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 report 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 asynchronous/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

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 without (squares) a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

18

Speed of synchronous generator 9 due to a three-phase fault with and without a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

19

Speed of synchronous generator 24 due to a three-phase fault with and without a wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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. . . . . . . . . . . . . . . . . 10

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

22

Participation factors associated with generator speeds and λ = −0.51±5.9i with wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

23

Participation factors associated with generator speeds and λ = −0.81±9.4i without wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

24

Participation factors associated with generator speeds and λ = −0.78±9.5i with wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

25

Participation factors associated with generator speeds and λ = −0.92±9.7i without wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

26

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

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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 different that the conventional generation they augment or replace due to their inherent intermittency and fundamental difference in their interface to the grid. Conventional synchronous 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 interface 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 implemented to study the impact of increased renewable energy on the behavior of the largescale 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 Generation (DFIG), variable-speed wind generation is added. This type of wind generation was selected due its increasing popularity and connection that utilizes an electrical converter. The report summarizes dynamic models for the power system, summarizes “established” 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 barrier 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 transmission lines, transformers and shunt devices; generators that inject power into the network; 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 ∈