Model Abstraction Techniques for Large-Scale Power Systems · Model Abstraction Techniques for Large-Scale Power Systems Prepared for the U.S. Department of Energy Office of Electricity

Model Abstraction Techniques for

Large-Scale Power Systems

Prepared for the

U.S. Department of Energy

Office of Electricity Delivery and Energy Reliability

Under Cooperative Agreement No. DE-FC26-06NT42847

Hawai‘i Distributed Energy Resource Technologies for Energy Security

Subtask 10.3 Deliverable 2

Report on System Simulation using High Performance Computing

Prepared by

New Mexico Tech New Mexico Institute of Mining and Technology

Submitted to

Hawai‘i Natural Energy Institute School of Ocean and Earth Science and Technology

University of Hawai‘i

October 2012

Acknowledgement: This material is based upon work supported by the United States Department of Energy under Cooperative Agreement Number DE-FC-06NT42847. Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference here in to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Final Report for Task Two

Project: Application of High Performance Computing to

Electric Power System Modeling, Simulation and Analysis

Task Two: Model Abstraction Techniques for Large-scale Power Systems

Sponsor: Hawaii Natural Energy Institute, University of Hawaii

Date: July 5, 2011

i

Abstract

This report presents techniques applicable to the analysis of large-scale electric

power systems. In particular, techniques were selected and implemented that lend

themselves to assessment of the impact of wind energy. The first part of the re-

port summarizes “established” techniques such as small-signal stability based on

eigenvalues and participation factors, trajectory sensitivities and tracking operating

conditions as wind speed and consumption vary. An example analysis is provided

for the IEEE 24-bus reliability test system with a wind farm integrated. The wind

farm is taken to be composed of variable-speed wind turbines, and doubly fed asyn-

chronous/induction generators (DFAG/DFIG) in particular. The second part of

the report summarizes nontraditional approaches based upon “probabilistic testing

for stochastic systems” and “stochastic safety verification using barrier certificates.”

These approaches were investigated for use in the study of electric power systems

with wind farms as stochastic systems, but scaleability and applicability remain in

question.

ii

Contents

1 Introduction 1

2 Model of Power System 3

2.1 Differential-algebraic model . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 DFAG model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 IEEE 24-bus reliability test system (RTS) . . . . . . . . . . . . . . . . . . 8

3 Techniques for Analysis 12

3.1 Wind-power-voltage curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Trajectory sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Small-signal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Participation factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Approaches for Stochastic Systems 24

4.1 Probabilistic testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Safety verification using barrier certificates . . . . . . . . . . . . . . . . . . 24

iii

List of Tables

1 Symbols associated with wind turbine generator model . . . . . . . . . . . 4

2 Number of components in IEEE 24-bus RTS . . . . . . . . . . . . . . . . . 9

3 Location of generator units in IEEE 24-bus RTS . . . . . . . . . . . . . . . 9

iv

List of Figures

1 Block diagram of wind turbine generator model . . . . . . . . . . . . . . . 4

2 One machine, infinite bus (OMIB) system . . . . . . . . . . . . . . . . . . 5

3 Table of eigenvalues for OMIB system with DFAG in power factor control

mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Table of eigenvalues for OMIB system with DFAG in voltage control mode 6

5 Response of DFAG (with power factor control) powers to three-phase fault 6

6 Response of DFAG (with power factor control) terminal voltage and rotor

speed to three-phase fault . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

7 Response of DFAG (with voltage control) powers to three-phase fault . . . 7

8 Response of DFAG (with voltage control) terminal voltage and rotor speed

to three-phase fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

9 IEEE 24-bus reliability test system (RTS) . . . . . . . . . . . . . . . . . . 8

10 IEEE 24-bus RTS in PowerWorld simulator . . . . . . . . . . . . . . . . . 10

11 IEEE 24-bus RTS with one generation station replaced by a wind farm . . 11

12 PV curve for two voltages and varied wind speeds . . . . . . . . . . . . . . 12

13 Initial sensitivities of bus voltages to voltage set-point of synchronous machine 14

14 Initial sensitivities of bus voltages to voltage set-point of DFAG . . . . . . 15

15 Steady-state sensitivities of bus voltages to voltage set-point of synchronous

machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

16 Steady-state sensitivities of bus voltages to voltage set-point of DFAG . . . 15

17 Eigenvalues associated with synchronous generators with (circles) and with-

out (squares) a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

18 Speed of synchronous generator 9 due to a three-phase fault with and with-

out a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

19 Speed of synchronous generator 24 due to a three-phase fault with and

without a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

v

20 Speed of synchronous generator 31 due to a three-phase fault with and

without a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

21 Participation factors associated with generator speeds and λ = −0.42±5.5i

without wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


with wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21




with wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22



26 Participation factors associated with generator speeds and λ = −1.39 ±10.9i with wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

vi

1 Introduction

Environmental, political and social factors continue to drive incorporation of renewable

energy into the electric power system. These sources of renewable energy are quite dif-

ferent that the conventional generation they augment or replace due to their inherent

intermittency and fundamental difference in their interface to the grid. Conventional syn-

chronous generation (e.g., from fossile fuel, nuclear and hydro-electric plants) has long

provided robust and reliable control of electrical power and frequency through an ability

to directly control mechanical power that in turn governs speed of the machine which is

equivalent to electrical frequency of the power produced. In contrast, sources of renewable

energy are subject to ever changing environmental conditions and their electronic inter-

face to the grid decouples the power conversion process from the electrical power input

to the grid.

Due to these two shifts in paradigms, techniques need to be developed and imple-

mented to study the impact of increased renewable energy on the behavior of the large-

scale electric power system. Longer-term, this analysis should be performed by viewing

the power system as a stochastic dynamic system using analysis methods applicable to

such systems (see, for example, the theory presented in [1]), but for now as a first step

researchers are treating the uncertainty of renewable generation and loads separately from

the power system’s fundamental, dynamic properties.

Intermittency and its impact on fluctuations in the state of the power system have

been addressed using Monte Carlo and probabilistic approaches [2–4]. Results of this work

are distributions for the magnitudes and phase angles of voltages at busses, active and

reactive powers injected into busses by generation and loads, active and reactive powers

on transmission lines, etc. These distributions provide the likelihood of unsafe conditions

such as areas that are under-voltage or transmission lines operating above capacity, and

in turn enable mitigation of related failures through planning.

Stability of a power system can be analyzed in a multitude of ways based upon linear

and nonlinear representations of its dynamics. These approaches require mathematical

models of all components in the electric power system to include generation (conventional

and renewable), network, loads and control. Models of the traditional components are

presented in [5, 6], for example, and models of the wide variety of sources of renewable

energy have been presented extensively in the literature. In particular, models of wind

farms composed of variable-speed wind turbines can be found in [7–9].

While analytical approaches for analysis of the nonlinear models of power systems

1

exist [10], they are difficult to apply in general. Therefore, most analysis of power systems

is based upon numerical studies of transient and voltage stability, or analytical studies that

utilize linearization of the dynamic models about an operating point to draw conclusions

locally [5,6,11]. Results of studies based upon linearization of power systems that include

variable-speed wind turbines can be found in references [8, 12, 13].

This report focuses on analysis of power systems at the transmission-level to which

Doubly-Fed Asynchronous Generation (DFAG), also known as Double-Fed Induction Gen-

eration (DFIG), variable-speed wind generation is added. This type of wind generation

was selected due its increasing popularity and connection that utilizes an electrical con-

verter. The report summarizes dynamic models for the power system, summarizes “estab-

lished” techniques for analysis, presents example results for the IEEE 24-bus reliability

test system with a wind farm, and proposes that nontraditional approaches based upon

“probabilistic testing for stochastic systems” and “stochastic safety verification using bar-

rier certificates” should be investigated further as means for more sophisticated analysis.

2

2 Model of Power System

The primary components of an electrical power system are the network made up of trans-

mission lines, transformers and shunt devices; generators that inject power into the net-

work; loads that consume power; and control and protection devices that monitor and

respond to the system’s behavior. The following sections give a summary of how each

component is represented to construct a system-wide dynamic model for simulation and

analysis.

2.1 Differential-algebraic model

Disregarding the discrete events associated with the switching of protection devices, power

systems can be represented as a set of differential-algebraic equations (DAE) of the form

x = f(x, y, ρ) (1)

0 = g(x, y, ρ) (2)

where x ∈ <n is the vector of (dynamic) state variables such as rotor angles, rotor angular

velocities, internal flux linkages, etc. associated with synchronous generators; active and

reactive power order, blade pitch angle, active current command, equivalent voltage for

VAR control command, etc. associated with variable speed wind turbines; and active and

reactive power recovery associated with an exponential recovery model of loads. y ∈ <m is

the vector of algebraic variables that represent the magnitude and phase angle of voltages

at buses where the network is assumed to be in quasi steady-state with little variation in

frequency; ρ ∈ <p is the vector of system parameters such as generator inertia, controller

gains and line admittances; f(·) is the vector function of ordinary differential equations

that represent the behavior of generators, loads and controllers; and g(·) is the vector

function that represents power balance at all buses.

In the context of this report conventional generators were modeled as synchronous

machines with IEEE-Type I DC exciters [5, 6]. These models yield seven dynamic states

for each conventional generator as well as many parameters.

The model for DFAGs was taken from references [7–9] and captures the major com-

ponents of the wind generator that include the generator/converter, electrical control,

turbine and turbine controls as shown in Figure 1. The DFAG can be operated in voltage

control model with Vrfq the set-point for the voltage at the point of interconnection or in

power factor control model where φref is the set-point for the DFAG’s power factor angle.

Key variables for the model of the DFAG are defined in Table 1.

3

Figure 1: Block diagram of wind turbine generator model

Symbol Definition

β blade pitch angle

E′′qcmd equivalent voltage command for VAR control

Pg active power output

Pord active power order

Qg reactive power output

Vreg voltage at point of interconnection

Vterm voltage at terminal bus

vw wind speed

ωg generator rotor speed

Table 1: Symbols associated with wind turbine generator model

Loads are taken in this report to be represented by the exponential recovery model

which is a first-order representation of aggregated load behavior. It is typically fit empir-

ically to measurements recorded at a substation [14].

The network is taken to be in quasi steady-state assuming frequency varies little such

that transmission lines, transformers and capacitive compensation can be represented

4

by impedances. In addition, the network is assumed to be balanced so that only one

(positive-sequence) phase is studied. This model is developed by summing currents, and

ultimately power, at each bus such that the topology and electrical characteristics of the

network can be captured with an admittance matrix [5, 6].

It is noted that the steady-state (equilibrium) solution (x0, y0) of the DAE model (1),

(2) can be found by assuming an equilibrium and setting (1) equal to zero. This leads to

0 = f(x0, y0, ρ) (3)

0 = g(x0, y0, ρ) (4)

which can be solved for (x0, y0) with appropriate choices for some x0, y0, ρ to get a

consistent set of equations. The iterative solution of (4) is typically called the load/power-

flow solution and enforces balance of complex power at each bus.

2.2 DFAG model validation

The DFAG model is quite detailed, so it was validated against results presented in refer-

ence [8] and implemented in the commercial power system simulator PSS/E. The simple

one machine, infinite bus (OMIB) system shown in Figure 2 was utilized to isolate the

dynamics of the DFAG and validate its model through fault responses and eigenvalue

analysis.

Figure 2: One machine, infinite bus (OMIB) system

Figures 3 and 4 show the eigenvalues computed and published for the OMIB in power

factor and voltage control modes, respectively, and shows a favorable comparison. The

eigenvalues were computed via the linearization process described later in the report.

The three-phase fault response of the OMIB was also simulated and compared to

that published for a DFAG in both the power factor and voltage control mode. For a

three-phase fault at bus 2 that lasted 100ms the voltage, speed and power responses are

5

Figure 3: Table of eigenvalues for OMIB system with DFAG in power factor control mode

Figure 4: Table of eigenvalues for OMIB system with DFAG in voltage control mode

shown in Figures 5 - 8 with the published results overlaid. Once again the results compare

favorably to those published.

Figure 5: Response of DFAG (with power factor control) powers to three-phase fault

6

Figure 6: Response of DFAG (with power factor control) terminal voltage and rotor speed

to three-phase fault

Figure 7: Response of DFAG (with voltage control) powers to three-phase fault

Figure 8: Response of DFAG (with voltage control) terminal voltage and rotor speed to

three-phase fault

The eigenvalues and fault responses of the OMIB system were almost identical to those

published; thus, there is confidence that the DFAG model was implemented correctly

and will facilitate realistic system studies when incorporated into larger, multi-generator,

multi-load system models. A larger system is presented in the next subsection, and used

for subsequent results.

7

2.3 IEEE 24-bus reliability test system (RTS)

In order to study the impact of DFAGs on a transmission-level power system, the IEEE

24-bus reliability test system (RTS) shown in Figure 9 and described in reference [15] was

selected for analysis.

A summary of the system is provided in Table 2 where multiple conventional generation

units combine to form generation stations at busses. The numbers of the units and busses

to which they are attached are provided in Table 3 and shown in Figure 10. Figure 10 was

captured from the PowerWorld simulator which was used to verify the load-flow solution

achieved via calculations in Matlab.

Figure 9: IEEE 24-bus reliability test system (RTS)

In order to contrast the characteristics and behavior with and without a wind farm

composed of DFAGs, the conventional generation station at bus 1 was interchanged with

a wind farm as shown in Figure 11. The wind farm was sized such that its rating was the

same as the conventional generation it replaced.

8

Component Quantity

Buses 24

Generation Stations 10

Generation Units 32

Loads 17

Lines 38

Table 2: Number of components in IEEE 24-bus RTS

Bus No. Generator Unit Nos.

1 1 - 4

2 5 - 8

7 9 - 11

13 12 - 14

15 15 - 20

16 21

18 22

21 23

22 24 - 29

23 30 - 32

Table 3: Location of generator units in IEEE 24-bus RTS

9

Figure 10: IEEE 24-bus RTS in PowerWorld simulator

10

Figure 11: IEEE 24-bus RTS with one generation station replaced by a wind farm

11

3 Techniques for Analysis

This section describes techniques selected for analysis of electric power systems with wind

farms. They were chosen due to their applicability to large-scale systems and ability

to provide insight into stability and behavior of the power system as wind farms are

incorporated.

3.1 Wind-power-voltage curves

Voltage stability is often studied using PV and QV curves where a power (active or

reactive) is increased for a load and additional generation is dispatched accordingly [5,6].

As the power consumed is varied in (4), magnitudes of voltages at busses of interest are

calculated and plotted versus the increase in power to create the PV or QV curve. The

curve shows how the magnitude of voltage decreases with the increase in load until a loss

of stability, i.e., loss of solvability of (4). The slope of the curve at any point also provides

a sensitivity of the voltage’s magnitude to a further increase in load.

The difficulty with wind farms in general is that their power is difficult to dispatch and

dependent on wind speed. One means of capturing the impact of wind and load increases

(assuming the power system is in equilibrium) is to use a 3D plot to display the solution

of (4). An example of such a curve is shown in Figure 12 for the 24-bus RTS with one

wind farm (see Figure 11).

Figure 12: PV curve for two voltages and varied wind speeds

The magnitudes of voltages at busses 9 and 12 are plotted in response to an overall

12

increase in load (total active and reactive power consumed) and various wind speeds. As

the wind speed increases and output of the wind farm approaches capacity (above wind

speeds of 13m/s) the curves can be seen to extend further indicating an overall increase

in voltage stability. The concern here would be the shorter distance to low-voltage or

voltage collapse when wind speeds are low.

3.2 Trajectory sensitivities

Trajectory sensitivities capture the variation in system variables x and y due to small

changes in the parameters, ρ [16]. Differentiating (1) and (2) with respect to parameters

of interest yields

xρ = fx(x, y, ρ)xρ + fy(x, y, ρ)yρ + fρ(x, y, ρ) (5)

0 = gx(x, y, ρ)xρ + gy(x, y, ρ)yρ + gρ(x, y, ρ) (6)

where the notation hz = ∂h/∂z results in a matrix and xρ = ∂x/∂ρ is the trajectory

sensitivity vector that captures the variation in the states due to a small change in pa-

rameters. The order of differentiation was interchanged since parameters are assumed to

not vary with time, but rather will only be changed once. xρ can be found by solving the

DAE sensitivity equations (5) and (6) along with the system equations (1) and (2), by

numerical approximation of the derivative via a finite difference of two trajectories where

one is the result of a small parameter change, and also solving for the steady-state value

by assuming (5) and (6) are in equilibrium (xρ = 0).

The simple approach of a finite difference is taken here such that

xρ ≈x(t, ρ0 + ∆ρ)− x(t, ρ0)

∆ρ(7)

for ρ0 the nominal value and ∆ρ small. Trajectory sensitivities at two particular values

of time are of interest: a small amount of time ∆t after the time t = t0 at which the

parameter value is changed and the final time as t → ∞. xρ(t0 + ∆t) gives a measure

of the initial size and direction of state movement immediately after the parameters are

changed by ∆ρ at t0. xρ(∞) provides the amount and direction the state will move if the

system is left to evolve to an equilibrium. Conceptually, one can argue that the initial

sensitivity would be useful for control updates as “best” changes in ρ could be determined

on a fast time-scale, while in contrast, the steady-state sensitivity provides guidance as

to how the parameter should be changed once for longer-term impact.

The 24-bus RTS with all conventional generation (see Figure 9) and one wind farm in

place of one conventional generation station (see Figure 11) was selected for a comparison

13

of the impact of voltage controllers in each type of generation. Initial and final sensitivities

were computed due to a small change in the generator’s set-point for the voltage at bus

1. When the generator at bus 1 is a conventional synchronous machine, the set-point Vref

in the automatic voltage regulator is taken as the parameter, and when the generator is

a wind farm the set-point Vrfq of the voltage controller is taken as the parameter.

Initial sensitivities are shown in Figure 13 for the magnitudes of all bus voltages due

to a small change in the conventional generator’s voltage set-point at bus 1.

Figure 13: Initial sensitivities of bus voltages to voltage set-point of synchronous machine

Initial sensitivities are shown in Figure 14 for the magnitudes of all bus voltages due

to a small change in the wind farm’s voltage set-point for bus 1. Note there are additional

busses in this case due to inclusion of a unit transformer and collection transformer for

the wind farm. The sensitivities of these additional busses are high due to their direct

connection to the DFAGs that make up the wind farm. The other 24 busses in the grid

are of primary interest for comparison to the case with all conventional generation. The

magnitude of the sensitivities at the 24 busses within the grid are quite comparable which

indicates both voltage controllers have a similar initial impact on the grid’s voltages when

changed. This implies either is equally effective in manipulating voltages in the short-

term.

Steady-state sensitivities are shown in Figure 15 for the magnitudes of all bus voltages

due to a small change in the conventional generator’s voltage set-point at bus 1.

Steady-state sensitivities are shown in Figure 16 for the magnitudes of all bus voltages

due to a small change in the wind farm’s voltage set-point for bus 1. Note once again

there are additional busses in this case due to inclusion of a unit transformer and collection

transformer for the wind farm, and that the sensitivities of these additional busses are

high due to their direct connection to the wind farm. The other 24 busses in the grid are

of primary interest for comparison to the case with all conventional generation. There

14

Figure 14: Initial sensitivities of bus voltages to voltage set-point of DFAG

Figure 15: Steady-state sensitivities of bus voltages to voltage set-point of synchronous

machine

is a substantial difference between steady-state sensitivities for the two cases with much

larger values for the wind farm. This provides an indication that should low-voltage

conditions occur and reactive power support be required, wind farms are more effective

than conventional generation. This is most likely due to the converter and electronic

control in the DFAG.

Figure 16: Steady-state sensitivities of bus voltages to voltage set-point of DFAG

15

3.3 Linearization

Performing a Taylor series expansion of the DAE model (1), (2) about an equilibrium

(assumed here to be a desirable operating point) (x0, y0) and neglecting the higher-order

terms yields the following linear differential-algebraic equations [6].

∆x = fx(x0, y0, ρ)∆x+ fy(x0, y0, ρ)∆y (8)

0 = gx(x0, y0, ρ)∆x+ gy(x0, y0, ρ)∆y (9)

where ∆x = x − x0 and ∆y = y − y0. ∆y can be found in (9) and substituted into (8)

to find the following linearized dynamics that describe the behavior of the state variables

∆x.

∆x = (fx − fyg−1y gx)︸︷︷︸

A

∆x (10)

= A∆x (11)

where A ∈ <n×n is commonly referred to as the state matrix, and the solution of (11) is

∆x = eAt∆x0 where eAt is a matrix exponential defined by a Taylor series. The linearized

dynamics can now be studied using a variety of techniques from linear systems and control

theory to draw conclusions about local (near (x0, y0)) stability and behavior.

3.4 Small-signal stability

Small-signal stability centers about the eigenvalues of the state matrix A. Eigenvalues of

A are denoted by λ and are scalar quantities that satisfy

det(A− λI) = 0 (12)

where I is the n× n identity matrix and det(·) is the determinant of a matrix argument.

If distinct eigenvalues are assumed, then there are n of them.

Lyapunov’s first method states the system (11) will be asymptotically stable if all the

eigenvalues of A have negative real parts [5]. In addition, the real part of the eigenvalue

is directly related to the rate of convergence of each mode and the imaginary part is

directly related to the frequency of oscillation associated with each mode. Researchers

have utilized small-signal stability to assess the impact of variable-speed wind turbines

on the overall stability of a power system, and in most cases have found beneficial effects

on electromechanical modes [12,13,17].

16

The eigenvalues associated with the electromechanical modes of the conventional gen-

erators (other than that at bus 1) are plotted in Figure 17. Here again the two sets of

eigenvalues are for the IEEE 24-bus RTS without wind generation (see Figure 9) and

with one wind farm in place of the conventional generator at bus 1 (see Figure 11). It is

noted that eigenvalues have shifted both left and right which indicates the wind farm has

beneficial and detrimental effects, respectively. In addition, the bigger shifts are to the

left, so one can argue that the net effect of the wind farm tends to be beneficial.

Figure 17: Eigenvalues associated with synchronous generators with (circles) and without

(squares) a wind farm

The tendency of the wind farm to improve stability of the electromechanical modes

is verified through the system’s response to a disturbance. Here a three-phase fault was

induced for a short time and the response of the conventional generator’s speed recorded

for both with and without the wind farm. Selected responses for the speed of three

conventional generators are shown in Figures 18, 19 and 20. The rate of damping can

be seen to be roughly the same or slightly better when the wind farm is present which

supports the trends seen in the eigenvalues.

The relationship between eigenvalues and modes, and states is not obvious as each

state ∆x of (11) is made up of a weighted sum of modes. Participation factors are

introduced in the next subsection as a means of measuring the tie between them.

17

Figure 18: Speed of synchronous generator 9 due to a three-phase fault with and without

a wind farm


a wind farm


a wind farm

3.5 Participation factors

Participation factors are an extension of small-signal stability and make use of the lin-

earized system, and eigenvalues and eigenvectors of the state matrix A. It will be assumed

18

that the matrix A has a full set of eigenvectors which corresponds to distinct eigenvalues.

Participation factors provide a measure of the participation of each mode in each state,

and vice versa [11,18].

A vector vi ∈ <n associated with the i-th eigenvalue λi that satisfies

Avi = viλi (13)

is called the right eigenvector. Similarly, the vector wi ∈ <n that satisfies

wTi A = λiwTi (14)

is called the left eigenvector. [·]T denotes the transpose of a matrix/vector quantity and

a standard normalization is assumed such that

wTi vi = 1 (15)

The right eigenvectors can be used as columns of the matrix V = [v1, v2, . . . , vn], the

left eigenvectors can be used as the columns of the matrix W = [w1, w2, . . . , wn] and

eigenvalues can be placed along the diagonal of a matrix Λ = diag(λ1, λ2, . . . , λn). This

permits (13) and (14) to be rewritten in the following matrix forms relating all eigenvalues

and right and left eigenvectors.

AV = V Λ (16)

W TA = ΛW T (17)

A coordinate transformation

∆x = V∆z (18)

diagonalizes the linear dynamics (11) via

V∆z = AV∆z (19)

⇒ ∆z = V −1AV∆z (20)

= W TAV︸︷︷︸Λ

∆z (21)

where W T = V −1 is a property of matrices with distinct eigenvalues.

The solution to (21) is

∆z = eΛt∆z0 (22)

19

where eΛt is a matrix exponential defined by a Taylor series (diagonal in this case) and

∆z0 = V∆x0 is the initial condition. Applying the inverse coordinate transformation

∆z = V −1∆x = W T∆x to (22) yields the solution

∆x = V eΛtW T∆x0 (23)

=n∑i=1

vieλitwTi ∆x0 (24)

to (11). Each term in (24) represents the contribution of a specific mode to the system’s

response. wTi ∆x0 gives the contribution of the initial condition to the i-th mode and vi

represents the amount of activity associated with the i-th mode.

To isolate the amount the i-th mode participates in the k-th state, a non zero initial

condition can be specified for only the k-th state, i.e., ∆x0 = [0, 0, . . . , 0,∆xk0, . . . , 0]T so

that

∆xk =n∑i=1

wkivkieλit∆xk0 (25)

=n∑i=1

pkieλit∆xk0 (26)

where pki = wkivki is defined as the participation factor. From (26) the participation

factor can be seen to describe the participation of the i-th mode in the k-th state.

When placed in matrix form, the participation factors provide a concise overview of the

strength of relationships between modes and states. In general the participation matrix

will have the form

λ1 λ2 . . . λn

P =

p11 p12 . . . p1n

p21 p22 . . . p2n

......

. . ....

pn1 pn2 . . . pnn

∆x1

∆x2

...

∆xn

(27)

where rows and columns indicate modes and states as annotated. In practice the absolute

values of the participation factors are typically used for display and comparison.

Three electromechanical modes were selected for a participation factor-based study

of the IEEE 24-bus RTS without (see Figure 9) and with (see Figure 11) a wind farm

comprised of DFAGs. The three modes were selected by viewing the electromechanical

eigenvalues in Figure 17 and selecting those that were both dominant (to the right) and

had moved.

20

Participation factors for the first mode selected are shown in Figures 21 and 22. Note

participation of the modes in generator units 1-4 disappears due to those synchronous

generators being replaced by the wind farm. In addition, for the most part, the same

modes contribute to same generators’ speeds with the exception of generator 21 having

stronger dependence.

Figure 21: Participation factors associated with generator speeds and λ = −0.42 ± 5.5i

without wind farm


with wind farm

Participation factors for the second mode selected are shown in Figures 23 and 24.

Once again conventional generation units 1-4 were replaced by a wind farm, and thus

their states are removed. Notice in this case a significant change occurred as the mode

contributes to far fewer generators’ speeds when the wind farm is in place. This suggests

a fundamental change in dynamic behavior for the conventional generators involved.

Participation factors for the third and final mode selected are shown in Figures 25

and 26. This mode remains tightly coupled with a small group of generators’ speeds,

but the group changes. This indicates a fundamental change in dynamic behavior for the

conventional generators involved.

21


without wind farm


with wind farm


without wind farm

Participation factors provide insight into the contribution of modes to particular states

based upon a linearization of the nonlinear model of the power system. The reason for,

22

Figure 26: Participation factors associated with generator speeds and λ = −1.39± 10.9i

with wind farm

and implications of, the shifts in contribution as was seen in two cases above remains

unclear and warrants further investigation.

23

4 Approaches for Stochastic Systems

While many approaches are available to study different aspects of electric power systems,

none have progressed to a point where the complete nonlinear, stochastic nature of the

system can be addressed with one unifying theory. The fact of the matter is that the

complexity of the power system’s behavior makes such a theory difficulty to develop and

apply. Two promising approaches from the broader controls and systems communities are

briefly described below. They were investigated as part of the project, but much remains

to be done to determine their applicability.

4.1 Probabilistic testing

Given a stochastic system one can explore its behavior by taking a certain set of initial

conditions and simulating the system starting from them to “test” the likelihood that a

trajectory will enter an unsafe region of the state-space. This approach is appealing due to

its simplicity so long as the computational time is available to run an appropriately large

number of simulations to cover all trajectories. An intermediate approach to probabilistic

testing is proposed in reference [19] where reachability/safety properties are inferred by

using trajectories of an approximate deterministic system and a stochastic bisimulation

that establishes a bound on the quality of the approximation.

The key idea of the approach is that a neighborhood of initial conditions can be

shown to have the same reachability/safety properties, i.e., probability of entering an

unsafe region. This enables rigorous conclusions to be made with far fewer simulations

and trajectories as only one needs to be run from each neighborhood of initial conditions.

Although the constraints on a stochastic bisimulation function are clear, construction of

such a function for the dynamics of a power system turns out to be challenging. These

functions and their application to power systems will continue to be a focus of ongoing

research.

4.2 Safety verification using barrier certificates

Barrier certificates, also known as altitude functions, are proposed in references [20, 21]

to address reachability questions for stochastic systems. The methodology is similar to

stability analysis via Lyapunov functions in that conclusions can be made without com-

puting trajectories for the system. In contrast to the probabilistic testing discussed above,

this approach is symbolic/analytic meaning conclusions are drawn without simulation.

24

The idea is to find a measure of “altitude” for the system such that if the initial states

of the system are in a “low” region, unsafe states are in a “high” region and the expected

rate of change of altitude along the system trajectories is non-increasing, then it is unlikely

for the system to enter unsafe conditions. One nice property of the approach is that an

upper bound is placed on the probability of a trajectory entering the unsafe region.

The approach was investigated for electric power systems with stochastic dynamics due

to fluctuations in wind speeds, but determination of the altitude functions, like Lyapunov

functions, can be difficult. Effort in this area will continue as well.

25

References

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26

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Documents

Model Abstraction Techniques for Large-Scale Power Systems · Model Abstraction Techniques for Large-Scale Power Systems Prepared for the U.S. Department of Energy Office of Electricity