribution Networks

This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSG.2017.2772224, IEEETransactions on Smart Grid

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Small Signal Stability Analysis of DistributionNetworks with Electric Springs

Diptargha Chakravorty, Student Member, IEEE, Jinrui Guo, Student Member, IEEE,Balarko Chaudhuri, Senior Member, IEEE, and Shu Yuen Ron Hui, Fellow, IEEE

Abstract—This paper presents small signal stability analysisof distribution networks with electric springs (ESs) installed atthe customer supply points. The focus is on ESs with reactivecompensation only. Vector control of ES with reactive compen-sation is reported for the first time to ensure compatibility withthe standard stability models of other components such as theinterface inverter of distributed generators (DGs). A linearizedstate-space model of the distribution network with multiple ESsis developed which is extendible to include inverter-interfacedDGs, energy storage, active loads etc. The impact of distance ofan ES from the substation, proximity between adjacent ESs andthe R/X ratio of the network on the small signal stability of thesystem is analyzed and compared against the case with equivalentDG inverters. The collective operation of ESs is validated throughsimulation study on a standard distribution network.

Index Terms—Distributed Generation, Dynamic Model, Dis-tribution Network, Electric Spring, Small Signal Stability, StateSpace, Vector Control

I. INTRODUCTION

THe concept of Electric Spring (ES) was introduced toregulate the voltage at the point of coupling (PoC) in the

face of intermittency in renewable (wind, solar) power [1]. Theeffectiveness of ES is demonstrated through proof-of-conceptexperiments with laboratory prototypes [2] and simulationstudies with ES installed in typical distribution networks [3].The contributions of ES in grid frequency regulation [4], powerquality improvement [5], phase imbalance mitigation [6] andresilience enhancement [7] are also reported making ES anattractive prospect in future smart grids.

An ES is a power electronic compensator that injects acontrollable series voltage to decouple the loads from thefeeder/supply. The loads could either be a cluster of customers(domestic/industrial) [8] or individual high power loads (heat-ing, lighting, appliances etc.) [4]. Either way, mass installationof ES would result in a very high penetration of powerelectronic converters in the distribution network. Adverseinteraction between the distributed generator (DG) inverters

This work was supported in part by the Engineering and Physical Sci-ence Research Council, U.K., under the Autonomic Power Systems grantEP/I031650/1, China Scholarship Council and Imperial College London (CSCNo. 201604100122) and the Hong Kong Research Grant Council under theTheme-based project T23-701/14-N.

D. Chakravorty, J. Juo, B. Chaudhuri and S. Y. R. Hui are with theDepartment of Electrical and Electronic Engineering, Imperial College Lon-don, London, SW7 2AZ U.K. (e-mail: [email protected];[email protected]; [email protected]; [email protected]).S.Y.R. Hui is also with the Department of Electrical and Electronic Engineer-ing, The University of Hong Kong.

Supporting data is available on request: please contact [email protected].

and the distribution network is known to cause small signalstability problems. This has been reported in the context ofmicrogrids [9] and distribution networks [10]. Hence, thereis a concern that large penetration of ES, often in closeelectric proximity, could affect the small signal stability of thedistribution networks. Distributed control of multiple ElectricSprings (ESs) for the purpose of voltage and/or frequencyregulation has been studied previously [3], [4], [11], [12].In these studies, an ES installed in series with a load (orcluster of loads) was represented as a controllable current orpower source. However, a framework for small signal stabilityanalysis of a distribution network with multiple electric springsis yet to be reported. As use of ES for voltage/frequencycontrol attracts increasing attention, it is important to studythe overall small signal stability of a distribution networkwith dense penetration of ESs under different scenarios whichmotivates this paper.

This work follows on from previous papers [13] & [14]on modelling and stability analysis of ES (Electric Springwith only reactive compensation (ES-Q), also termed as ESversion 1 in previous papers). The dynamic model in [13] isnot based on the standard vector control modelling framework.Hence, the tuning of the controller gains, unlike standardvector control [15] approach which is widely used for powerelectronic converters, are not based on analytical methods.Moreover, it lacks an inner current control loop which isnecessary for limiting the inverter current within safe limits.However, there are unique challenges in developing vectorcontrol of ES-Q due to two important factors: (a) it has a seriesconfiguration unlike PV inverters and (b) only reactive powerexchange is allowed. These two factors impose a quadratureconstraint between the load current, filter capacitor current andthe inverter current due to which it is not appropriate to alignthe d-axis with the PoC voltage (which is commonly done forshunt converters).

Based on the work done in [13], a linearized state spacemodel is developed in [14]. However, three important lim-itations of [14] are: (i) the frequency domain model is notbased on the vector control and hence not compatible withthe standard stability model for other elements such as DGinverters etc., (ii) the paper did not report the effect of changein distribution network parameters on small signal stability and(iii) the inferences are based on the results from time-domainsimulation without delving into the root cause through modalanalysis. This paper addressed all these issues by proposing away of using vector control for ES-Q which can be integratedin the stability model of a distribution network and microgrid



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with other DG inverters. A linearized state space model of adistribution network with vector controlled ES-Q is presentedfor stability analysis. The state-space approach is adoptedinstead of the impedance-based [16], [17] approach as thelatter, although simple, does not allow participation factor ormode shape analysis to identify the root cause of the stabilityproblem.

The original contributions of this paper are as follows:1) Vector control of ES with only reactive power compen-

sation (ES-Q) is reported for the first time which can beintegrated in the stability model of distribution networksand microgrids with other DG inverters.

2) Linearized state-space model of the distribution net-work with vector controlled ES-Q is developed usingthe Component Connection Method [18] which can beeasily scaled up to include more electric springs and DGinverters.

3) The developed model is used to analyse the smallsignal stability of the distribution network with ES-Q and compared against the stability with equivalentpenetration of DG inverters. The analysis is presentedfor a range of scenarios covering low- to mediumvoltage (LV/MV) levels and sparse rural to dense urbandistribution networks.

The study reported in this paper is based on a single phasenetwork and does not consider the switching model of theconverters, voltage unbalance or harmonics and neglects thedynamics of the DC link of ES-Q (in state-space model)which would require a different modelling approach (e.g.dynamic phasor). State-space model of a three-phase ES [6]could be derived from that of a single-phase ES and is notconsidered separately. Henceforth, an ES with reactive powercompensation (ES-Q) is simply referred to as ‘Electric Spring(ES) and small signal stability is termed as just ‘stability’.

II. VECTOR CONTROL OF ELECTRIC SPRING

The configuration of an ES connected in series with aresistive-inductive (RL-type) load is shown in Fig. 1 [1]. TheRL-type load is denoted by RNC & LNC and the corre-sponding voltage across it is given by Vnc. The ES injects acontrolled series voltage Ves across the filter capacitance (Cf )to regulate the voltage (Vpoc) at the point of coupling (PoC)with the supply. The equivalent inductance and resistance ofrest of the network including other passive loads is denoted byL1 and RC , respectively. The source impedance is representedby the inductance Lg .

A. Current Control Loop

The current control loop is designed based on the dynamicsof the inverter current (Iinj), filter capacitor voltage (Ves) andload current (I) which can be represented in a synchronouslyrotating (dq) reference frame as follows [15]:

LfdIinjddt

= Vinvd − Vesd −RfIinjd + ω0LfIinjq (1)

LfdIinjqdt

= Vinvq − Vesq −RfIinjq − ω0LfIinjd (2)

GND

Lf

Vpoc

IinjIcap

II

VinvVes

II

VncLNC

Idc

VdcCdc

L1

GND

Lg

Source

GND

ILIgIg

Cf RC

RNC

Rf

Electric Spring

Figure 1. Electric Spring (ES) configuration

CfdVesddt

= Id + Iinjd + ω0CfVesq (3)

CfdVesqdt

= Iq + Iinjq − ω0CfVesd (4)

LNCdIddt

= Vpocd − Vesd −RNCId + ω0LNCIq (5)

LNCdIqdt

= Vpocq − Vesq −RNCIq − ω0LNCId (6)

The current flowing out of the inverter (Iinj) is assumed to bepositive. Considering Iinjd & Iinjq as state variables, Vinvd& Vinvq as control inputs and Vesd & Vesq are the disturbanceinputs, the current control loop is designed following the stan-dard method outlined in [15]. The closed loop time constant(τ ) is chosen to be 0.1 ms to achieve a fast enough responsewhile ensuring that the controller bandwidth ( 1

τ ) is at least 10times smaller than the switching frequency.

B. Choice of Reference Frame (d-axis)

For regulating the point of coupling (PoC) voltage usingvector control of the shunt connected inverters (such as DGinverter or STATCOM), the d-axis is normally aligned with thePoC voltage to achieve decoupled control. In this paper, thefocus is on ES with only reactive power compensation (ES-Q) which injects a voltage (Ves) across the filter capacitor inquadrature (lead or lag) with the load current (I). In this case,aligning the d−axis with the PoC voltage (Vpoc) results in non-zero inverter current (Iinj) on both d and q axes as shown inFig. 2.

Vnc

I

Ves

Vpoc = Vpocd

d-axis

q-axis

Vncd

Iq

Vncq

Id

Iinj

Icap

Iinjq

Icapq

IcapdIinjd

Vnc

I

Ves

Vpoc = Vpocd

d-axisd-axis

q-axisq-axis

Vncd

Iq

Vncq

Id

Iinj

Icap

Iinjq

Icapq

IcapdIinjd

Figure 2. Phasor diagram with d−axis aligned with PoC voltage



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Thus, it is not possible to force the d-axis component of theinverter current to zero in order to ensure zero active power(neglecting filter and inverter losses) exchange in steady-state.To get around this problem, the d-axis could be aligned eitherwith the filter capacitor voltage (Ves) or the load current (I).The latter is ruled out due to the possible implementationchallenges posed by high harmonic content in the load current.So the d-axis is aligned with the filter capacitor voltage (Ves)as shown Fig. 3.

Vnc

Ves

Vpoc

d-axisq-axis Vncd

Iq

Vncq

Iinjq

Icapq

Ves = Vesd

Figure 3. Phasor diagram with d−axis aligned with filter capacitor voltage

This enables the d−axis control loop to maintain the DClink voltage Vdc while the q−axis loop can be used forregulating the PoC voltage (Vpoc) or the load voltage (Vnc).However, it is necessary to maintain a minimum filter capacitorvoltage to allow satisfactory operation of PLL, as discussedlater in Section III-B. In Fig. 3, the filter capacitor voltage(Ves) leads the load current (Iq) by 90◦ providing inductivecompensation. The ES can provide capacitive compensation ifVes is made to lag I by 90◦. The compensation mode dependson the polarity of Ves, unlike a STATCOM, where it is decidedby the phase angle of the injected current with respect to thePoC voltage. For capacitive compensation, Vesd is aligned with-ve d−axis and Iq becomes slightly more than Iinjq so thatIcapq now aligns with -ve q−axis, thereby leading Vesd by90◦.

C. PoC Voltage Control LoopThe inverter current references (Iinjdref & Iinjqref ) are

obtained from the voltage control loop described here. Anapproximate small-signal model for PoC voltage dynamicsis presented first based on Fig. 1. The feeder resistance isneglected for simplicity in tuning the PoC control loop. Thed−axis component of the source current can be expressed asIgd ≈ ILd (as the d−axis load current component Id = 0)while the q−axis current is given by Igq = ILq+Iq . Assumingan ideal PLL, the PoC voltage along d−axis is given by (7),where the first two terms represent the load effect and the lastterm shows the control effect. As the current through the filtercapacitor is negligibly small, an incremental change in loadcurrent is almost equal to a change in the inverter current,i.e. ∆Iq ≈ ∆Iinjq. Hence, the second term is considered ascontrol effect.

∆Vpocd = Lgd∆ILddt

− Lgω0∆ILq − Lgω0∆Iq (7)

Similarly, the q−axis component of the PoC voltage can begiven by (8) which shows that ∆Vpocq depends on the rate ofchange of the load current.

∆Vpocq = Lgd∆Igqdt

+ Lgω0∆Igd

= Lgd∆ILqdt

+ Lgω0∆ILd + Lg∆Iqdt

(8)

The voltage control loop is typically much slower thanthe current control loop. Hence the rate of change of smalldeviation in the load current (d∆Iq

dt ) is quite small such thatthe effect of the small change in control variable ∆Iinjq canbe neglected in case of ∆Vpocq . Therefore, the control loop forthe PoC voltage regulation can be represented as in Fig. 4.Thecurrent controller shown in Fig. 4 is a simple representationneglecting the feed forward and decoupling terms.

The compensator (Kvpoc) for the PoC voltage regulation isa proportional-integral (PI) controller with kpvpoc = 100 andkivpoc = 4. The source inductance is Lg = 0.0011H whichincludes the transformer leakage and network inductance. Aconservative phase margin of 90◦ is kept to ensure robustnessin the face of varying source impedance and account for theunmodelled dynamics of the filter, loads and PLL.

D. Phase-Locked Loop (PLL)

For the single-phase PLL considered here, an additionalfictitious phase was created by introducing a phase delay

of1

4of a fundamental cycle. A range of different PLL

configurations have been reported [19]. However, in stabilitystudies, the dynamics of PLL is either completely neglected[9], [20] or simplified by a proportional-integral (PI) controller[21], [22]. For a PI representation of a PLL, the tuning of kpand ki has little effect on the overall phase margin in the lowfrequency range which prompted us to adopt PI based PLLin this study. The compensator gains are chosen as kp = 0.3and ki = 0.5 to allow adequate gain and phase margins at lowfrequencies.

E. DC Link Control Loop

The DC link control loop is designed following the stan-dard procedure [15]. A proportional-integral compensator isused with gains kpdc = −2Cdcζω and kidc = −Cdcω2,where the DC link capacitor Cdc = 1.5mF, ζ = 0.707 andω = 24.16rad/s.

F. Performance Validation

The performance of the voltage control loop is tested usingthe system shown in Fig. 5, modelled in Matlab Simulink.The simulation model uses an average model for the converterbased on a controlled voltage source on the AC side and acontrolled current source on the DC side. Switching actionwithin the converter is not modelled explicitly. The dynamicsof the AC and the DC side are coupled through the ideal powerbalance equation i.e. Pac = Pdc. A 1.8 kVA resistive load Rnewis inserted to reduce the PoC voltage by approximately 5 V.



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Vpocref Kvpoc(s)

Vpoc

Iinjq Qref Iinjqref

ILq

-Lgω0

Vncref

-+-+

Vnc

Kvnc(s)

sLg

ILd

Vpocd

()2

()2Vpocq

()1/2

Vesdmax

2Vesdmax

2Vesdmax

21/(sLf+Rf)Kq(s)

Iqerror

Iinjq

Vinvq Vinvq

q axis current controller

-+-

+ -+-+ ++

++

++

++

++

++

1 + 0.1e-3

s

1

1 + 0.1e-3

s

1

1 + 0.1e-3

s

1

Figure 4. PoC voltage control scheme

The transformer and the feeder parameters taken from [23]correspond to typical figures for distribution networks in theUK. The R/X ratio of the lines is 4.26 (line type D in [23])and the transformer is rated at 200 kVA.

GND

Cf

Lf

Vpoc

IinjIcap

II

VinvVes

II

VncLNC

Idc

VdcCdc

GND

200 kVA

transformer

GND

I1I1

GND

L1

InewInew

I3I3I2I2

L2L2 L3L3

1.8 kVA

load

R1 R2

Rnew

R3

RC

Rf

RNC

Rtr

Ltr

Figure 5. Test system for validation of PoC voltage controller

Electric Spring (ES) with reactive compensation does notexchange any active power between the ac and the dc sideof the converter (except for supplying filter losses). To ensurethis condition, the filter voltage (Ves) and the load current(I) always maintains a quadrature relationship. The choice ofmode of operation depends on the system requirement. In caseof under voltage disturbance, the ES switches from inductiveto capacitive mode to provide voltage support. From the phasordiagram in Fig. 3, it is evident that in the inductive modeof operation Vesd is aligned with the +d-axis and the loadcurrent Iq is aligned with the -q-axis. In order to switch to thecapacitive mode, Vesd needs to realign with the -d-axis whileIq needs to align with the +q-axis. To enable this transitionprocess, the ES outputs a zero voltage for 3 cycles after adisturbance.

Fig. 6 shows the dynamic response following reductionin PoC voltage at t =1 s. The compensator switches frominductive mode (I lagging Ves by 90◦) to capacitive mode(I leading Ves by 90◦) by making the inverter current (Iinj)slightly less than the load current (I) such that the filtercapacitor current (Icap) now aligns with the negative q−axis.This operation can be better understood from the phasordiagram in Fig. 3.

Fig. 7(a) shows that I lags Ves by 90◦ i.e. the compensatorworks in inductive mode before the disturbance while after thedisturbance it switches to capacitive mode where I leads Vesby 90◦ (Fig. 7(b)).

1 1.1 1.2 1.3

Time(s)

-500

-300

-100

100

300

500

Vo

ltag

e(V

)

(a)

Vpoc

Vnc

Ves

1 1.1 1.2 1.3

Time(s)

-20

-10

0

10

20

Cu

rren

t(A

)

(b)

Iinj I I

cap

0.5 1 1.5 2

Time(s)

205

210

215

220

Vp

ocr

ms

(V)

(c)

0.5 1 1.5 2

Time(s)

390

395

400

405

410

Vd

c(V

)

(d)

Figure 6. Dynamic response of (a) PoC, load and ES voltages; (b) inverter,load and filter capacitor current; (c) RMS value of PoC voltage and (d) DClink voltage

0.8 0.85 0.9

Time(s)

-150

-100

-50

0

50

100

150

Vo

ltag

e/C

urr

ent

(a) before disturbance

Ves

I

1.5 1.55 1.6

Time(s)

-150

-100

-50

0

50

100

150

Vo

ltag

e/C

urr

ent

(b) after disturbance

Figure 7. ES voltage (Ves) and load current (I) (a) before the disturbance(inductive mode) and (b) after the disturbance (capacitive mode)

From Fig. 6(c), it is clear that the PoC voltage could notbe regulated at its pre-disturbance level due to violation of theinverter current limit which is set according to the allowablevoltage variation across the load [11]. This limit for Vnc isfixed at ±0.1 per unit in this particular case. Deviation in DCbus voltage after the disturbance, seen in Fig. 6(d), is causedby additional losses in the filter resistor (Rf ) due to changein Iinj .



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III. LINEARIZED STATE SPACE MODEL (SSM)The linearized state-space model (SSM) of a distribution

network with ES is developed here. First, the SSM is devel-oped separately for the following sub-systems (a) PLL (b) PoCvoltage control loop (c) current control loop (d) filter and (e)network and loads. Then, the Component Connection Method(CCM) [18] is used to obtain the overall SSM. For simplicity,only resistive loads are considered and the DC link dynamicsis neglected.

A. PLLThe PI compensator based single phase PLL has one state

variable associated with the compensator (∆x1) and anotherone corresponding to the integrator (∆θ). The input and outputvariables are ∆Vesq and ∆ω, ∆θ where ∆Vesq denotes theincremental change in q−axis component of the filter voltage,∆ω denotes the change in PLL frequency and ∆θ representsthe change in PLL angle. The SSM of PLL is given by (9)and (10) where kp and ki are proportional and integral gains,respectively.[

∆x1

∆θ

]=

[0 0ki 0

] [∆x1

∆θ

]+

[1kp

] [∆Vesq

](9)

[∆ω∆θ

]=

[ki 00 1

] [∆x1

∆θ

]+

[kp0

] [∆Vesq

](10)

B. PoC Voltage Control LoopThe block diagram of the voltage control loop is shown in

Fig. 8 which is divided into: outer PI controller (Fig. 8 (a))and inner integral controller (Fig. 8 (b)). The state variableassociated with the outer loop is denoted by ∆x21. The inputsignal Vpocmag is the measured PoC voltage which can beexpressed in terms of dq−components as

√V 2pocd + V 2

pocq .This non-linear term has been linearized about a stable operat-ing point Vpocmag0 which results in (

Vpocd0

Vpocmag0∆Vpocd) for an

incremental change in d− axis component. Similar expressioncan be obtained for an incremental change in the q−axiscomponent. Thus, the SSM of the outer loop (Fig. 8 (a))is given by (11) and (12). kpv and kiv are the proportionaland integral gains of the voltage control loop compensator,respectively. In Fig. 8 (a) the symbol for the sum is considered−Vpocmagref + Vpocmag based on the operating principle ofelectric spring. An increase in PoC voltage magnitude (dueto disturbance) results in a positive error thus increasing thevoltage reference set point across the load (Vncref ). This waythe power consumption (both active and reactive in case ofnon-unity power factor load) of the load increases to regulatethe PoC voltage back to its reference value.

[∆ ˙x21

]=[0] [

∆x21

]+[

Vpocd0

Vpocmag0

Vpocq0

Vpocmag0

] [∆Vpocd∆Vpocq

](11)

[∆Vncref

]=[kiv] [

∆x21

]+[kpvVpocd0

Vpocmag0

kpvVpocq0

Vpocmag0

] [∆Vpocd∆Vpocq

](12)

-+Vpocmagref

Vpocmag

Vncref

(a)

Vncref

Vnc0 Vncmag

Iinjqref

(b)

skpv + kivs

skpv + kivs

skpv + kivs +

+ +-

2.5s

2.5s

Figure 8. Block diagram of (a) PoC and (b) load voltage control loops

The state variable associated with the inner loop is denotedby ∆x22. The outer loop sets the reference for the inner loopwhich is updated by the maximum load voltage Vnc0, (i.e.minimum voltage across the filter capacitor) corresponding tothe operating point Vpocmag0. A minimum voltage is requiredacross the filter capacitor to ensure satisfactory operation ofthe PLL. The minimum filter capacitor voltage is consideredto be 50 V in this study. The tuning of the PLL is carriedout considering this minimum voltage. The choice of 50V isprompted by the fact that a sufficiently high value would easethe tuning of the PLL parameters to achieve an acceptabledynamic response. However, in practice this minimum filtervoltage can be well below 50V depending on the type of PLLand the tuning of its compensator.

For a resistive load, Vnc0 =√V 2pocmag0 − 502 which is

linearized as before and is expressed as Vpocd0√V 2pocmag0−502

∆Vpocd

for an incremental change in d− axis component. Similarexpression can be obtained for an incremental change in theq−axis component. The load voltage magnitude feedback termVncmag is also linearized in a similar way around Vncmag0.The SSM of the inner loop (Fig. 8 (b)) is given by (13) and(14). In (14) (and other equations henceforth), zero in boldletter denote a matrix of appropriate dimension.

[∆ ˙x22

]=[0] [

∆x22

]+

1−Vncd0

Vncmag0−Vncq0

Vncmag0Vpocd0√

V 2pocmag0−502

Vpocq0√V 2pocmag0−502

T ∆Vncref∆Vncd∆Vncq∆Vpocd∆Vpocq

(13)

[∆Iinjqref

]=[2.5] [

∆x22

]+[0]

∆Vncref∆Vncd∆Vncq∆Vpocd∆Vpocq

(14)

C. Current Control Loop

The structure of the current control loop adopted here isstandard for any power electronic converter. The state variablesassociated with the PI controllers in d and q axis loops aredenoted by ∆γd and ∆γq , respectively and the controllergains (same for d and q axis loops) are kpc and kic. TheSSM of the current controller is given by (15) and (16).The incremental changes in the current control loop reference



6

values are denoted by ∆Iinjdref & ∆Iinjqref while thechange in the inverter currents and filter voltages are expressedas ∆Iinjd & ∆Iinjq and ∆Vesd & ∆Vesq , respectively.

[∆γd∆γq

]=[0] [∆γd

∆γq

]+

1 00 1−1 00 −10 00 0

T ∆Iinjdref∆Iinjqref

∆Iinjd∆Iinjq∆Vesd∆Vesq

(15)

[∆Vinvd∆Vinvq

]=

[kic 00 kic

] [∆γd∆γq

]

+

kpc 00 kpc−kpc ω0Lf−ω0Lf −kpc

1 00 1

T ∆Iinjdref∆Iinjqref


(16)

D. Filter

The SSM of the filter part of the ES can be obtained fromthe dynamics of the inverter current (Iinj) and filter capacitorvoltage (Ves) given by (1), (2), (3) and (4). The state variablesare the d− and q− axis components of the inverter current(Iinjd, Iinjq) and the filter capacitor voltage (Vesd, Vesq). Theequations are linearized around Iinjd0, Vesd0 (and similarlyfor the q− axis) to derive the SSM of the filter as in (17)and (18). The filter resistance, inductance and capacitance aregiven by Rf , Lf and Cf , the load resistance is given by RNCand ω0 denotes the nominal frequency of the network.

∆ ˙Iinjd∆ ˙Iinjq∆ ˙Vesd∆ ˙Vesq

=

−Rf

Lfω0 − 1

Lf0

−ω0 −Rf

Lf0 − 1

Lf1Cf

0 0 ω0

0 1Cf

−ω0 0


+

Iinjq0

1Lf

0 0 0

−Iinjd0 0 1Lf

0 0

Vesq0 0 0 1CfRNC

0

−Vesd0 0 0 0 1CfRNC

∆ω∆Vinvd∆Vinvq∆Vncd∆Vncq

(17)


=

1 0 0 00 1 0 00 0 1 00 0 0 1


+[0]

∆ω∆Vinvd∆Vinvq∆Vncd∆Vncq

(18)

E. Network and Loads

The simple network shown in Fig. 5 with three segmentsand an ES is considered here. The load Rnew used for creatingdisturbance is modelled as a current source Is with a high

resistance (Rhigh) in parallel. The transformer impedance (Rtrand Ltr) is merged within R1 and L1 in (19) and (20). The lineresistance R3 is merged with the equivalent load (RC) whilethe inductance L3 is neglected. The state variables are the d−and q− axis components of the line currents flowing throughL1 and L2. The SSM of the network and the load is given by(21) and (22) with Aline, Bline, Cline and Dline expressedas in (19), (20), (23) and (24). The incremental change inthe source voltage (and the disturbance current) is denoted by∆Vsd & ∆Vsq (∆Isd & ∆Isq).

∆ ˙IL1d

∆ ˙IL1q

∆ ˙IL2d

∆ ˙IL2q

= Aline

∆IL1d

∆IL1q

∆IL2d

∆L2q

+Bline

∆ω∆Vsd∆Vsq∆Isd∆Isq∆Vesd∆Vesq

(21)

∆IL1d

∆IL1q

∆IL2d

∆IL2q

∆Id∆Iq

= Cline

∆IL1d

∆IL1q

∆IL2d

∆L2q

+Dline

∆ω∆Vsd∆Vsq∆Isd∆Isq∆Vesd∆Vesq

(22)

Cline =

1 0 0 00 1 0 00 0 1 00 0 0 10 0 RC

RC+RNC0

0 0 0 RC

RC+RNC

(23)

Dline =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 − 1

RC+RNC0

0 0 0 0 0 − 1RC+RNC

(24)

Once the SSM is developed for the individual sub-systems (9)-(24), their A, B, C, D matrices are arrangedtogether to form block-diagonal matrices. As an example,for four sub-systems, a block-diagonal state matrix A =diag(A1, A2, A3, A4) is formed (and similarly for B, C, Dmatrices). Further, a matrix LT is formed to capture thealgebraic relationship between the inputs and outputs of all thesubsystems. Finally, the state matrix of the whole system (F ) iscomputed as (25). More details of this Component ConnectionMethod (CCM) can be found in [18].

F = A+BLT (I −DLT )C−1 (25)

The SSM for a simple network with one ES can be easilyscaled up to include any number of network segments and ESsusing CCM by simply repeating the equations for individualsubsystems. This is used for the study case discussed next



7

in Section IV where four network segments and two ESs areconsidered, as shown in Fig. 10.

Presence of more than one converter requires that all thevariables are expressed in a common reference frame. This isachieved by the frame transformation method provided in [20].As an example, the filter voltage Ves can be expressed on acommon reference frame ‘DQ’ according to (26), where TSand TC are the transformation matrices given by (27) & (28)and ∆δ is the incremental change in the phase angle differencebetween the reference frame of the respective inverter and thecommon reference frame ‘DQ’. In (27) & (28), Vesd0, Vesq0& δ0 represent the corresponding values at a stable operatingpoint. [

∆VesD∆VesQ

]= TS

[∆Vesd∆Vesq

]+ TC

[∆δ]

(26)

TS =

[cos(δ0) −sin(δ0)sin(δ0) cos(δ0)

](27)

TC =

[−Vesd0sin(δ0)− Vesq0cos(δ0)Vesd0cos(δ0)− Vesq0sin(δ0)

](28)

F. Validation of Linearized State Space model

The developed SSM is validated by comparing the dynamicresponse of the state variables with respect to that of the non-linear model developed in Matlab Simulink. At t = 1sec, a100 W disturbance was introduced through Rnew (Fig. 5) for 3cycles. Fig. 9 shows the comparison for six state variables as-sociated with the filter capacitor and the network inductances.The responses match closely confirming the validity of theSSM.

IV. FREQUENCY-DOMAIN ANALYSIS

The modelling framework discussed in Section III is usedto obtain the SSM of the study system shown in Fig. 10. Thedq-frame corresponding to the first electric spring (ES1) isconsidered to be the common frame and all other quantities arereferred to this using the previously discussed transformationtechnique (26), (27) & (28).

Eigen value analysis of the state matrix (25) of the studysystem shown in Fig. 10 confirms that there are 32 state

variables. There are two high-frequency modes of 472.31Hz & 369.63 Hz and a third mode (56.07 Hz) close to thefundamental frequency. Participation factor analysis revealsthat the filter capacitor voltage of both ESs (Ves1 & Ves2)and the current through the network segment 1 (IL1) and 2(IL2) have dominant participation in the two high-frequencymodes (472.31 Hz & 369.63 Hz) while the 56.07 Hz mode ismainly associated with the filter capacitor voltage.

Varying the network parameters, such as the length of theline segments and R/X ratio, over a certain range showsmovement of the critical system modes obtained from theeigen values of the system state matrix. This study helpsto ascertain the overall small signal stability of the systemunder different scenarios. Although the study system shownin Fig. 10 is simple, varying the length of line segments 1, 2and 3 and the R/X ratios cover a wide range of distributionnetwork scenarios. For instance, the length of line segment1 and 2 indicates the location of the ES with respect tothe substation while the length of line segment 3 signifiesthe proximity of adjacent ESs. Similarly, a range of R/Xratio can be chosen to represent a dense urban (R/X ≈ 4)to sparse rural (R/X ≈ 10) low-voltage (LV) network aswell as the medium-voltage (MV) level (R/X ≈ 1) withindustrial/commercial customers.

Stability of the distribution network with ES (inverter inseries with the customer load) is compared against the casewhere PoC voltage is regulated by shunt (connection withrespect to the customer load) converters such as STATCOMor DG inverters. The inverters in both cases are chosen to beequivalent in terms of PoC voltage regulation capability. TheSSM of the network with DG inverters is fairly well reportedand hence, not included in this paper. The readers can referto [9] for details. The SSM of the study system shown inFig. 10, but with the two ESs replaced by equivalent DGinverters, reveals the presence of four modes with frequenciesof 4798.3Hz, 4695Hz, 1124.46Hz and 996.23Hz which aremuch higher than the case with ES.

A. Distance from Substation

The impact of distance of the ES or shunt converter fromthe substation is investigated by varying the length of linesegments 1 and 2 which are separated by a unity power factor

Aline =

−R1+Rhigh

L1ω0

Rhigh

L10

−ω0 −R1+Rhigh

L10

Rhigh

L1Rhigh

L20 − 1

L2(R2 +Rhigh + RNCRC

RNC+RC) ω0

0Rhigh

L2−ω0 − 1

L2(R2 +Rhigh + RNCRC

RNC+RC)

(19)

Bline =

IL1q0

1L1

0 −Rhigh

L10 0 0

−IL1d0 0 1L1

0 −Rhigh

L10 0

IL2q0 0 0Rhigh

L20 − 1

L2

RC

RC+RNC0

−IL2d0 0 0 0Rhigh

L20 − 1

L2

RC

RC+RNC

(20)



8

0 0.5 1 1.5 2-0.5

0

0.5 V

esd

(V)

Time domain

State Space

0 0.5 1 1.5 2-0.1

0

0.1

Ves

q(V

)

0 0.5 1 1.5 2-0.05

0

0.05

IL

1d

(A)

0 0.5 1 1.5 2-0.1

0

0.1

IL

1q

(A)

0 0.5 1 1.5 2

Time(s)

-0.01

0

0.01

IL

2d

(A)

0 0.5 1 1.5 2

Time(s)

-0.01

0

0.01 I

L2

q(A

)

Figure 9. Comparison of dynamic response of state variables in linearizedstate space model and non-linear model due to a small disturbance

Vpoc1

GND

200 kVA

transformer

GND

IL1IL1

GND

IsIs

IL4IL4IL2IL2

Vpoc2

Vc

IL3IL3

GND GND

I2I2

Vnc1Vnc1 Vnc2Vnc2

Ves1Ves1 Ves2Ves2

I1I1

GNDGND

I1

GND

2 km2 km 2 km2 km 1 km1 km 1 km1 km

Line parameters

R=0.32Ω/km

L=0.238e-3

H/km

R/X=4.26

R=0.012Ω

L=0.114e-3

HES1 ES2

R1 R2 R3 R4L1 L2 L3 L4

Rhigh

Rtrn

Ltrn

RNC1 RNC2

Rc

Figure 10. Study system with two ESs used for frequency-domain analysis

load. The length of the remaining line segments are kept thesame as the base case (as in Fig. 10) and the R/X ratioof all segments are fixed at 4.26. The impact of varying thelengths between 0.1km to 2km on the 472.31Hz & 369.63 Hzmodes is shown in Fig. 11(a) for ES. The frequency of bothmodes is significantly higher for ESs located farther away fromthe substation. However, the real part remains well beyond−4000s−1 (i.e. far away in the left half of s-plane) withoutposing any threat to network stability.

Fig. 11(b) shows the impact with ESs replaced by equivalentDG inverters. In this case, increasing distance from the sub-station causes the high frequency modes (> 4 kHz) to moveaway from the imaginary axis. It is worth noting that with ESthe modes are much farther away in the left half of the s-planecompared to the case with DG inverters.

B. Electrical Proximity

The length of line segment 3 is varied to represent theelectrical proximity between adjacent ES or DG inverters. Asingle ES will typically be installed at the PoC of a clusterof customers instead of individual customers to exploit theload diversity. Depending on the number of customers withina cluster, the distance of separation between adjacent ESs willvary. Fig. 12(a) shows that separation distance has hardly anyeffect on the frequency of the mode while the real part remainsbeyond −3500s−1, thus posing no threat to system stability.

Figure 11. Root locus with (a) ES and (b) DG inverters when length of L1& L2 is varied from 0.1km to 2km (red to blue) while keeping the other lineparameters constant

Fig. 12(b) shows that DG inverters in close electricalproximity will result in very high frequency modes (> 10kHz). With increasing separation, the frequency reduces butthe modes migrate closer to the imaginary axis.

Figure 12. Root locus with (a) ES and (b) DG inverters when length ofL3 is varied from 0.1km to 2km (red to blue) while keeping the other lineparameters constant

C. Distribution Network Voltage Level

The R/X ratio of the line segments 1 and 2 is variedbetween 1 (MV level) to 10 (LV level-sparse rural network).The state variables associated with line segments 1 and 2have dominant participation in the 472.31 Hz & 369.63 Hzmodes present in the case of ES. This results in a large shiftin those modes seen in Fig. 13(a). For higher R/X ratios, themodes move farther to left of the s-plane due to larger dampingintroduced by higher resistance. Similar effect is observed inFig. 13(b) in case of DG inverter except that the frequenciesincrease slightly for higher R/X unlike the case with ES whereit reduces slightly.

Fig. 14 shows the effect of varying the R/X ratio of linesegment 3 while keeping other line parameters constant. Thestate variable associated with this line segment shows verylimited impact on the three system modes (Fig. 14(a)) com-pared to varying the R/X of line segment 1 & 2 (Fig. 13(a)).For the case with DG inverter, varying the R/X ratio of linesegment 3 actually shows movement of the high frequencymode whereas, limited movement is observed for the lowerfrequency mode compared to Fig. 13(b).

From the discussion in Sections IV-A, IV-B and IV-C, itcan be concluded that the installation of ESs at different



9

Figure 13. Root locus with (a) ES and (b) DG inverters when R/X ratiosof L1 & L2 are varied from 1 to 10 (red to blue) while keeping the otherline parameters constant

Figure 14. Root locus plot with (a) ES and (b) DG inverters when R/Xratio of L3 is varied from 1 to 10 (red to blue) while keeping the other lineparameters constant

locations with respect to the substation, with varying electricalseparation and at MV and LV levels will not pose any threatto distribution network stability. In fact, the situation is morestable (modes are farther to the left of s-plane) with ES thanequivalent penetration of DG inverters. Unlike the case of amicrogrid [9], the choice of voltage droop gain did not havea major influence on stability in this case due to the stiff (orstrong) upstream system mimicked by the voltage source inFig. 5 & 10.

The small signal analysis presented here is based on asimple network. However, it covers a wide range of scenariosencountered in practical distribution networks. For example,the impact of different distance of an ES from the substationis captured by varying the length of the line segments 1 and 2while maintaining the R/X ratio and the length of other linesconstant. Similarly, the impact of electrical proximity betweenadjacent ESs is investigated by varying the length of the linesegment 3 while keeping the length and R/X ratio of otherlines constant. Voltage level (e.g. MV or LV) and topology(e.g. dense urban, spare rural network) of distribution networkis accounted for in the study by varying the R/X ratio of theline segments. The results presented in this section are thusextendible for the simulation study presented in the followingsection on a typical urban distribution network.

V. SIMULATION STUDY

The capability of a group of ES to regulate the PoCvoltage is demonstrated on a typical urban distribution networkadopted from [23]. There are four main feeders and severallateral feeders as shown in Fig. 15. The total load on the

substation transformer (11.0.4 kV, 550 kVA) is 431.3 kW (at1 p.u. voltage) which is equally divided among the residentialcustomers. The feeder lengths and the rating of the loadclusters are mentioned in Fig. 15 together with the cable type[23] for each segment.

11kV 0.4kV

500kVA

transformer

ZT= 2.04 +j9.28

(p.u. on 100MVA)

C

27m

U16

(8 cust)

U17

(8 cust)

L L

E E

40m 17m

D 82m

D

26m

E 90m

L U18

(6 cust)

E

53m

L U19

(4 cust)

D

15m

D

80m

U14

(15 cust)

E

85m

L

U15

(15 cust)

L

A

90m

B

41m

E 93m

U8

(12 cust)

C

35m

LU12

(12 cust)

A

32m

E 73m

L

U3

(12 cust)

L

U4

(12 cust)

L

LU7

(6 cust)

U5

(14 cust)

U6

(9 cust)

L L

E E

24m 52m

E 67m

U9

(9 cust)

L

E E

12m 53m

E 70m

E41m

E

32m

U11

(6 cust)

L

U10

(6 cust)

L

D

33mE 68m

E

17m

L

E 40m

U13

(9 cust)

D

78.6m

E

E

36m

85m

L

L

U1

(15 cust)

U2

(12 cust)* L type line length = 30m

190 customersTotal load = 431.3kW (at 1p.u.)

Figure 15. Low voltage network for simulation study [23]

For this study, load clusters U1, U11, U13 and U18 are eachequipped with one ES each. An under-voltage disturbance isintroduced at the location marked by an arrow in Fig. 15. Thisreduces the PoC voltages as shown in Fig. 16(a). The voltageinjected by the ESs reduces the deviation in PoC voltage fromthe nominal value as shown in Fig. 16(b). This is achieved atthe expense of reduced voltage across the loads which are stillmaintained within the stipulated limits. The ESs at U11 andU13 are unable to restore the PoC voltages back to the nominalvalue as their combined capacity is not adequate consideringthe severity of the voltage disturbance.

0.4 0.6 0.8 1 1.2 1.4

Time(s)

0.94

0.96

0.98

1

Vpocrm

s(p

.u.)

(b) with ES

0.4 0.6 0.8 1 1.2 1.4

Time(s)

0.94

0.96

0.98

1

Vpocrm

s(p

.u.)

(a) without ES

U1

U1

U11

U13

U11U13

U18

U18

Figure 16. Variation of PoC voltage (a) without ES, (b) with ES

VI. CONCLUSION

This paper shows that Electric Springs installed in distri-bution networks are not likely to threaten the small signalstability of the system. In fact, the stability margin with ES



10

installed is more than the case with equivalent penetrationof shunt converters such as DG inverters. The distance ofan ES from the substation, proximity between adjacent ESinstallations and the R/X ratio of the distribution feederinfluence the stability. However, for a wide range of distance,proximity and R/X ratios representing low- to medium (MV)voltage levels and typical dense urban networks to sparse ruralfeeders, the distribution network was found to be stable. Thelinearized state-space model of vector controlled ES presentedin this paper could be incorporated into the stability modelsof microgrids to study how droop control of ES (required forappropriate sharing of voltage control burden [24]) impactsthe overall stability.

REFERENCES

[1] S. Y. R. Hui, C. K. Lee, and F. F. Wu, “Electric springs - A new smartgrid technology,” IEEE Transactions on Smart Grid, vol. 3, no. 3, pp.1552–1561, 2012.

[2] C. K. Lee, B. Chaudhuri, and S. Y. R. Hui, “Hardware and controlimplementation of electric springs for stabilizing future smart grid withintermittent renewable energy sources,” IEEE Journal of Emerging andSelected Topics in Power Electronics, vol. 1, no. 1, pp. 18–27, 2013.

[3] X. Luo, Z. Akhtar, C. K. Lee, B. Chaudhuri, S. C. Tan, and S. Y. R.Hui, “Distributed voltage control with electric springs: Comparison withSTATCOM,” IEEE Transactions on Smart Grid, vol. 6, no. 1, pp. 209–219, 2014.

[4] D. Chakravorty, B. Chaudhuri, and S. Y. R. Hui, “Rapid frequencyresponse from smart loads in great britain power system,” IEEE Trans-actions on Smart Grid, vol. PP, no. 99, pp. 1–1, 2016.

[5] Y. Shuo, S. C. Tan, C. K. Lee, and S. Y. R. Hui, “Electric springfor power quality improvement,” in IEEE Applied Power ElectronicsConference and Exposition (APEC), 2014, pp. 2140–2147.

[6] S. Yan, S. C. Tan, C. K. Lee, B. Chaudhuri, and S. Y. R. Hui, “Electricsprings for reducing power imbalance in three-phase power systems,”IEEE Transactions on Power Electronics, vol. 30, no. 7, pp. 3601–3609,July 2015.

[7] L. Liang, Y. Hou, D. Hill, and S. Y. R. Hui, “Enhancing resilience ofmicrogrids with electric springs,” IEEE Transactions on Smart Grid,vol. PP, no. 99, pp. 1–1, 2017.

[8] D. Chakravorty, B. Chaudhuri, and S. Y. R. Hui, “Estimation ofaggregate reserve with point-of-load voltage control,” IEEE Transactionson Smart Grid, vol. PP, no. 99, pp. 1–1, 2017.

[9] N. Bottrell, M. Prodanovic, and T. C. Green, “Dynamic stability of amicrogrid with an active load,” IEEE Transactions on Power Electronics,vol. 28, no. 11, pp. 5107–5119, Nov 2013.

[10] F. Andrn, B. Bletterie, S. Kadam, P. Kotsampopoulos, and C. Bucher,“On the stability of local voltage control in distribution networks witha high penetration of inverter-based generation,” IEEE Transactions onIndustrial Electronics, vol. 62, no. 4, pp. 2519–2529, April 2015.

[11] D. Chakravorty, Z. Akhtar, B. Chaudhuri, and S. Y. R. Hui, “Comparisonof primary frequency control using two smart load types,” in 2016 IEEEPower and Energy Society General Meeting (PESGM), July 2016, pp.1–5.

[12] Z. Akhtar, B. Chaudhuri, and S. Y. R. Hui, “Smart loads for voltagecontrol in distribution networks,” IEEE Transactions on Smart Grid,vol. 8, no. 2, pp. 937–946, March 2017.

[13] N. R. Chaudhuri, C. K. Lee, B. Chaudhuri, and S. Y. R. Hui, “Dynamicmodeling of electric springs,” IEEE Transactions on Smart Grid, vol. 5,no. 5, pp. 2450–2458, 2014.

[14] Y. Yang, S. S. Ho, S. c. Tan, and S. Y. R. Hui, “Small-signal modeland stability of electric springs in power grids,” IEEE Transactions onSmart Grid, vol. PP, no. 99, pp. 1–1, 2017.

[15] A. Yazdani and R. Iravani, Voltage-Sourced Converters in Power Sys-tems, 1st ed. John Wiley INC., 2010.

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[18] Y. Wang, X. Wang, F. Blaabjerg, and Z. Chen, “Small-signal stabilityanalysis of inverter-fed power systems using component connectionmethod,” IEEE Transactions on Smart Grid, vol. PP, no. 99, pp. 1–1,2017.

[19] Q. Huang and R. Kaushik, “An improved delayed signal cancellationpll for fast grid synchronization under distorted and unbalanced gridcondition,” IEEE Transactions on Industry Applications, vol. PP, no. 99,pp. 1–1, 2017.

[20] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis andtesting of autonomous operation of an inverter-based microgrid,” IEEETransactions on Power Electronics, vol. 22, no. 2, pp. 613–625, March2007.

[21] N. Kroutikova, C. a. Hernandez-Aramburo, and T. C. Green, “State-space model of grid-connected inverters under current control mode,”IET Electric Power Applications, vol. 1, no. 3, pp. 329–338, May 2007.

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[23] Cigre Working Group C4.605, Rep.5, “Modeling and Aggregation ofLoads in Flexible Power Networks,” Tech. Rep., Feb. 2014.

[24] C. K. Lee, N. R. Chaudhuri, B. Chaudhuri, and S. Y. R. Hui, “Droopcontrol of distributed electric springs for stabilizing future power grid,”IEEE Transactions on Smart Grid, vol. 4, no. 3, pp. 1558–1566, 2013.

Diptargha Chakravorty (S’14) received theM.Tech degree in Energy Engineering in 2012 fromIndian Institute of Technology Delhi, India. He hasworked as a research assistant for a year withthe Institute of Electrical Power Systems and HighVoltage Engineering at the Technische UniversittDresden, Germany. He is currently pursuing thePh.D. degree from Imperial College London, U.K.His research interests include smart grid, demandside management, renewable energy integration andpower quality.

Jinrui Guo (S’16) received the B.Sc. and M.Sc.degrees from North China Electric Power University,Beijing, China, in 2012 and 2015, respectively. Heis currently pursuing the Ph.D. degree from ImperialCollege London, U.K. His research interests includepower system stability, smart grid, and renewableenergy integration.

Balarko Chaudhuri (M’06-SM’11) received thePh.D. degree in Electrical and Electronic engineeringfrom Imperial College London, London, U.K., in2005 where he is currently a Reader in PowerSystems at the Control and Power Research Group.His research interests include power systems stabil-ity, grid integration of renewables, HVDC, FACTS,demand response and smart grids. Dr Chaudhuri isan editor of the IEEE Transactions on Smart Gridand an associate editor of the IEEE Systems Journaland Elsevier Control Engineering Practice. He is a

Fellow of the Institution of Engineering and Technology (IET) and a memberof the International Council on Large Electric Systems (CIGRE).



11

Shu Yuen Ron Hui (M’87-SM’94-F’03) receivedthe B.Sc. (Hons.) degree in Electrical and Electronicengineering from the University of Birmingham,Birmingham, U.K., in 1984, and the D.I.C. andPh.D. degrees from Imperial College London, Lon-don, U.K., in 1987. He currently holds the PhilipWong Wilson Wong Chair Professorship with theUniversity of Hong Kong, Hong Kong. Since 2010,he has concurrently held a part-time Chair Pro-fessorship in PowerElectronics at Imperial CollegeLondon. He has published over 200 technical papers,

including about 170 refereed journal publications and book chapters. Over 55of his patents have been adopted by industry. Prof. Hui was the recipient ofthe IEEE Rudolf Chope Research and Development Award from the IEEEIndustrial Electronics Society and the IET Achievement Medal (CromptonMedal) from the Institution of Engineering and Technology in 2010. He is aFellow of the Australian Academy of Technological Sciences and Engineering.He is also the recipient of the 2015 IEEE William E. Newell Power ElectronicsAward. He is an Associate Editor of the IEEE TRANSACTIONS ON POWERELECTRONICS and the IEEE TRANSACTIONS ON INDUSTRIALELEC-TRONICS.

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Small Signal Stability Analysis of Distribution Networks