dynamic modeling and analysis of large-scale power systems...

Dynamic Modeling and Analysis of Large-scale Power Systems in theDQ0 Reference Frame

Juri Belikov

Tallinn University of Technology

[email protected]

December 12, 2017

Juri Belikov (TUT) Modeling and Identification December 12, 2017 1 / 26

Sources of energy

Fossil fuel (non-renewable) energy sources:

I Oil, gas, coal, etc.

I Limited and can eventually run out

Renewable energy sources:

I Sun, wind, biomass, tides, waste, etc.

I Unlimited


Motivation

Fossil fuel problems:

I Non-renewable

I Environmental hazards: Greenhouse gas emissions (carbon, nitrogen, and sulfur dioxide,etc.), air and water pollution

I Price fluctuations

I Overdependence

I Resources are running out: Fossil fuels are finite

Possible solution:

Shift energy production from fossil to renewable energy sources


Power systems: Current trends

I increasing interconnection

I more renewable sources

I more small and distributed power sources

I a shift from a centralized approach to adistributed approach

≈310 MW of wind energy by 2016 ≈44983 MW of wind energy by 2016


Renewable energy goals [EU]: present/future

Main directives:

I 2009/72/EChttp://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:

32009L0072&from=EN

I COM(2016) 767/F2https://ec.europa.eu/transparency/regdoc/rep/1/2016/EN/

COM-2016-767-F2-EN-MAIN-PART-1.PDF

EU Goals on renewable energy source (from COM(2016) 767/F2):10.4% by 200717% by 2015>27% by 2030 (Current estimation is 24.3%. EU countries has some work to do). Reaching thistreshhold is in accordance with Paris agreement 2016(http://unfccc.int/paris_agreement/items/9485.php).

National plans by countries:https://ec.europa.eu/energy/en/topics/renewable-energy/national-action-plans

Estonia Goals on renewable energy:5.1% by 2010 (real 9.7%)25% by 2020: https://www.mkm.ee/et/eesmargid-tegevused/arengukavad

27% by 2030: https://elering.ee/taastuvenergia-0

Real time online system: https://dashboard.elering.ee/en/system/production-renewable


http://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32009L0072&from=EN

http://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32009L0072&from=EN

https://ec.europa.eu/transparency/regdoc/rep/1/2016/EN/COM-2016-767-F2-EN-MAIN-PART-1.PDF

https://ec.europa.eu/transparency/regdoc/rep/1/2016/EN/COM-2016-767-F2-EN-MAIN-PART-1.PDF

http://unfccc.int/paris_agreement/items/9485.php

https://ec.europa.eu/energy/en/topics/renewable-energy/national-action-plans

https://www.mkm.ee/et/eesmargid-tegevused/arengukavad

https://elering.ee/taastuvenergia-0

https://dashboard.elering.ee/en/system/production-renewable

Distributed approach: Challenges

How do we manage and control many independent energy sources and make them work together?

I Security

I Efficiency

I Reliability

I Dynamics & Stability

I Design

I Sensing


Modeling 1: Transient models

Network: linear model

d

dtx = Ax + BV

I = Cx + DV

Units: nonlinear models

d

dtξ = f (ξ, I )

V = g(ξ, I )

Advantage: detailed and accurateDisadvantage: too complex


Modeling 2: Quasi-static models (time-varying phasors)

Network: Power-flow equations

Pn(t) =N∑

k=1

|yn,k ||Vn(t)||Vk (t)|

× cos(∠yn,k + δk (t)− δn(t))

Qn(t) = −N∑

k=1

|yn,k ||Vn(t)||Vk (t)|

× sin(∠yn,k + δk (t)− δn(t))

Units: nonlinear but time-invariant

αd2

dt2δ = Pm(t)− 3P(t)− Kd

d

dtδ

V = g(ξ, I )

Advantage: simple models and well-defined operating point ⇒ small-signal stability analysisDisadvantage: models are only valid under assumption of slowly varying signals


Mathematical Tools: DQ0 transformation

Let x represent the quantity to be transformed (current, voltage, or flux), and use the compactnotation xabc = [xa, xb, xc ]T, xdq0 = [xd , xq , x0]T. The dq0 transformation with respect to thereference frame rotating with the angle ωs t can be defined as

xdq0 = Tωs xabc , (1)

with

Tωs =2

3

cos (ωs t) cos(ωs t − 2π

3

)cos(ωs t + 2π

3

)− sin (ωs t) − sin

(ωs t − 2π

3

)− sin

(ωs t + 2π

3

)12

12

12

, (2)

where ωs = 2πfs and fs ∈ {50, 60} Hz being the system nominal frequency.

symmetric 6= balancedJuri Belikov (TUT) Modeling and Identification December 12, 2017 9 / 26

DQ0 Transformation (cont.)

Advantages:

+ Sinusoidal (AC) signals are mapped into constant (DC) or slowly varying signals atsteady-state

+ Inherits advantages of both quasi-static and abc models

+ The analysis and controller design are significantly simplified

Disadvantage:

– Network is assumed to be symmetric

Table: Comparison of approaches for dynamic modeling

Model Operating Small- High Non-symmetric

point signal frequencies networks

time-varying phasors X X X X

abc X X X X

dq0 X X X X


Elementary passive components: Inductor L

Consider a network with a single three phase inductor in the native abc reference frame.

unit

Lva(t) ia(t)

Lvb(t) ib(t)

Lvc (t) ic (t)

A model of the symmetric three-phase inductor is given by

Ld

dtIabc,12 = Vabc,1 − Vabc,2. (3)


Elementary passive components (cont.): Inductor L

The differentiation of Idq0 = Tωs Iabc results in

d

dtIdq0 =

dTωs

dtIabc + Tωs

d

dtIabc , (4)

which after simple algebraic manipulations yields

d

dtid,12 = ωs iq,1 +

1

L

(vd,1 − vd,2

),

d

dtiq,12 = −ωs id,1 +

1

L(vq,1 − vq,2) ,

d

dti0,12 =

1

L(v0,1 − v0,2) .

(5)

This equation describes a state-space model of a symmetric three-phase inductor.


Elementary passive components: Capacitor C and resistor R

The model of a symmetric three-phase capacitor C is given as

d

dt

(Vdq0,1 − Vdq0,2

)=W

(Vdq0,1 − Vdq0,2

)+

1

CIdq0,12. (6)

And for a symmetric three-phase resistor R the model is given by the simple static relations

Vdq0 = I3RIdq0, (7)

where I3 denotes the 3× 3 identity matrix.


Transmission network: Frequency domain model

In symmetric power networks, a dynamic model based on dq0 signals can be described asId (s)Iq(s)I0(s)

=

N1(s) jN2(s) 0−jN2(s) N1(s) 0

0 0 Y bus (s)

Vd (s)Vq(s)V0(s)

,where Y bus (s) is the frequency dependent nodal admittance matrix and

N1(s) :=1

2

(Y bus (s + jωs ) + Y bus (s − jωs )

),

N2(s) :=1

2

(Y bus (s + jωs )− Y bus (s − jωs )

).

Remark: If the general Y (s − jωs) can be approximated by a constant matrix when s → 0, thenthe dynamic model is quasi-static, and the network may be modeled by means of time-varyingphasors.


Transmission network: More details

Network topology (by MatPower1)

yi

shunt

element

bus i bus k

yk

shunt

element

Lik Rikideal

transformer

τik : 1

Figure: Standard branch connecting buses i and k.

Nodal admittance matrix:

Yik (s) =

C̃i s

1 + C̃i R̃i s+

1

Li s+

1

Ri+∑

k∈Fi

1

L`1`2s + R`1`2

+∑k∈Ti

1

τ2`1`2

(L`1`2

s + R`1`2

) if i = k,

−1

τ`1`2

(L`1`2

s + R`1`2

) if i 6= k.

1R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady-state operations, planning, and analysistools for power systems research and education,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 12–19, Feb. 2011.


Synchronous machine: Simplified model

The dynamic behavior of the angle δ such that δ = θ − ωs t + π/2 is described by

d2

dt2δ =

poles

2Jωs

(−P3φ + 3Pref −

1

D

d

dtδ

), (8)

which is the classic swing equation with the droop control mechanism. The term J is the rotormoment of inertia, poles is the number of machine poles (must be even), Pref is the single-phasereference power, and D represents the droop control sloop parameter. The three-phase power canbe computed by

P3φ =3

2(vd id + vq iq + 2v0i0) . (9)

Let δ = φ1, then the state equations become

d

dtδ = ω − ωs ,

d

dtω =

poles

2Jωs

(−

3

2Ve (cos(δ)id + sin(δ)iq) + 3Pref −

1

D(ω − ωs )

),

(10)

and the outputs are defined byvd = Ve cos (δ)

vq = Ve sin (δ)

v0 = 0.

(11)


Synchronous machine: Physical model

Recall a more sophisticated (physical) model of a synchronous machine. The model presentedherein captures the interaction of the direct-axis magnetic field with the quadrature-axis mmf,and the quadrature-axis magnetic field with the direct-axis mmf, as well as the effects ofresistances, transformer voltages, field winding dynamics, and salient poles.

Table: Nomenclature: Synchronous machine

λd , λq , λ0 flux linkages

λf field winding flux linkage

vd , vq , v0 stator voltages

id , iq , i0 stator currents

vf , if field winding voltage and current

Ld , Lq , L0 synchronous inductances

Laf mutual inductance between the field winding and phase a

Lff self-inductance of the field winding

Ra, Rf armature and field winding resistance

J rotor moment of inertia

Tm mechanical torque


Synchronous machine: Physical model (cont.)

The state equations of a synchronous machine in the dq0 reference frame (with respect to ωs t)are given by

d

dtφ1 = −

2RaLff

L2β

φ1 + φ2φ5 +2RaLaf

L2β

φ4 + sin(φ6)vd − cos(φ6)vq ,

d

dtφ2 = −

Ra

Lqφ2 − φ1φ5 + cos(φ6)vd + sin(φ6)vq ,

d

dtφ3 = −

Ra

L0φ3 + v0,

d

dtφ4 =

3Rf Laf

L2β

φ1 −2Rf Ld

L2β

φ4 + vf ,

d

dtφ5 =

poles

2J

(Tm +

3L2β − 6Lff Lq

2L2βLq

φ1φ2 +3Laf

L2β

φ2φ4

),

d

dtφ6 = φ5 − ωs ,

(12)

where L2β = 2Ld Lff − 3L2

af . In this model, the state variables are selected as φ1 = λd , φ2 = λq ,

φ3 = λ0, φ4 = λf , φ5 = ω, δ = φ6 and the inputs as vd , vq , v0, vf , Tm.


Synchronous machine: Physical vs. Simplified


Examples: State-space representations (matrix form)

9-bus system x ∈ R45, u, y ∈ R27

Aξ

Cξ

Bξ

Dξ

200-bus x ∈ R1119, u, y ∈ R600

Aξ

Cξ

Bξ

Dξ

US state of Illinois

2383-bus x ∈ R15675, u, y ∈ R7149

Aξ

Cξ

Bξ

Dξ

Polish system: winter 1999-2000 peak

4 5 6 9 14 24 30 39 57 118

2383

2736

90

95

100

Sparsity

(%)

abc

dq0

4 5 6 9 14 24 30 39 57 118 2383

2736

Number of buses

100

101

102

103

104

105

Nonzero

elem

ents

abc dq0

4 5 6 9 14 24 30 39 57 118 2383

2736

90

95

100Sparsity

(%)

abc

dq0

4 5 6 9 14 24 30 39 57 118

2383

2736

Number of buses

100

101

102

103

104

105

Non

zero

elem

ents

abc dq0


Examples: 118-bus network (single-line diagram)


Examples: 118-bus network (matrices)


Examples: 118-bus network (Scenario 1)

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

Real

-400

-200

0

200

400

Imag

Figure: Eigenanalysis: root locus of largest eigenvalues whenactive power consumption is changed. Diamonds (�) and crosses(×) correspond to quasi-static and dq0 models, respectively.

0 5 10 15 20 25 30

Time [s]

-0.05

0

0.05

0.1

0.15

i d,27[A

/MW

]

Figure: Comparison of time domain responses. The linescorrespond to quasi-static (’ ’), abc (’ ’), and dq0 (’ ’)models.

Table: Largest Eigenvalues: Increase in Active Power Consumption

Model Eig. # Initial (4242 MW) Step (50%)

dq0 1 −0.1629 −0.0848

qs 1 −0.1628 −0.0848

dq0 2 −0.2390 −0.1908

qs 2 −0.2389 −0.1907

dq0 3, 4 −0.2941± 314.1393j −0.2940± 314.1394j

qs 3 −0.6200 −0.4121


Examples: 118-bus network (Scenario 1)

-30 -20 -10 0

Real

-200

-100

0

100

200

Imag

100%

-30 -20 -10 0

Real

-200

-100

0

100

200

−25%

-30 -20 -10 0

Real

-200

-100

0

100

200

−50%

Figure: Eigenanalysis: root locus of largest eigenvalues whenactive power consumption is changed. Diamonds (�) and crosses(×) correspond to quasi-static and dq0 models, respectively.

0 1 2 3 4 5 6 7 8

Time [s]

-0.05

0

0.05

0.1

i d,27[A

/MW

]

8.2 8.3 8.4 8.5

0.05

0.1

Figure: Comparison of time domain responses. The linescorrespond to quasi-static (’ ’), abc (’ ’), and dq0 (’ ’)models.

Table: Largest Eigenvalues: Changes in Damping Factor

Model Kd Eigenvalues

dq0100%

−0.1629 −0.2390 −29.356± 185.681j

qs −0.1628 −0.2389

dq0 −25%−0.1630 −0.2390 −13.916± 185.437j

qs −0.1629 −0.2389

dq0 −50%−0.1631 −0.2390 2.668± 183.11j

qs −0.1630 −0.2389


Software package

“Toolbox for Modeling and Analysis of Power Networks in the DQ0 Reference Frame”

MATLAB Central File Exchangehttps://www.mathworks.com/matlabcentral/fileexchange/58702

Manual & Tutorialhttps://a-lab.ee/projects/dq0-dynamics

Currently, the package contains:

I Construct the minimal state-space model of a power network from given ??(??) matrix

I Construct state-space models of common units

I Derive feedback-connected system

I Small-signal stability analysis

I Compute step response of very large systems ¿104 states

I Various examples of different networks ranging from 2 to 2736 buses (mainly based onMatPower)

I Graphical user interface/Tutorial

I Etc.


https://www.mathworks.com/matlabcentral/fileexchange/58702

https://a-lab.ee/projects/dq0-dynamics

Thank you very much for your attention!

Any questions?


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