Dynamic phasor based stability analysis of aninverter connected to grid

A. Sawant, P. Jagtap, J. Aute, U. MumbaikarDepartment of Electrical Engineering

VJTI, MumbaiEmail: savant.apurva@yahoo.com

Abstract—Increasing electricity demand has always pushedthe power grid to operate at its peak. However penetration ofrenewable energy, though intermittent, will help in easing thestress through the development of microgrids. Since microgridsinvolve better and efficient control of power flow bidirectionally,intelligent devices which can provide this function are essential.The solid state transformer provide a means of serving thisobjective, apart from replacing the conventional low frequencytransformer. This paper focuses on the analysis of the inverterstage of the SST when it is connected to the grid at the low voltagedistribution side. A comparison of two modeling approaches isinvestigated, the averaged model and the dynamic phasor basedapproach. The later proves to be computationally faster and alsotakes into account the ripple in the states which is neglected in theformer approach. By linearizing the dynamic phasor equations,a PI controller is designed which maintains the output voltageof the inverter at its desired value.

Keywords-Dynamic phasors (DP), grid, inverter, PI con-troller, solid state transformer, stability analysis.

TABLE INOMENCLATURE

Rfi Filter ResistanceLfi Filter InductanceCfi Filter Capacitance

Vdcl Input VoltageRgi Grid ResistanceLgi Grid InductanceVg Grid Voltage

S1 or D Duty CycleR Real partI Imaginary part

I. INTRODUCTION

Electrical Power System symbolizes a system with dynamicoperating characteristics where transformer plays a major rolein distribution system. To withstand different complexities indistribution system, new technology is implemented whichreplaces old conventional transformer with solid state trans-former (SST) in microgrid[5]. SST is also known as a three-port energy router and power exchanger. The characteristicsfeature of a three-port solid state transformer is that it givesbetter performance than conventional transformer. Three stageSST is widely used at the distribution level to supply powerto the consumers[5].As shown in Figure 1 three stages of SST are explained below:

i) At rectifier stage, high voltage AC is converted to highvoltage DC which is fed to the dual active bridge (DAB)converter.

ii) At second stage, dual active bridge (DAB) converter,converts high voltage DC to low voltage DC. Generallythis stage is called as DC/DC converter.

iii) Third stage is inverter stage, which converts DC voltageto the regulated low AC voltage[5].

Fig. 1. Schematic diagram of three stage solid state transformer

This situs also gives perfect control of voltages and currentson both primary and secondary side. Rectifier (AC to DC)stage acts as a power factor improvement device thus powerfactor is always at unity. Power electronic devices act as acircuit breaker thus fault detection as well as protection of asystem takes place through solid state transformer[5].

Dynamic phasor (DP) representation of a system’s dynamicquantity in the form of phasors which is further used inresolving system’s matrix to study stability issues of therespective system. Dynamic phasor approach, in which phasorrepresentation can be carried out from switched model systemwith fourier series representation. Phasor representation is aclass of mathematical transformation to eliminate the funda-mental frequency component from the system’s equations[1].Equations are formed with reference to the state variablesof that system. Performing linearization on those equations,considering small perturbations in the system’s dynamicalquantity, system’s matrix is formed which is further used forconducting stability analysis.

To maintain synchronisation in the power system, the sta-bility of small signal model is analysed by providing smallperturbations in the control signal for dynamic response. Thesystem dynamics are controlled using a PI controller. PIcontroller designing includes deciding values of the gain kpand ki on a trial and error basis. Recursive loop for differentvalues of kp and ki continues till a stable closed loop systemis obtained[7].

This work addresses application of PI controller to maintainoutput of grid connected inverter as constant and perform smallsignal stability analysis using Lyapunov’s first method andcontrol the model with PI controller. Second section introducesnecessity of dynamic phasors. DP based modelling of gridconnected inverter with linearization model of the inverterwhich is described in section three.

II. NECESSITY OF DYNAMIC PHASORS

To evict the fundamental frequency component from thesystem computation, phasor representation is carried out,which is categorize in analytic transmutation. Phasor repre-sentation of a system is classified in the analytical transmu-tation to evict the fundamental frequency component fromthe computation. For analyzing steady state operation ofvoltages and currents of a system, phasor representation ofsinusoidal signal is realized. Phasors are widely used for theanalysis of stationary case of the power system[3]. With afourier series representation, approximate complex switchedwaveform x(τ )in the interval of τ ∈ (t-T, t] is given in theform of,

x(τ) = Re∑k∈K

Xk(t)ejkωτ (1)

Xk(t) =1

T

∫ t

t−Tx(τ)e−jkωτdτ = 〈x〉k(t) (2)

where,ω = 2π/TXk(t) is the kth time varying fourier coefficient, which is alsocalled dynamic phasor.K is the set of fourier coefficients.

K contributes a better equivalent value of the systemwaveform (e.g. K = 0, 1, 2)[3]. DP approach propose anumber of benefits over traditional forms. In frequency do-main, selection of K gives larger bandwidth. It’s selectionand variation also gives choice of displaying combinationbetween various parameters and denotes specific problems atdifferent frequencies[3]. DP makes simulation very faster thanswitched model because variations of DP are a lot slower thanthe spontaneous parameters. So they can be used to evaluatethe fast electromagnetic transients with larger step size. DPapproach approximates a periodically switched system witha continuous system as it permits an analytical judgment[3].Some important characteristics of DP are:

i) As given in (3), relationship between the derivativesof x(τ ) and the derivatives of Xk(t), where the timeargument t has been neglected for accuracy. This can beeasily justified by differentiating the equation given in(1)[3].

〈dxdt〉k

=dXk

dt− jkωXk (3)

ii) The product of two switched variables equals to thediscrete time convolution of the two DP sets of variables,which is given in (4)[3].

〈xy〉k =

∞∑`=−∞

(Xk−`Y`) (4)

III. DP BASED MODELLING OF INVERTER

A. Switched modelling of inverter

A full bridge single-phase inverter model, where Q1, Q2,Q3 and Q4 represent IGBT switches as shown in Figure 2.The aim of developing average equivalent model is to avoidmodelling of high frequency switching transients[1]. Withreference to Figure 2 a single phase inverter with a RLC filter,Vdcl with output of DAB is an input to inverter, and resistanceRg , inductance Lg and output voltage Vg are grid parameters.

Fig. 2. Inverter with grid

Average model of grid connected inverter is shown in Figure3. Development of a dynamic phasor (DP) model of a single-phase inverter is achieved by average model of Figure 3.

Fig. 3. Average model of inverter connected to the grid

Representation of inverter in switched model by applyingmesh and nodal analysis is done as,

Lfid〈if 〉dt

= 〈S1〉Vdcl − 〈Vcf 〉 −Rfi〈if 〉 (5)

Lgd〈ig〉dt

= 〈Vcf 〉 − Vg −Rgi〈ig〉 (6)

Cfid〈Vcf 〉dt

= 〈if 〉 − 〈ig〉 (7)

Above equations are written with fundamental frequency com-ponent as,

d〈if 〉dt

= −jωif +〈S1〉Vdcl −Rfi〈if 〉 − 〈Vcf 〉

Lfi(8)

d〈Vcf 〉dt

= −jωVcf +〈if 〉 − 〈ig〉

Cfi(9)

d〈ig〉dt

= −jωig +〈Vcf 〉 − Vg −Rgi〈ig〉

Lgi(10)

B. DP modelling of inverter

Dynamic phasor representation of inverter obtained fromswitched model equations by removing fundamental frequencycomponent is written as,

d〈if 〉R

dt= ω〈if 〉I +

〈S1〉RVdcl −Rfi〈if 〉R − 〈Vcf 〉R

Lfi(11)

d〈if 〉I

dt= −ω〈if 〉R +

〈S1〉IVdcl −Rfi〈if 〉I − 〈Vcf 〉I

Lfi(12)

d〈Vcf 〉R

dt= ω〈Vcf 〉I +

〈if 〉R − 〈ig〉R

Cfi(13)

d〈Vcf 〉I

dt= −ω〈Vcf 〉R +

〈if 〉I − 〈ig〉I

Cfi(14)

d〈ig〉R

dt= ω〈ig〉I +

〈Vcf 〉R −Rgi〈ig〉R − VgLgi

(15)

d〈ig〉I

dt= −ω〈ig〉R +

〈Vcf 〉I −Rgi〈ig〉I − VgLgi

(16)

C. Linearized model of inverter

For dynamic system, linearization is the best way to achievesmall signal stability. Thus linearize equations in switched andDP of the grid connected inverter with small perturbations.Linearized Equations are obtained in the form of,

∆x = A∆x+B∆d (17)∆y = C∆x+D∆d (18)

where,A is the state matrix of size n× n

B is the control or input matrix of size n× r

C is the output matrix of size m× n

D is the feedforward matrix of size m× r

∆x is the state vector of dimension n

∆y is the output vector of dimension m

∆d is the input vector of dimension r

Linearized switched model equations are written in formof (17 and 18),

ddt

∆〈if 〉∆〈Vcf 〉∆〈ig〉

=

−Rfi

Lfi

−1Lfi

o

1Cfi

0 −1Cfi

0 1Lgi

−Rgi

Lgi

∆〈if 〉

∆〈Vcf 〉∆〈ig〉

+

VdclDLfi

0

0

∆d

Linearized dynamic phasor equations are written in formof (17 and 18),

ddt

∆〈if 〉R

∆〈if 〉I

∆〈Vcf 〉R

∆〈Vcf 〉I

∆〈ig〉R

∆〈ig〉I

=

−Rfi

Lfiω −1

Lfi0 0 0

−ω −Rfi

Lfi0 −1

Lfi0 0

1Cfi

0 0 ω −1Cfi

0

0 1Cfi

−ω 0 0 −1Cfi

0 0 1Lgi

0 −Rgi

Lgiω

0 0 0 1Lgi

−ω −Rgi

Lgi

×

∆〈if 〉R

∆〈if 〉I

∆〈Vcf 〉R

∆〈Vcf 〉I

∆〈ig〉R

∆〈ig〉I

+

Vdcl cos 2πDLfi

−Vdcl sin(2πD−1)Lfi

0

0

0

0

∆d

As above switched and DP equations are written in form ofeq.(17 and 18). So separate all matrices from equations andrewrite as,

In switched model,

A =

−Rfi

Lfi

−1Lfi

0

1Cfi

0 −1Cfi

0 1Lgi

−Rgi

Lgi

; B =

VdclDLfi

0

0

C = [ 0 1 0 ]T

D= 0

With dynamic phasors,

A =

−Rfi

Lfiω −1

Lfi0 0 0

−ω −Rfi

Lfi0 −1

Lfi0 0

1Cfi

0 0 ω −1Cfi

0

0 1Cfi

−ω 0 0 −1Cfi

0 0 1Lgi

0 −Rgi

Lgiω

0 0 0 1Lgi

−ω −Rgi

Lgi

B =

Vdcl cos 2πDLfi

−Vdcl sin(2πD−1)Lfi

0

0

0

0

C = [ 0 0 0 0 1 0 ]T

D=0

IV. SMALL SIGNAL STABILITY ANALYSIS OF GRIDCONNECTED INVERTER

Small signal stability analysis is to maintain synchronismof power system when system is affected by small pertur-bations[7]. Linearizing switched model as well as dynamicphasors (DP) equations for analysis of small signal stabilityof grid connected inverter.

There are two methods for analysis of small signal stability.

i) Lyapunov’s first methodii) Lyapunov’s second method or direct method

In Lyapunov’s first method, small signal stability is deter-mined by the Eigen values of matrix A which is obtainedby linearization of the equations. Lyapunov’s second methoduses state functions to obtain small signal stability. For gridconnected inverter, small signal stability is achieved by usinglyapunov’s first method by making use of the linearized matrixA[7]. Criteria for analysis of the stability from lyapunov’s firstmethod is as follows:

i) System is asymptotically stable when eigenvalues con-sist of negative real part.

ii) If system has at least one Eigen value lying on right halfof the S-plane the system will be unstable.

Small signal stability can also be achieved usingparticipation matrix. Combination of right and lefteigenvectors with association between state variablesand modes is called as participation matrix[7].

P =[P1 P2 ..... Pn

]with Pi =

Φ1iΨi1

Φ2iΨi2

.

.ΦniΨin

where,Φi is a right eigen vector andΨi is a left eigen vector

Term Pi = ΦiΨi is known as participation factor andin association with eigenvector normalization, the sum ofparticipation factors is 1.

V. STABILITY ANALYSIS USING PI CONTROLLER

PI controller is most commonly used for controlling dy-namic systems as they have plain structure, easy to designand economical. Regardless of these advantages, PI controllerfails when controlled element is highly nonlinear and unpre-dictable[6]. As grid connected inverter have semiconductorswitches, PI controller eliminate forced oscillations as wellas harmonics and steady state error which is resulting whileon-off controller[6]. PI controller gives pessimistic effect onresponse speed of the system and overall stability of the systemas it works with integral mode. On the other hand derivativemode has ability to predict contingencies of the errors. Thusreaction time of the controller is decreased[6].

PI controller is used when,

i) Where fast response of the system is not needed.ii) If there is only one energy storage element (capacitor or

inductor) in process.iii) When course of process includes large disturbances and

noise.iv) There are large response delays in the system.

VI. RESULTS

Simulations results of all three methods are given in thefollowing figures.

A. Simulation Results

Simulink simulations have been executed to demonstrate theswitched model and DP based inverter connected to the grid.Figure (4), (5), (6) and (7) show output of inverter with grid,ripple is approximately 240 volts peak to peak and current43 A peak to peak across load/grid in both switched and DPmodels.

Fig. 4. Output voltage in switched model of a grid connected inverter

Fig. 5. Load current in switched model of a grid connected inverter

Fig. 6. Output voltage in DP of a grid connected inverter

Fig. 7. Load current in DP of a grid connected inverter

B. Eigen values and participation matrix

Table II and III shows the results when stability analysis ismeasured through Eigen values. All eigenvalues of switchedmodel as well as DP system are observed in left half ofS-plane. Also from analysis of participation matrix, righteigenvalues and left eigenvalues have real negative part. Thus itshows that the system of an inverter connected to grid is stable.Corresponding measure of adjacency to voltage instability isprovide by eigenvalues associated with a mode of voltage andreactive power variation[8].

TABLE IIEIGENVALUES OF SWITCHED MODEL OF GRID CONNECTED INVERTER

Sr.No Eigenvalues of switched model

1 (1.0e+ 03) × (−3.1867 + 3.0228i)2 (1.0e+ 03) × (−3.1867 − 3.0228i)3 (1.0e+ 03) × (−0.3864 + 0.0000i)

TABLE IIIEIGENVALUES IN DP OF GRID CONNECTED INVERTER

Sr.No Eigenvalues in DP

1 (1.0e+ 03) × (−3.1867 + 3.3998i)2 (1.0e+ 03) × (−3.1867 − 3.3998i)3 (1.0e+ 03) × (−3.1867 + 2.6458i)4 (1.0e+ 03) × (−3.1867 − 2.6458i)5 (1.0e+ 03) × (−0.3864 + 0.3770i)6 (1.0e+ 03) × (−0.3864 − 0.3770i)

C. Frequency analysis plot

Figure (8), (9), (10) and (11) show the results of inverterwithout controller and inverter with controller of switchedmodel and DP model based on gain margin and phase margin.

A systematic comparision is given by simulating thesedifferently modeled components in MATLAB based commonsimulation framework. Switched model and Reduced orderdynamic Phasor model of component results are compared.

Fig. 8. Switched model of a grid connected inverter without controller

Fig. 9. Switched model of a grid connected inverter with controller

Fig. 10. DP model of a grid connected inverter without controller

Fig. 11. DP model of a grid connected inverter with controller

Table IV and V shows comparison of results obtained inswitched model and DP model of grid connected inverter withand without using PI controller. As the gain and phase marginincreases, the stability is improved.

TABLE IVCONTROLLER PARAMETERS OF SWITCHED MODEL

Sr.No parameter Without controller With controller

1 Gain Margin Infinity InfinityFrequency Infinity Infinity

2 Phase Margin 5.41 60Frequency 6.86×104rad/s 4.06×103rad/s

TABLE VCONTROLLER PARAMETERS OF DP

Sr.No parameter Without controller With controller

1 Gain Margin (dB) -13.8 14Frequency (rad/s) 0 4.19×103

2 Phase Margin (degree) -139 60Frequency (rad/s) 2.68×103 1.27×103

VII. CONCLUSIONS

A basic introduction to various methods of inverter model-ing is explored in this paper. The focus of the paper is on thedesign of a PI controller for the linearized DP model of the gridconnected inverter. A comparison between the switched modeland the DP model simulations is performed which prove thatthe later is a better tool when ripple has to be considered.It is further proposed to use Model Predictive Control fordeveloping the controller in the DP domain which will trulypropagate the advantages of the DP approach.

VIII. APPENDIX

TABLE VISYSTEM PARAMETERS OF GRID CONNECTED INVERTER

f 60HzRfi 11ΩLfi 1.7mHCfi 30µF

Vdcl 400VRgi 35ΩLgi 121mH

S1 or D 0.6Vcf 240V

ACKNOWLEDGMENT

The authors would like to acknowledge the support ofTEQIP-II through Centre of Excellence in Complex andNonlinear Dynamic Systems(CoE-CNDS), VJTI, Matunga,Mumbai, India. We are also thankful for continuous technicalinputs provided by S.R.Wagh, N.M.Singh and A.Stankovicthroughout the completion of the research.

REFERENCES

[1] Adarsh. Nagarajan and Raja. Ayyanar, ”Dynamic phasor model of single-phase inverters for analysis and simulation of large power distributionsystems”, Arizona State University Tempe, AZ 85287.

[2] Tiefu. Zhao, Jie. Zeng, Subhashish. Bhattacharya, Mesut E. Baran andAlex Q. Huang, ”An average model of solid state transformer for dynamicsystem simulation”, IEEE Member.

[3] T. Demiray, G. Andersson and L. Busarello, ”Evaluation study for thesimulation of power system transients using dynamic phasor models”

[4] Seth R. Sanders, J. Mark Noworolski, Xiaojun Z. Liu, andGeorge C. Verghese, ”Generalized averaging method for power conver-sion circuits”, IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL. 6. NO. 2, APRIL 1991

[5] Shilpkala G. Bansode, Prasad M. Joshi, ”Solid state transformers: Newapproach and new opportunity”, GCE, Karad.

[6] K. Smriti Rao, Ravi. Mishra ”Comparative study of P, PI and PIDcontroller for speed control of VSI-fed induction motor”, IJEDR 2014,Volume 2, Issue 2, ISSN:2321-9939.

[7] P. Kundur, Textbook on ”Power system stability and control”, McGraw-Hill,Inc.

[8] Dr. Enemuoh F. O., Dr. Onuegbu J. C. and Dr. Anazia E. A. ”Modalbased analysis and evaluation of voltage stability of bulk power system”,International Journal of Engineering Research and Development, Volume6, Issue 12 (May 2013), PP. 71-79