Assessment of Dynamic Phasors Modelling Technique for Accelerated Electric Power System Simulations

Tao Yang, Serhiy Bozhko, Greg Asher UNIVERISITY OF NOTTINGHAM

Nottingham NG7 2RD Nottingham, UK

E-Mail: eexty4@nottingham.ac.uk

Acknowledgements

The author gratefully acknowledges the EU FP7 funding via the Clean Sky JTI – Systems for Green Operations ITD.

Keywords « Dynamic phasor », « synchronous generator », « transmission line », « fault », «unbalanced »

Abstract This paper investigates application of dynamic phasors for accelerated study of electric power system dynamics. Four different modelling techniques are applied to simulate the example system that includes synchronous generator, transmission line and load, under both normal and faulty operation. Effectiveness of dynamic phasor concept for electric power systems modelling is shown, and the ways to achieve the best simulation performance have been discussed. Introduction The more electric aircraft (MEA) concept has been seen as the direction of aircraft power system technology [1]. This concept means more electric loads based on electronic converters which make the system more complex. To simulate and analyse such complex system, efficient time-saving models are of great importance. The functional modelling technique based on state-space vector representation of three-phase quantities (DQ0 technique) for MEA applications has shown great performance during the balanced operation situation [2-4]. However, for unbalanced and faulty regimes this technique loses its efficiency and is time-consuming [5].

Dynamic phasor approach provides a middle ground between sinusoidal quasi-steady-state representation and time-domain representation for electric power system modelling [6]. This approach has been used successfully to analyze induction motor, double-fed induction generator (DFIG), flexible AC transmission system (FACTS) devices etc. in balanced conditions [6-11]. For the unbalanced situation, the authors in paper [12,13] implemented positive and negative sequences into dynamic phasor and the models for synchronous generator, DFIG and transmission lines have been established.

In this paper we investigate application of dynamic phasors based technique for accelerated electric power system (EPS) simulations under both normal and faulty/unbalanced conditions. All the dynamic phasor models are derived from correspondent 3-phase ABC and DQ0 frame based time-domain models. The unbalanced situation is achieved with a fault injector and no positive and negative sequences decomposition is required in proposed dynamic phasor model, which is more convenient for handling unbalanced systems compare to the methods used in [12,13]. The performance of dynamic phasor models is compared against standard three-phase ABC model and DQ0 model. The simulation results have shown the potential advantages of dynamic phasors approach for accelerated EPS simulation studies under faulty/unbalanced operation, and further implementation aspects are discussed.

Concepts and Definitions The generalized averaging method that we used is based on the fact that the time-domain waveform x(t) can be represented on the interval (t-T, t] using a Fourier series of the form [14],

tjkw

kk

setXtx ∑∞

−∞=

= )()( (1)

Where ωs=2π/T. Xk(t) is the kth time-varying Fourier coefficient in complex form, also called dynamic phasor and is determined,

k

t

Tt

tjkwk Xdtetx

TtX s == ∫

−

−)(1)( (2)

k is called dynamic phasor index and the selected set of k will define the accuracy of approximation of original waveform, e.g. K=0,1,2. Some important properties of dynamic phasors are revealed below:

Derivative property:

ks

k

k

ijkwdtid

dtdi +=

(3)

Convolution property:

∑ −

=i

ikkkyxxy

(4) The above two properties play key roles when transferring time-domain models to dynamic phasor models.

Dynamic Phasor Models In this part, the dynamic phasor models of main power system elements will be introduced, which include synchronous generator, transmission line, fault injector etc. All these models are derived through transforming the time-domain models, both in ABC and DQ0 reference frame, to dynamic phasor models by using the two properties shown in (3) and (4).

Synchronous Generator The time-domain models of synchronous generator in ABC and DQ0 coordinates are well established [15]. For the purpose of this study the generator is considered as having one damper winding in both d axis and q axis on the rotor side. The direction of positive stator current is out of the terminals, however, the positive direction for rotor winding current is into the terminals.

• DP-ABC Model

The dynamic phasor model of a synchronous generator in ABC frame (DP-ABC) can be written as,

dtd

iRV kabcskabcsskabcs

λ+−=

(5)

dt

diRV kqdr

kqdrrkqdr

λ+=

(6)

∑∑ −−+−=

iikqdrisr

iikabcsiskabcs iLiL λ

(7)

∑∑ −−+=

iikqdrir

iikabcsirskqdr iLiL λ

(8) Where (fabc)T=[fa, fb, fc], (fqdr)T=[fkq, ffd, fkd], Rs=diag[rs, rs, rs], Rr=diag[rr, rr, rr]. The s and r subscripts denote the variables associated with stator and rotor windings respectively. As mentioned above, the dynamic phasor set K used for approximation defines the accuracy of DP-ABC model and has to be decided at the start of modelling. In the balanced situation, since the

variables in stator side are sinusoidal, it’s enough to choose K=1 for approximating these variables. For the rotor side variables, since they are dc-like, it would be accurate enough to assume K=0. Meanwhile, during the unbalanced situation, the flux will be distorted and include the 3rd harmonic that will induce the 3rd harmonics in both voltage and current at the stator. Considering the unbalanced case, the index sets used in DP-ABC model of synchronous generator are revealed in table I.

Table I: Dynamic phasor indexes used for DP-ABC model of synchronous generator

Variables Dynamic phasor index set K=… Vabcs, Iabcs, λabcs 1,3 λqdr,Vqdr, Iqdr 0,2,4 Parameters

Lss 0,2 Lrs, Lsr 1

Lrr 0

• DP-DQ0 Model

The transformation matrix converting the ABC three-phase model of synchronous generator to DQ0 model is given here:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+−−−−

+−=

5.0 5.0 5.0 )3/2sin( )3/2sin( sin

)3/2cos( )3/2cos( cos

32 πθπθθ

πθπθθ

sK

(9) The DQ0 model of synchronous machine in [15] is a base for deriving the corresponding DP-DQ0 model given below,

ksqdksdq

ksdqsksdq dt

diRV 0

000 λω

λ++−=

(10)

dt

diRV kqdr

kqdrrkqdr

λ+=

(11)

ksqdsrksqdssksqd iLiL 000 −′+′=λ (12)

kqdrrksqdrskqdr iLiL ′+′= 0λ (13)

The dynamic phasor index k set for electrical quantities of stator side is chosen with K=0,2,4 in d- and q-axes. As analysed in ABC frame, when the system is unbalanced, the flux, terminal voltage and stator current include fundamental and 3rd harmonics. Transforming these into DQ0 frame will result in DC, 2nd and 4th harmonics components in stator voltage and current. Therefore the dynamic phasor index sets for DP-DQ0 model shall include these components as it is shown in table II.

Table II: Dynamic phasor indexes used for DP-DQ0 model of synchronous generator Variables Dynamic phasor index set K=…

Vdqs, Idqs, λdqs 0,2,4 V0s, I0s, λ0s 1,3 λqdr,Vqdr, Iqdr 0,2,4 Parameters

L’ss ,L’

rs, L’sr, L’

rr 0

Transmission Line The transmission line in each phase is represented by equivalent RLC circuit in original three-phase ABC model, as shown in Fig.1.

L R

I s,abc

Vs,abc Vr,abc

Ir,abc

Fig.1 Transmission line in ABC representation

The phasor model for transmission line in ABC frame can be easily established as follows,

kabcskabcskabcs

kabcrkabcs iLjiRdt

idLvv ><⋅⋅⋅+><+

><=><−>< ,,

,,, ω

(14)

kabcrkabcr

kabcrkabcs vCjdt

vdCii ><⋅⋅⋅+

><=><−>< ,

,,, ω

(15)

The DQ0 model of transmission line is shown in [4]. Using dynamic phasor transformation on that model, the DP-DQ0 model for transmission line can be given:

kdqoskdqskdqskdqs

kdqrkdqs iLjiLJiRdt

idLvv ><⋅⋅⋅+><⋅⋅⋅+><+

><=><−>< ,0,0,

0,0,0, ωω

(16)

kdqrkdqrkdqr

kdqrkdqs vCJvCJdt

vdCii ><⋅⋅⋅+><⋅⋅⋅+

><=><−>< 0,0,

0,0,0, ωω

(17) Where J= [0 1 0; -1 0 0; 0 0 0]. The approximation index set K is the same as that used in dynamic phasor model of synchronous machine. Fault Injector When EPS line fault occurs, the system will become imbalanced. To model the impact of line faults, a general three-phase fault, illustrated in Fig.2, can be represented by a conductance matrix [16]

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++−

−++−

−−++

=

11 1 1 1

1 111 1

1 1 111

ccbcacbcac

bcbcbbabab

acabacabaa

f

RRRRR

RRRRR

RRRRR

Y

(18)

The three-phase fault current associated with the general fault can be given by

abcffabcf VYI ⋅= (19) The above equation can be expressed in DQ0 frame as

fdqsfsfdq VKYKI 01

0 ⋅= −

(20)

The dynamic phasor model for fault injector on both ABC and DQ0 frame then can be given as:

kabcfkfkabcf VYI = (21)

ikisdqisfsksdq VKYKI

−

−∑= 01

0

(22)

Fig.2: A general three-phase fault configuration The index set K is the same with that chosen in dynamic phasor model of synchronous generator. In the mean time, the transformation matrix Ks has to be transferred as well. As shown in equation (9), the matrix only includes fundamental sinusoidal and constant values, so the index set is chosen as K=0,1.

Model Comparison This section is aimed to compare the accuracy and computational efficiency of the different modelling techniques, which are:

• Standard three-phase domain modelling (ABC) • DQ0 modelling using EPS components library reported in [5](DQ0) • Dynamic phasors (DP) model based on three-phase system equations (DP-ABC) • DP model based on EPS DQ0 equations (DP-DQ0)

For this purpose we used an example EPS with the system elements described above as depicted in Fig.3. The EPS parameters are given in the Appendix. Simulations were run on Pentium(R) 4 CPU/3.40GHz/3.00GB of RAM using Modelica/Dymola v.7.4 software. The Dassl algorithm was chosen in the solver and the tolerance was set to 1e-4. To evaluate the computation efficiency of these modelling techniques, the CPU time taken for the simulation run has been used. The simulation accuracy has been evaluated by comparing the simulation results in the figures below and seems to be good enough for the purpose of this study.

Fig.3: The example EPS scheme

Two types of faults have been studied: line-to-ground fault and line-to-line fault. To simulate the line-to-ground fault, phase A is connected to ground through a 1mΩ resistor. Line-to-line fault is simulated by connecting phase A and phase B with a 1mΩ resistor. All the faults are assumed to occur at t=0.2s. Simulation results from four models are overlaid upon each other.

Fig.4 and Fig.5 show the simulation results with line-to-ground fault occurs at t=0.2s. Fig.4 compares the currents flowing through load resistors. Fig.5 shows the currents before the fault point. As one can see, all four models deliver virtually identical results in both balanced and line-to-ground fault conditions.

Fig.6 and Fig.7 show the simulation results under line-to-line fault situation. Fig.6 depicts the current passing through load resistors. Fig.7 shows the current before fault point. Good agreement among the four models can be seen.

Fig.4: Phase currents flowing through resistor comparison among four modelling methods, with Line-to-ground fault occurs at t=0.2s

Fig.5: Phase currents before fault point comparison among four modelling methods, with Line-to-Ground fault occurs at t=0.2s

Fig.6: Phase currents flowing through resistor comparison among four modelling methods, with Line-to-Line fault occurs at t=0.2s

0.19 0.195 0.2 0.205 0.21 0.215 0.22-400

-200

0

200

400

Ia(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-1000

-500

0

500

Ib(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-500

0

500

Time(s)

Ic(A

)

ABC DQ0 DP-ABC DP-DQ0

0.19 0.195 0.2 0.205 0.21 0.215 0.22-6000

-3000

0

3000

6000

Ia(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-1000

-500

0

500

Ib(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-500

0

500

1000

Time(s)

Ic(A

)

ABC DQ0 DP-ABC DP-DQ0

0.19 0.195 0.2 0.205 0.21 0.215 0.22-400

-200

0

200

400

Ia(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-400

-200

0

200

400

ABC DQ0 DP-ABC DP-DQ0

0.19 0.195 0.2 0.205 0.21 0.215 0.22-500

0

500

1000

Time(s)

Fig.7: Phase currents before fault point comparison among four modelling methods, with Line-to-Line fault occurs at t=0.2s

The consumed CPU time comparisons of the four modelling methods under different fault conditions are given in Tables III and IV. As one can conclude, DQ0 model is the best performing before the fault occurs. This is due to the fact that all the variables are dc-like in DQ0 frame under balanced operation, therefore much larger simulation time step can be applied by the solver. Meanwhile, when the system is unbalanced, the DP-DQ0 model becomes the most efficient.

Fig.8 and Fig.9 indicate CPU time taken by four modelling techniques with line-to-ground and line-to-line fault respectively. It’s shown that ABC model nearly keeps the same pace before and after fault. DQ0 model is the fastest before the fault but become the slowest after fault clearance. For the dynamic phasor model, both DP-DQ0 and DP-ABC show better performance during fault situation, with nearly the same slope in unbalanced condition.

Table III: CPU time consumed comparison with line-to-ground fault occurs at t=0.2s

ABC DQ0 DP-ABC DP-DQ0 t<0.2s 0.750s 0.094s 0.297s 0.204s t=0.4s 1.703s 1.313s 1.032s 0.782s

Fig.8: Elapsed CPU time comparison among different modelling methods

0.19 0.195 0.2 0.205 0.21 0.215 0.22-4000

-2000

0

2000

4000

Ia(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-4000

-2000

0

2000

4000

Ib(A

)

0.19 0.195 0.2 0.205 0.21 0.215 0.22-500

0

500

1000

Time(s)

Ic(A

)

ABC DQ0 DP-ABC DP-DQ0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Simulation Time(s)

CP

U T

ime

Ela

psed

(s)

Fault Occurs

DQ0

ABC

DP-ABC

DP-DQ0

Table IV: CPU time consumed comparison with line-to-line fault occurs at t=0.2s

ABC DQ0 DP-ABC DP-DQ0 t<0.2s 1.000 s 0.141s 0.531s 0.266s t=0.4s 2.422s 2.719s 1.890s 1.094s

Fig.9: Elapsed CPU time comparison among different modelling methods

The above discussion leads to a conclusion that the best simulation performance can be achieved by creating a mixed model employing DQ0 technique during normal conditions and switching over to the DP-DQ0 model automatically when the fault or unbalanced operation is detected. This will be the focus of our future studies.

Conclusion In this paper four modelling methods, which are ABC, DQ0, DP-ABC and DP-DQ0, were compared using an example EPS in both balanced and unbalanced condition. The results showed that in balanced condition the DQ0 model is of the best efficiency, however, under unbalanced conditions the dynamic phasors based models are much more efficient. Based on this work, in our future studies we will aim for a hybrid model that will combine DQ0 and DP-DQ0 models, with automatic switching between them depending on EPS operation condition.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.5

1

1.5

2

2.5

3

Simulation Time(s)

CP

U T

ime

Ela

psed

(s)

Fault occurs

ABC

DP-DQ0

DP-ABC

DQ0

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APPENDIX. PARAMETERS OF THE EXAMPLE POWER SYSTEM STUDIED Generator: Rs=4.4mΩ, Lls=19.8μH, Lmd=0.221mH, Lmq=0.162mH, Rf=68.9mΩ, Llf=32.8μH, Rkd=14.2mΩ, Llkd=34.1μH, Rkq=3.09Ω, Llkq=0.144mH, f=400Hz, ωr=2512s-1 Transmission Line: R=0.1Ω, L=2e-6H, C=1μF

Resistor: R=1Ω.