DEGREE PROJECT, IN , SECOND LEVEL ELECTRIC POWER SYSTEMS STOCKHOLM, SWEDEN 2015 Impact of Large Amounts of Wind Power on Primary Frequency Control A TECHNICAL AND ECONOMIC STUDY NAKISA FARROKHSERESHT KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING
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DEGREE PROJECT, IN , SECOND LEVELELECTRIC POWER SYSTEMS
STOCKHOLM, SWEDEN 2015
Impact of Large Amounts of WindPower on Primary Frequency Control
A TECHNICAL AND ECONOMIC STUDY
NAKISA FARROKHSERESHT
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ELECTRICAL ENGINEERING
Impact of Large Amounts of WindPower on Primary Frequency Control: a
Technical and Economic Study
Author:
Nakisa Farrokhseresht
Supervisor:
Prof. Mohammad Reza Hesamzadeh
Prof. Hector Chavez
Examiner:
Prof. Mohammad Reza Hesamzadeh
A thesis submitted in fulfilment of the requirements
for the degree of Master of Science
in the
Electricity Market Research Group
Electric Power Systems Department
School of Electrical Engineering
KTH Royal Institute of Technology
Abstract
Renewable energy sources help reaching the environmental, social and economic goals
of producing electrical energy in a clean and sustainable matter. Among the various
renewable resources, wind power is assumed to have the most favorable technical and
economic prospects and offers significant potential for reducing greenhouse gas (GHG)
emissions. As wind power installations are more and more common in power systems,
additional research is needed in order to guarantee the quality and the stability of the
power system operation.
Maintaining the frequency as close as possible to its rated level is one of the most im-
portant tasks for grid operators in order to maintain a stable electricity grid. However,
the significant penetration of wind generation in power grids has raised new challenges
in the operational and planning decisions of power systems. Wind turbine units almost
always include power converters decoupling the frequency dynamics of the wind power
generators from those of the grid. This decoupling causes a reduction in the total system
inertia, affecting the system’s ability to overcome frequency disturbances.
To study the impact of wind power on the system inertia, first the Nordic 32-A System,
representing a scaled version of the Swedish grid, is implemented in PSS/E. A system
identification of model parameters with actual data follows. This ad-hoc identification
method determines the dynamic parameters of the governors and prime movers in the
model. The two metrics of primary frequency control; the instantaneous minimum fre-
quency and the rate of change of frequency (ROCOF) are simulated using the identified
power system, and via an extrapolation, the maximum wind power penetration in Swe-
den is found, considering that the system has to comply with the instantaneous minimum
frequency requirements and also that the tripping of the generators’ ROCOF relays is
prevented.
The second part of the work focuses on an economic study of the cost to guarantee
an adequate frequency response, particulary the Primary Reserve (PR). The Primary
Reserves is the capacity of the generators that is reserved for the governors to use for Pri-
mary Frequency Control (PFC). Primary Reserves also include the ramping capability
requirement of power plants for regulating power imbalances caused by contingencies.
Recent studies have shown that having more renewable resources, such as wind with
no PFC capability as well as an electricity market design with no incentive for PFC,
are important drivers for a decline in the frequency response in the system. One so-
lution is the careful design of a PFC ancillary service market by introducing suitable
constraints to ensure the adequacy of Primary Frequency Control. However, applying
these constraints will increase the generation cost especially when more and more wind
power is integrated. This work proposes the use of an adequacy constraint to evaluate
i
the economic impact of wind integration with respect to its influence on guaranteeing
an adequate PFC. To analyze the cost increment for maintaining an adequate frequency
response in the presence of wind power, an optimal power flow (OPF) problem is de-
signed with an objective function of the generation cost minimization and considering a
PFC adequacy constraint. The results show that the inclusion of the new constraints in
the optimal dispatch OPF leads to a higher dispatch cost.
Keywords: Inertial response, primary frequency control, power system simulation, sys-
tem identification, wind power integration, power system optimization optimal power
flow
ii
Acknowledgements
There are several people I would like to thank for helping me not only to complete this
master thesis project but generally, in my life in Sweden and Belgium.
I would like to begin by thanking Associate Professor Mohammad Reza Hesamzadeh
for creating the thesis project, and giving me the opportunity to work on it. Professor
Hesamzadeh has been my main supervisor, and has given much appreciated and con-
tinuous support, encouragement and positive interaction. Also I would like to express
my gratitude to my other supervisor, Assistant Professor Hector Chavez, for his support
and guidance. His valuable comments, insight and encouragement were always of the
greatest assistance to me.
I would like to give a special acknowledgement to my master program coordinator Pro-
fessor Johan Driesen for his guidance and kind help. In addition, a special thanks goes
to EIT/KIC-InnoEnergy for funding my two years master program: Energy in Smart
Cities. I owe Bert Willems a lot of thanks. You were not only my financial coordinator
but more importantly you became one of my best friends. I sincerely want to thank
Hossein Shahrokni, my course instructor. Thank you for your kind help and guidance.
I would like to thank my colleagues of the Electricity Market Research Group of KTH,
especially Mahir Sarfati, the people in the Electric Power Systems department of KTH,
particulary Dr. Ebrahim Shayesteh, and my friends in the ELECTA group of the depart-
ment of Electrical Engineering of KULeuven, in particular Dr. Priyanko Guha Thakurta.
I am also grateful to my boss, Mr. Hassan Khamseh of the BIDEC company, where I
have been working for almost four years as a mechanical engineer and never forget his
support and his warm encouragement. Many thanks go to my lovely friends in Iran for
cheering me up when I needed it; Mina Safari, Zohreh Kashi, Shahrzad Mohammadpour,
Pegah Tiba and Farzad Farkhondehkalam.
I will never forget all the kindness from the Jacqmaer family; Frans, Hilde and Pieter.
Thanks for everything, for your kind wishes, your prays and for all the candles you
lighted for my success. During these years, you were with me either in times of joy or
difficulties and you help me sincerely and I thank you from the bottom of my heart.
Last, but not least, I would like to express my gratitude to my lovely family and my
uncle Ali Farrokhseresht for their financial and emotional support. Thank you Babaee
to stimulate the love for nature in me and teach me to dare to dream and hold on to
my dreams. Thanks to Giti, my tree of life, I learned that the goal is not of the greatest
importance, but that the path leading to the goal is more valuable! Finally my little
sister, my cute classmate! Thank you for sharing this fascinating journey with me!
5.4 The control diagram for the Simplified Excitation System . . . . . . . . . 43
5.5 The dynamic control model for STAB2A . . . . . . . . . . . . . . . . . . . 44
5.6 The block diagram for the HYGOV governor . . . . . . . . . . . . . . . . 44
6.1 Measured frequency in the Nordic system (NORDEL) after a sudden trip-ping of 530, 800 and 1100 MW generation . . . . . . . . . . . . . . . . . . 47
6.2 Procedure for calculating the COI frequency . . . . . . . . . . . . . . . . . 51
6.3 Frequency response of the original Nordic 32-A grid and the measuredfrequency response after a actual contingency . . . . . . . . . . . . . . . . 52
6.4 Algorithm for the ad-hoc model identification method . . . . . . . . . . . 54
6.5 Algorithm for calculating the ROCOF and frequency Nadir after wind isintegrated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.6 Frequency response after contingency of 550 MW for different amountsof integrated wind power production . . . . . . . . . . . . . . . . . . . . . 58
6.7 Linear extrapolation of the predicted ROCOF and frequency Nadir behavior 58
vii
6.8 Result of identification: comparison of real frequency data and the simu-lated center-of-inertia frequency . . . . . . . . . . . . . . . . . . . . . . . . 61
6.9 Calculating the ramp rate ci as the slope of the mechanical power over time 62
6.10 Method to determine the change in generation cost when the PFC ade-quacy constraints are imposed . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.11 Inertia and cost difference when wind is integrated in the system . . . . . 70
Figure 1.1: World cumulative installed capacity of wind power [3]
from wind power has increased significantly from 0.5 % in 2003 to 4.4 % in 2012 [4].
Over the period 2003 - 2012, the production of electricity from wind power has been
increased more than tenfold (Figure 1.3). Sweden also has a planning framework for
wind power, projecting a production of 17 GW (6 GW onshore, 11 GW offshore) by
2030 [5].
As wind power installations are more and more common in power systems, additional
research is needed in order to guarantee the quality and the security of the power system
operation in view of the increased presence of this new energy source which has different
characteristics from traditional sources.
2
1.1 Background Chapter 1
0 20,000 40,000 60,000 80,000 100,000120,000
Europe
Asia
NorthlAmerica
LatinlAmerica
PacificlRegion
AfricalandlMiddlelEast
Series1
0 50,000 100,000 150,000 200,000 250,000
Europe
Asia
NorthlAmerica
LatinlAmerica
PacificlRegion
AfricalandlMiddlelEast
Europe AsiaNorth
AmericaLatin
AmericaPacificRegion
AfricalandMiddlelEast
Endl2012 109,817 97,715 67,748 3,530 3,219 1,165
Endl2013 121,474 115,927 70,811 4,764 3,874 1,255
Global wind power capacity (MW)
Figure 1.2: Regional distribution of the globally installed wind power capacity (MW)for the end of 2012 and 2013 [3]
0
1
2
3
4
5
6
7
8
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Win
d p
rod
uct
ion
(TW
h)
Figure 1.3: Electricity production by wind power in Sweden [4]
1.1.2 Power Systems Stability
Power systems stability has been recognized as an important issue for a secure system
operation [6]. Power system stability is the ability of a power system to regain an equi-
librium state after being subjected to a physical disturbance [7]. The study of power
systems stability can be divided into the following topics: the study of rotor angle sta-
bility, of frequency stability and of voltage stability [7]. This classification of power
systems stability is shown in Figure 1.4. This work focuses on the frequency stability.
3
1.1 Background Chapter 1
power system stability
frequency stability
small distrubance
angle stability
rotor angle stability
voltage stability
transientstability
largedisturbance
voltage stability
smalldisturbance
voltage stability
Figure 1.4: Classification of power systems stability [7]
Maintaining the frequency as close as possible to its rated level is one of the most im-
portant tasks for grid operators in order to maintain a stable electricity grid [8]. If the
frequency deviates significantly from its scheduled value, Under Frequency Load Shed-
ding (UFLS) is more likely to occur. Also the possibility of tripping of the over-frequency
generator protection relays increases which can lead to a blackout [8].
The definition of frequency stability given by CIGRE and IEEE is the following: “Fre-
quency stability refers to the ability of a power system to maintain a steady frequency
following a severe system disturbance, resulting in a significant imbalance between gen-
eration and load” [9]. Primary Frequency Control (PFC) is the leading mechanism of
the frequency control system to ensure reliable operation [10]. The PFC is defined by
ENTSO-E as “the power delivered by the rotating masses of the synchronous machines
in response to frequency drops” and also as “the governor response that acts to arrest
frequency decays” [11].
1.1.3 Generation Scheduling
In order to have the secure and the stable power system, the socioeconomic cost should
be minimized. This can be done by scheduling the generation well. The generation
scheduling normally consists of three time frames: the day-ahead market (DAM), the
intraday market and the real-time market (RTM). The day-ahead market (DAM) usually
opens in the morning, on the day before the actual dispatch of the generation units [12].
It is also called the planning period and in this period market participants submit their
bids and offers to the market based on forecasts of the loads. Then, the market operator
sets the forecast price and the forecast dispatch level. The next phase is the intraday
4
1.2 Problem Definition Chapter 1
market which takes place one hour before the actual day (in the case of NORDEL).
Additional information such as updated forecasts and units’ availability help the market
participants to make adjustments in their trading [13]. Finally, in the real time market
which is also known as the operating period, the market operator defines the real price
and the real dispatch for all the participants. The operation of the real-time markets
varies from bidding zone to bidding zone, and can be done over a time period of 5 min-
utes, up till periods of one hour before the actual dispatch [12].
However, the load forecast in the planning period may not always be correct. There-
fore in the real-time market, generators may have to change their production instantly.
Hence, the frequency control system is required to control the generator outputs. But
this frequency control system needs “reserve capacity” in order to operate adequately.
One of the important parts of these reserves is the primary reserve (PR). The “primary
reserve” is the capacity that is required by the primary frequency control system and it
is employed to stabilize the frequency deviation in the entire interconnected grid [14].
1.2 Problem Definition
There are two main types of wind turbine generators: fixed-speed and variable-speed.
In the fixed-speed wind turbine, the generator is coupled via a transformer immediately
to the grid. But a more common turbine generator is the variable-speed wind turbine
due to its advantages; it generates an almost constant torque, it can absorb wind fluc-
tuations, and it can improve the power quality of the grid. However, this type of wind
turbine has negative effects on the PFC [15]. Since in modern variable-speed wind tur-
bines power electronic converters are employed to decouple the generator from the grid,
the moving parts of the wind turbines are not synchronized with the system frequency.
Also, a large penetration of wind power implicates a reduction of the power supplied by
conventional synchronous generators, so the contribution of wind machines to the total
system inertia is low to zero [15] [16].
In addition, recent studies have shown that having more renewable resources, such as
wind with no PFC capability as well as an electricity market design with no incentive
for providing PFC, are important drivers for a decline in the frequency response in the
system [8] [17]. Moreover, a classical optimal power flow problem without additional
5
1.3 Thesis objectives Chapter 1
system constraints including the provision of a reserve capacity can not guarantee an
adequate operation of the grid in the presence of a large amount of wind power. Fi-
gure 1.5 summarizes the problem definition. This thesis seeks to investigate the problem
and provide solutions.
Large penetration of wind
Reduce the power suppliedby conventional
generators
- Less overall inertia in the grid- Less PFC capability- Less reserve capacity
Insufficientfrequency stability
Figure 1.5: Problem when great amounts of wind power are integrated in a powergrid
1.3 Thesis objectives
This thesis aims to perform a technical and economic analysis of the effect on the primary
frequency control of the Swedish grid, when a high amount of wind power is integrated in
the system. In the technical part of the thesis, the penetration level of wind generation is
determined that leads to insufficient PFC for the case of the Swedish grid. The following
items are discussed in this part:
• Review of the NORDEL grid code requirements for primary frequency control,
• Implementing the Nordic 32-A test system as the representation of the Swedish
grid in PSS/E,
• Identification of the grid’s model parameters with actual data,
• Including wind turbines into the model,
• Evaluating the impact of wind penetration on a few important primary frequency
control metrics,
• Determining the amount of wind generation that leads to an inadequate primary
frequency control.
6
1.5 Published papers Chapter 1
The second part of the work is economic study of the influence of large amount of wind
on PFC. The economic part has the aim to carefully design an OPF for the economic
operation of the system so that an adequate primary frequency control is guaranteed.
However, introducing additional constraints will increase the generation cost, especially
if more and more wind power is integrated. The following items are considered in this
part:
• Applying governors with ramp rate capability,
• Performing “stress test” to calculate ramp rate capability of each generator in the
gird,
• Including wind turbines into the identified model,
• Formulating an optimal power flow including constraints to guarantee an adequate
PFC operation,
• Evaluating the economic impact of wind integration when the developed mini-
mal requirements to ensure primary frequency adequacy are added to the control
system.
1.4 Resources/Tools Used for that Purpose
In this work the Nordic 32-A test system representing the real Swedish grid is im-
plemented in the power system analysis software PSS/E. The procedure for running
power flows, performing dynamic simulations and integrating wind turbines in the grid
in PSS/E are explained in the Appendix A. An ad-hoc system identification method
is applied in this work which requires PSS/E to be automated. In order to automate
PSS/E, calculations are performed in Matlab, which calls Python to execute PSS/E. All
the Matlab and Python codes for the PSS/E automation are provided in Appendix C.
Moreover, the optimization problem in the second part of the thesis is solved by the
KNITRO solver in the General Algebraic Modeling System (GAMS) platform [18]. The
GAMS codes are provided in Appendix D.
7
1.6 Outline of this work Chapter 1
1.5 Published papers
• N. Farrokhseresht, H. Chavez, M. R. Hesamzadeh, Determination of Acceptable
Inertia Limit for Ensuring Adequacy Under High Levels of Wind Integration, In-
ternational Conference on European Energy Market, Krakow, Poland, 28-30 May
2014.
• N. Farrokhseresht, H. Chavez, M. R. Hesamzadeh, Economic Impact of Wind Inte-
gration on Primary Frequency Response, IEEE PowerTech Conference, Eindhoven,
the Netherlands, 29 June-2 July 2015.
1.6 Outline of this work
The report is written in 7 chapters with the following descriptions:
• Chapter 1 has been specified to describe the thesis. This includes the background
of the thesis and a description of problem definitions and different thesis steps.
• Chapter 2 focuses on frequency control for power systems and the concepts of pri-
mary, secondary and tertiary control are briefly explained. Two important metrics
for primary frequency control are introduced. In the next part, the requirements
for primary frequency control in the UK, Ireland and Sweden are presented. This
Chapter ends with a brief explanation about the power system analysis software
PSS/E.
• Chapter 3 concentrates on wind power and two main wind turbine technologies:
the fixed-speed and variable-speed wind turbine are presented. Also in this Chap-
ter, the doubly-fed induction generator (DFIG), as one kind of the variable-speed
wind turbines which is commonly used, is explained.
• Chapter 4 discuses DC Power flow, Economic Dispatch (ED) and Optimal Power
Flow (OPF). The last Section of this Chapter presents the new adequacy con-
straints in a classical OPF for ensuring PFC adequacy.
8
1.6 Outline of this work Chapter 1
• Chapter 5 illustrates first, a case study consisting of the Nordic 32-A test system,
representing a scaled version of the Swedish grid. Then the modification on the
original Nordic 32-A test system which are needed in order to have capability of
dynamic simulation will be explained.
• Chapter 6 explains the method for studying the impact of wind integration on
the adequacy of primary frequency control , and also the economic impact of the
wind integration on the generation cost, where as a test grid the CIGRE Nordic
32-A system was taken. This Chapter includes an ad-hoc identification method
which required that PSS/E was automated. It discusses wind integration and
presents an OPF with PFC adequacy constraints. In the next part of this chapter,
two different scenarios are studied; the first scenario considers the PFC adequacy
constraint in the OPF, while in the second scenario, the PFC constraint is not
included. This way, the economic cost of this constraint can be found. At the end
of each Section, the simulation results and a summary are given in detail.
• Chapter 7 summarizes the main conclusions and provides some recommendations
for future research.
9
Chapter 2
Frequency Control for Power
Systems
2.1 Introduction
Nowadays, the demands on the quality and security of supply of the voltage and fre-
quency are higher and automatic controllers and regulators were introduced in order
to meet these requirements [19]. Therefore there is a need for an ancillary service1 to
supply these control actions. The task of the control systems of a power system is to
keep the system within acceptable operating limits in such a way that the security of
supply is maintained and the quality of the power, such as the voltage magnitude and
the frequency, is within specified limits.
In this chapter, first the basics of frequency control including the concepts of primary,
secondary and tertiary control are briefly explained. After describing the inertial re-
sponse and primary frequency control in detail, some important metrics to describe the
primary frequency control adequacy are provided. The primary frequency control re-
quirement for UK, Ireland and Sweden are discussed in the next part. The chapter ends
with a brief explanation about the power system analysis software PSS/E which is used
in this work to model a power system and analyze the frequency control.
1Ancillary services are defined as all services required by the transmission or distribution systemoperator to enable them to maintain the integrity and stability of the transmission or distributionsystem as well as the power quality [20]
10
2.3 Frequency Control Systems Chapter 2
2.2 Frequency Control Systems
The response of the power system and its generators to a frequency change can be
divided into four phases (Table 2.1) [10] [21].
Table 2.1: Different frequency controls levels
No. Control Name Time frame Control objectives
1 Inertial response 0-2 s Transient frequency dip minimization
2 Primary control 2-20 s Arrest frequency decays
3 Secondary control 20 s - 2 min Steady-state frequency
4 Tertiary control 15 min Economic-dispatch
In the first phase, which takes place during the first seconds after the frequency changes,
the rotor of the generators releases or absorbs part of its kinetic energy. This action is
mathematically described by the swing equation and is called “Inertial Response”. The
inertial response is inherently provided by conventional generators in power systems and
no control is activated within this phase.
If the frequency signal deviates from the set value, a signal is produced that will influ-
ence the valves, gates, servos, etc, in order to bring the frequency back to an acceptable
value. That is the purpose of “Primary Control”. All the generators are participating
in the primary control irrespective of the location of the disturbance. A typical time
response for this primary control is in the order of a few seconds (2-20 s).
In the “Secondary Control” phase, the remaining frequency error which is still present
after the primary frequency response phase is compensated by adjusting the power set-
points of the generators. The secondary control acts in a time response period of a few
seconds to minutes, typically 20 s-2 min.
Finally the “Tertiary Control” level occurs in a time frame of minutes (typically 15
minutes) and modifies the set-points of the active power in the generators to achieve
a desired economically optimal global power system operation strategy. Not only fre-
quency and active power controls are considered, but also voltage and reactive power
are controlled in this stage.
This work focuses on first two levels which will be explained in detail in the next parts.
11
2.3 Frequency Control Systems Chapter 2
2.3 Inertial Response
When there is a large contingency in the grid, the frequency begins to decline imme-
diately and the rate of this initial decline is mainly determined by the inertia of the
system. As the inertia is connected with the motion of synchronous devices, the swing
equation describes the inertia response well, thus it will be explained below.
2.3.1 Swing Equation
The net torque causing acceleration (or deceleration) when there is an unbalance between
the torques acting on the rotor is [22]:
Ta = Tm − Te (2.1)
Where
Tm the mechanical or shaft torque supplied by the prime mover lessretarding torque due to rotational losses, in N-m;
Te the net electrical or electromagnetic torque, in N-m;Ta the net accelerating torque, in N-m.
The differential equation describing the rotor dynamics based on law’s of rotation is:
Jd2θmdt2
= Tm − Te (2.2)
Where J is the total moment of inertia of synchronous machine (kg.m2), θm is the angu-
lar displacement of the rotor with respect to the stationary reference axis on the stator
(rad). It is more convenient to chose the angular reference relative to a synchronously
rotating reference frame moving with constant angular velocity ωsm, thus:
θm = ωsmt+ δm (2.3)
Where δm is the rotor position before disturbance at time t = 0. First derivative of
equation (2.3) gives the rotor angular velocity ωm as:
ωm =dθmdt
= ωsm +dδmdt
(2.4)
12
2.3 Frequency Control Systems Chapter 2
And the second derivative of equation (2.3) gives the rotor acceleration as:
d2θmdt2
=d2δmdt2
(2.5)
Substituting equation (2.5) in (2.2):
Jd2δmdt2
= Tm − Te (2.6)
Multiplied equation (2.6) by ωm:
Jωmd2δmdt2
= ωmTm − ωmTe (2.7)
Power is equal angular velocity times torque, thus;
Jωmd2δmdt2
= Pm − Pe (2.8)
The quantity Jωm is known as the inertia constant and is denoted by the M . The M is
related to kinetic energy Wk by:
Wk =1
2Jω2
m =1
2Mωm (2.9)
or
M =2Wk
ωm(2.10)
Since ωm does not change by a large amount before stability is lost, ωm ' ωsmM . Thus,
M =2Wk
ωsm(2.11)
The swing equation (2.8) in terms of M :
Md2δmdt2
= Pm − Pe (2.12)
If p is the number of poles of a synchronous generator, the electrical power angle δ is
related to the mechanical power angle δm by:
δ =p
2δm (2.13)
13
2.3 Frequency Control Systems Chapter 2
Also,
ω =p
2ωm (2.14)
Thus, swing equation (2.12) in terms of δ:
2
pMd2δ
dt2= Pm − Pe (2.15)
Equation (2.15) is divided by the base power Sbase in order to be normalized:
2
p
2Wk
ωsmSbase
d2δ
dt2=
PmSbase
− PeSbase
(2.16)
The quantity of per unit inertia constant H can be defined as:
H =Wk
Sbase(2.17)
The unit of H is in seconds and it has value in the ranges from 1 to 10 seconds, depending
on the size and the type of machine. Substituting equation (2.17) in (2.16):
2
p
2H
ωsm
d2δ
dt2= Pm(pu) − Pe(pu) (2.18)
According to (2.14), the swing equation can be written as:
2H
ωs
d2δ
dt2= Pm(pu) − Pe(pu) (2.19)
2.3.2 Center of Inertia
If a load suddenly increases by ∆PL at time t = 0 at bus k for a grid with multiple
machines , at t = 0+, each machine i will react according to its proximity to the change.
Each generator will then increase its generation according to the synchronizing power
coefficients PSiK . Generators that are closer to bus k, will contribute more, and genera-
tors that are farther away, will contribute less. PSiK is bigger if bus i is closer to bus k,
and smaller if bus i is farther from bus k, so the contribution from generator i is [21] :
∆Pei =(−PSik) (−∆PL)
n∑j=1
PSkj
=PSikn∑j=1
PSkj
∆PL (2.20)
14
2.3 Frequency Control Systems Chapter 2
where PSik =∂Pik∂δik|δik0 (2.21)
According to equation (2.19), the linearized swing equation for machine i is:
2Hi
ω0
d2∆δidt2
= −∆Pei (2.22)
Where δi is the rotor angle of generator i and ω0 is the nominal speed. The inertia
constant Hi has the dimension of time (s) and indicates the time that the system can
provide nominal power by using only the energy stored in its rotating masses. Substi-
tution equation (2.20) in (2.22):
2Hi
ω0
d2∆δidt2
= −
PSikn∑j=1
PSkj
∆PL (2.23)
Taking Hi to the right hand side of equation (2.23), we have:
2
ω0
d2∆δidt2
= −[PSikHi
]∆PLn∑j=1
PSkj
. (2.24)
In order to eliminate the term PSiK , first we use ∆ωi instead of ∆δi for all generators
i = 1, 2, . . . , n, then sum up the equations for each i:
2
ω0
dH1∆ω1
dt= − PS1k
n∑j=1
PSkj
∆PL
.
.
.
+2
ω0
dHn∆ωndt
= − PSnkn∑j=1
PSkj
∆PL
⇒ 2
ω0
n∑i=1
dHi∆ωidt
= −
n∑i=1
PSik
n∑j=1
PSkj
∆PL = −∆PL
(2.25)
In steady state, the speed will be the synchronous speed but during transients, the
speeds of the generators and hence the bus frequencies, differ. Now the ”Center Of
15
2.4 Frequency Control Systems Chapter 2
Inertia (COI)” of the system can be defined as:
ω ≡
n∑i=1
Hiωi
n∑i=1
Hi
or ∆ω ≡
n∑i=1
Hi∆ωi
n∑i=1
Hi
. (2.26)
Differentiating ∆ω with respect to time:
d∆ω
dt≡
n∑i=1
d(Hi∆ωi)dt
n∑i=1
Hi
, (2.27)
orn∑i=1
d (Hi∆ωi)
dt=
[n∑i=1
Hi
] [d∆ω
dt
]. (2.28)
Now substitute equation (2.28) into (2.25):
2
ω0
[n∑i=1
Hi
] [d∆ω
dt
]= −∆PL, (2.29)
Thus:d∆ω
dt=−∆PLω0
2n∑i=1
Hi
, (2.30)
And finally:d∆f
dt=−∆PLf0
2n∑i=1
Hi
≡ mf , (2.31)
Where mf can be evaluated at time instant 0, immediately after the contingency and is
then called the initial rate-of-change-of-frequency (ROCOF) [15].
2.4 Primary Frequency Control (PFC)
A power system needs a closed loop control system to regulate the frequency of the
system. If the system frequency decreases (increases), the primary frequency control
system sends instructions to the generators to increase (decrease) the power output.
Primary frequency control is mainly provided by generators’ governors. A governor is
the feedback controller that senses the system frequency and acts on generator’s prime
16
2.4 Frequency Control Systems Chapter 2
Figure 2.1: Ideal steady-state characteristics of a governor with speed droop [23]
movers (such as steam or water turbines) to regulate frequency deviations. Governors
with a speed-droop characteristic have several settings:
• Droop R
The governor droop R is defined as the variation in power output in steady state
with respect to the variation in system frequency (Figure 2.1). R is calculated as
the ratio of the speed deviation ∆ω or the frequency deviation ∆f to a change
in the valve/gate position or the power output ∆P . It is normally expressed in
percent:
percent R =percent speed or frequency change
percent power output change× 100
=
(ωNL − ωFL
ω0× 100
) (2.32)
where
ωNL steady-state speed at no load;ωFL steady-state speed at full load.
For example, a 5% droop means that generator output will increase by 100% if
there is a frequency deviation of 5%. Looking at Figure 2.1, it can be seen that
governors for primary control are proportional controllers, with the droop R as
the controlling gain.
• Dead-band db
The dead-band is defined as “the total magnitude of the change in steady-state
17
2.5 Metrics for Primary Frequency Control Chapter 2
Figure 2.2: Common dead-band configurations [25]
speed within which there is no resulting change in the position of the governor-
controlled valves or gates” [24] and it is expressed in percent of the rated speed.
The most common types of the dead-band are shown in Figure 2.2 [25]. The droop
characteristic can show a discontinuous step at the borders of the dead-band, or
can be continuous. For instance, a maximum dead-band of 0.06% (0.036 Hz for
nominal frequency of 60 Hz) for a large steam turbine and 0.02% for hydraulic
turbines is specified by IEEE standard [24]. If there is a small frequency deviation
which lies entirely within the dead-band, the governor will be inactive.
2.5 Metrics for Primary Frequency Control (PFC) Ade-
quacy
There are two important metrics for PFC adequacy: ROCOF and frequency Nadir.
2.5.1 ROCOF
The initial slope of the frequency deviation versus time after a contingency is called
the rate-of-change-of-frequency (ROCOF). The frequency dynamics are governed by the
swing equation [23]:df (t)
dt=
1
MH(Pm(t)− Pe (t)) (2.33)
where
18
2.5 Metrics for Primary Frequency Control Chapter 2
f (t) system frequency (Hz);MH system inertia (MWs/Hz);Pm (t) system mechanical power (MW);Pe system electrical load (MW).
After the loss of a power plant of size Pl at time t = 0, the swing equation becomes as
follows at the moment of the contingency:
ROCOF =df
dt(0) =
1
MH(−Pl) (2.34)
The system inertia MH is calculated as:
MH =Pl
|ROCOF |(2.35)
It can be seen from equation (2.34) that the ROCOF depends mainly on the kinetic
energy stored in the rotational parts of the generators and loads. The more inertia in the
system, the smaller the ROCOF magnitude, and a slower and hence less severe frequency
drop will take place. The ROCOF magnitude should not be too large, otherwise the
islanding detection relay will be tripped and the generator will be disconnected from the
grid.
2.5.2 Frequency Nadir
The frequency Nadir is the lowest frequency reached after a contingency and it is the
main metric which determines Under Frequency Load-Shedding (UFLS). The UFLS
leads to disconnecting large groups of costumers at predetermined frequency set-points
and it is a drastic form of emergency frequency control. Loads that are disconnected
through UFLS must be reconnected via special procedures. Therefore UFLS is an emer-
gency operating measure and it should be avoided in normal situations [10].
The magnitude of the frequency Nadir is governed mainly by the size of the contingency,
the kinetic energy of the rotating machines, the number of generators participating in
the primary frequency control, the reserves and their distribution over the generators,
and the dynamic characteristics of the loads and machines such as ramp rate capability
[26]. Each of these characteristics should economically be stimulated to provide an ade-
quate Nadir and hence avoiding UFLS. The general condition for having PFC adequacy
19
2.6 PFC Requirements Chapter 2
is [27]:
fNadir ≥ fmin (2.36)
Where fmin is the minimum acceptable frequency and fNadir is the frequency Nadir.
2.6 PFC Requirements
The high penetration of wind power has an impact on the stability of the power sys-
tem. Some countries, such as UK and Ireland, have prepared specific grid codes for the
ROCOF relay to maintain continuity and security of the electric supply. This section
presents the requirements for primary frequency control in the UK and Ireland. Also
the requirements for two metrics of PFC adequacy (ROCOF and frequency Nadir) for
Sweden will be specified.
2.6.1 UK
The electric power system in the UK is operated by National Grid, and has a maximum
demand of about 60 GW and an installed capacity of 80 GW. The demand is met by
nuclear, coal fired and gas fired power plants and the annual electricity consumption
is around 360 TWh. National Grid is responsible for providing a sufficient frequency
responsive reserve by defining a “Mandatory Frequency Response”. All the generators
connected to the UK transmission grid should fulfil the requirement to have the capacity
of providing this “Mandatory Frequency Response”. Generators must have a 3-5%
governor droop characteristic and be capable to provide continuous modulation power
response through their governing systems. The National Grid ROCOF relays are set
at 0.125 Hz/s but [28] shows that the integration of wind power may lead to ROCOFs
close to 1 Hz/s.
2.6.2 Ireland
The Irish power system consists of two different TSOs: EIRGRID for southern Ireland
and SONI for northern Ireland. The maximum magnitude of the ROCOF relays settings
recommended in the Irish grid is 0.5 Hz/s. As the republic of Ireland has set an electric-
ity target of 40% from renewable resources by 2020 [29], wind capacity will continue to
20
2.7 Simulation tool PSS/E Chapter 2
grow significantly in this period. The technical and operational implications associated
with this high share of renewable energy in the power system of Ireland were studied in
[29]. The results of this study show that two issues are limiting the acceptable level of
wind integration: frequency stability after loss of generation and transient stability after
severe network faults. Some additional recommendations for system operation were de-
rived based on this study. For instance, the ROCOF relays in distribution networks are
to be replaced by alternative protection schemes or the threshold of the ROCOF relays
is to be increased.
2.6.3 Sweden
Sweden is part of the NORDEL system. NORDEL was established in 1963 and is a
body for co-operation between the transmission system operators in Denmark, Finland,
Iceland, Norway and Sweden. The aim of the NORDEL is to establish a Nordic electricity
market. The installed capacity of NORDEL is about 100.8 GW, of which about 8.9 GW
is wind power [30].
As the Nadir adequacy point of view, an automatic load shedding for Sweden is specified
at 49.4 [email protected] s [31]. But in the case of ROFOC, there is no requirement on the
maximum value for the magnitude of the ROCOF in the Nordic Grid Code. However, a
report by Elforsk defines 0.5 Hz/s as the maximum acceptable ROCOF magnitude [32].
Thus, the primary frequency control is adequate for Sweden when:
• The magnitude of the ROCOF is be less than 0.5 Hz/s,
• The frequency Nadir is larger than 49.4 Hz.
2.7 Simulation tool PSS/E
All the calculations in this work are performed with the professional software pack-
age PSS/E (Power System Simulator for Engineering). The PSS/E is used by many
power system utilities for stability studies [33]. It has an extensive library of power
systems components including generators, exciters, governor, stabilizer and protection
21
2.7 Simulation tool PSS/E Chapter 2
models [34]. The PSS/E consists of a complete set of programs for the study of the
power system with both steady-state and dynamic simulations.
2.7.1 Network Representation in PSS/E
The power system network is modeled in PSS/E using a description with the bus ad-
mittance matrix:
I = Y.U (2.37)
where I is vector of positive-sequence currents flowing into the network at its buses,
U is vector of positive-sequence voltages at the network buses and Y is the network
admittance matrix [34].
2.7.2 Power Flow Calculation
The following are the basic input data for power flow calculation in the PSS/E:
• Transmission line impedance and charging admittance,
• Transformer impedance and tap ratios,
• Admittance of shunt-connected devices such as static capacitors and reactors,
• Load-power consumption at each bus of the system,
• Real power output of each generator or generating plant,
• Either voltage magnitude at each generator bus or reactive power output of each
generating plant,
• Maximum and minimum reactive power output capability of each generating plant.
And the outputs of power flow calculation are:
• The magnitude of the voltage at every bus where this is not specified in the input
data,
• The phase of the voltage at every bus,
22
2.7 Simulation tool PSS/E Chapter 2
• The reactive power output of each plant for which it is not specified,
• The real power, reactive power, and current flow in each transmission line and
transformer.
2.7.3 Dynamic Simulation
After solving the steady state power flow, a dynamic simulation can be performed in the
PSS/E. It consists of all the functionality for transient, dynamic and long term stability
analysis. System disturbances such as faults, generator tripping, motor starting and
loss of field can be incorporated in this dynamic simulation. The program consists of
an extensive library of generator, exciter, governor and stabilizer models as well as relay
model including under-frequency, distance and over-current relays.
2.7.4 Program Automation
The PSS/E provides a mechanism to control the PSS/E execution other than via direct
user interaction [35]. There is the ability to specify a set of operations for the PSS/E to
perform in a file and to tell the PSS/E to use the instructions in that file as commands.
This controlling of the execution is done by the API (Application Program Interface).
There are two automation processes in the PSS/E based on the API; the Python in-
terpreter (Python programs) and the IPLAN simulator (IPLAN programs). This work
uses Python which is an interpreter, interactive, object-oriented programming language.
This issue will be explained in Chapter 6 and Appendix A.
23
Chapter 3
Wind Power
3.1 Introduction
Among the various renewable energy resources, wind power is assumed to have the most
favorable technical and economic prospects [1]. People have utilized wind energy from
the very early recorded history. The first accepted establishment of the use of windmills
was in the tenth century in Sistan, in the eastern part of Iran. The wind drives mills
and raises water from the streams in order to irrigate gardens [36].
First, this chapter introduces the physical laws describing the conversion from wind
energy to electrical energy [37], and then discusses two main wind turbine technologies:
the fixed-speed and variable-speed wind turbine. In the next section, the Doubly-Fed
Induction Generator (DFIG) is explained in detail as it is one kind of variable-speed wind
turbine which is commonly used [15]. The chapter ends with explaining the modeling
of wind turbines in PSS/E.
3.2 Power Extraction From The Air Stream
The kinetic energy in a flow of air with a density of ρ [kg/m3] and speed of V [m/s]
through a unit area perpendicular to the wind direction is expressed as 12ρV
2 per unit
volume. The mass flow rate of an air stream flowing through an area A is ρAV , and
thus
W = (ρAV )1/2V 2 = 1/2ρAV 3 (3.1)
24
3.2 Power Extraction From The Air Stream Chapter 3
Figure 3.1: Airflow over wind tunnel [37]
The air density ρ depends on the air pressure and the air temperature:
ρ = ρ0(288B
760T) (3.2)
Where ρ0 is the density of dry air at standard temperature and pressure (1.226 kg/m3
at 288 K,760 mm HG),T is the air temperature (K) and B is the barometric pressure
in mm Hg. As the pressure and the temperature are both function of the height above
sea level, taking an air density of 1.2 kg/m3, thus:
W = 0.6V 3 per unit area (3.3)
Only a proportion of the power W can be converted to useful energy by a wind turbine.
An ideal air flow through a wind turbine is shown in Figure 3.1. The mass flow rate is
the same at position 0, 1 and 2: upstream, at the rotor and downstream:
Mass flow rate, m = ρA0V0 = ρA1V1 = ρA2V2 (3.4)
The force of F on the blade is calculated as:
F = m(V0 − V2) (3.5)
The power W is given by the rate of change of kinetic energy:
W = m(1/2V 20 − 1/2V 2
2 ) (3.6)
25
3.3 Wind Turbine Chapter 3
From equations above, it can be found that:
V1 = 1/2(V0 + V2) (3.7)
The downstream velocity factor b is defined as the ratio of the upstream and downstream
wind speeds:
b =V2
V0(3.8)
Then,F
A1= 1/2ρV 2
0 (1− b2) (3.9)
Then by using equation (3.6) and (3.9):
W
A1= 1/2ρV 3
0 × 1/2(1− b2)(1 + b) (3.10)
The fraction of energy extracted by the wind turbine is called the coefficient of perfor-
mance Cp:
Cp =W
W1(3.11)
Because,
W1 = 1/2ρA1V3
0 (3.12)
Then,
Cp = 1/2(1− b2)(1 + b) (3.13)
The maximum value of the coefficient Cp is found for b equal to 1/3:
Cp,max =16
27or about 59% (3.14)
Cp,max is called Belts’ limit and it used for all types of wind turbines. Another coefficient
is the “Capacity Factor” which is defined as effective number of operating hours (kWh)
per installed capacity (kW) and it is typical in the range of 35-40% [37].
3.3 Wind Turbine
There are two main types of wind turbines: fixed-speed and variable-speed. In this
section, these two types of wind turbines will be discussed.
26
3.4 Doubly Fed Induction Generator Chapter 3
Figure 3.2: Fixed-speed wind turbine [40]
3.3.1 Fixed-Speed Wind Turbine
A fixed-speed wind turbine is shown in Figure 3.2. The induction generator is directly
connected to the grid. In this type, a capacitor bank is necessary for providing reactive
power which is absorbed by the induction generator. The gear box is present in order
to couple the low speed of the turbine to the high speed of the generator. Fixed-speed
turbines are simple, robust and cost-efficient and they were used by many manufacturers
in the 1980s and 1990s [38]. The main problem of this type is however that the fluctuation
in the wind speeds cannot be controlled [39]. Another disadvantages of this type are the
risk of loss of synchronism because of over-speed in case of voltage dips and increasing
of reactive power consumption, especially after fault clearance.
3.3.2 Variable-Speed Wind Turbine
The variable-speed wind turbine consists of a converter connected to the stator of the
induction or synchronous generator as shown in Figure 3.3. This type of wind turbine
can generate an almost constant generator torque. The wind fluctuations are absorbed
by changes in the generator speed [41]. An increased capture of energy, an improved
power quality and a reduced mechanical stress on the turbine are advantages of variable-
speed wind turbines. However, the drawback is the use of more components and the
complicated electrical system leads to a higher cost.
Variable-speed turbines with “partial scale converters” are known as doubly-fed induc-
tion generators (DFIGs). This work uses the DFIG in the simulations as this type is
widely used in wind farms [38].
27
3.5 Wind Turbine Model in PSS/E Chapter 3
Figure 3.3: Variable-speed wind turbine with a synchronous/induction generator [42]
Figure 3.4: DFIG with a Power Converter connected to the rotor terminals [43]
3.4 Doubly Fed Induction Generator (DFIG)
The schematic of a Doubly-Fed Induction Generator (DFIG) wind turbine is shown in
Figure 3.4. The main part of the DFIG consists of an induction generator with power
supply on the rotor as it can be seen in Figure 3.4. The stator is directly connected to the
grid, while the rotor circuit is connected via a power converter to the grid. The power
converter regulates the rotor current and hence controls the electromagnetic torque,
field and the stator output voltage. Figure 3.5 illustrates the general block diagram for
controlling a DFIG. The main parts are the generator and drive train, the turbine rotor,
the grid-side converter with DC-link capacitor, the pitch controller and the rotor-side
controller. The rotor-side converter controls the active and reactive power which the
rotor consumes or produces. The grid-side converter controls the voltage of the DC-link
capacitor [44].
28
3.5 Wind Turbine Model in PSS/E Chapter 3
Figure 3.5: Control block diagram of DFIG wind turbine [44]
3.5 Wind Turbine Model in PSS/E
The PSS/E provides dynamic simulation models for DFIG units and the models were
developed by GE Energy. For instance, there exist the GE 1.5, 3.6 and 2.5 MW models
[45].
Integrating a wind turbine of the GE 1.5 MW model which is used in this work in PSS/E
consists of two steps: doing a static power flow and next a dynamic power flow. The
two steps are discussed briefly below. The procedure of wind integration in PSS/E is
provided in Appendix A.
3.5.1 Running a Static Power Flow
The first step of integrating a wind machine in power flow models, is running a static
power flow. There, a wind turbine is treated as a conventional machine. The important
parameters of a typical GE 1.5 MW machine are found in [43] and presented in Table 3.1.
29
3.5 Wind Turbine Model in PSS/E Chapter 3
Table 3.1: Power flow parameters of GE 1.5MW [43]
Data GE 1.5
Generator Rating 1.67 MVA
Pmax 1.5MW
Pmin 0.07MW
Qmax 0.726 MVAr
Qmin -0.726 MVAr
Terminal voltage 690 V
Unit Transformer Rating 1.75 MVA
Unit Transformer Z 5.75 %
Unit Transformer X/R 7.5
Figure 3.6: Overall wind turbine model of DFIG [45]
3.5.2 Running a Dynamic Power Flow
The dynamic models of a GE 1.5 MW wind turbine consist of an aerodynamics model
(GEWTA1), an electrical control model (GEWTE2), a model for the generator and
power converter (GEWTG2), a pitch control model (GEWTP1) and a 2-mass model
for the turbine shaft (GEWTT1). The connectivity between these models are shown in
Figure 3.6. The wind model parameters used in this work are given in Appendix B.
30
Chapter 4
Optimal Power Flow (OPF) with
PFC Adequacy Constraints
4.1 Introduction
One of the important tools for the planning of electrical power systems is the power flow
calculation [46]. A Power Flow (PF) is the basic tool for security analysis, by identifying
unacceptable voltage deviations or potential component overloading, generally for both
natural load evaluation and sudden structural changes.
However, in active power optimization problems where voltage and reactive power are
of no importance, it is possible to use a DC power flow model instead of an AC power
flow [19]. There, the equations are linearized by assuming that the voltage magnitudes
are constant and equal to 1 pu, by assuming that the voltage angles are small, and
by neglecting the line resistance with respect to the line reactance. Therefore, in this
chapter, DC Power flow will first be introduced. Then, the next section discusses the
problem of Economic Dispatch (ED) and thirdly, the more general problem of Optimal
Power Flow (OPF) will be explained. The last section is devoted to introducing new
constraints in the classic OPF in order to ensure the adequacy of the PFC.
31
4.2 DC Power Flow Chapter 4
4.2 DC Power Flow
The active power flow Pij through a line which connects the sending node i to the
Vi magnitude of voltage at bus i;θi angle of voltage at bus i;Gij series conductance of the transmission line;Bij series susceptance of the transmission line.
The Vi and θi are the state variables of the problem. Equation (4.1) is called the AC
power flow equation. AC power flows require many calculations but represent the reality
of the power system accurately. The following assumptions are made in order to speed
up the calculation [48]:
1. The resistance of each branch is negligible compared to the reactance,
2. Vi = Vj = 1 pu,
3. Assume that (θi−θj) < 2π/9, and then cos(θi−θj) ≈ 1 and sin(θi−θj) ≈ (θi−θj).
Then, equation (4.1) reduces to:
Pij = Bij(θi − θj) (4.2)
Equation (4.2) is known as the DC power flow equation. Considering Bij = 1Xij
, equa-
tion (4.2) changes to:
Pij =1
Xij(θi − θj) (4.3a)
(θi − θj) = XijPij (4.3b)
32
4.2 DC Power Flow Chapter 4
This is a linear equation. The assumptions of DC power flow linearize the power flow
equation. Suppose a network consists of n nodes and m transmission lines, and defines
the following parameters:
A the node-branch incidence matrix ((n− 1)×m);X a diagonal matrix (m× n) containing the reactance of the lines;PF the vector (m× 1) containing the power flow through the lines;θ the vector ((n− 1)× 1) of bus voltage angles.
The network incidence matrix A does not include the row corresponding to the reference
or slack node and is defined as [12]:
Ali =
1, if i is the originating node for link l;
−1, if i is the terminating node for link l;
0, otherwise.
Equation (4.3b) can then be expressed in matrix form as:
AT θ = XPF (4.4a)
PF = (X−1AT )θ (4.4b)
Defining P as the ((n− 1)× 1) vector of the power injections at every node, then:
P = APF (4.5)
And thus,
P = (AX−1AT )θ = Bθ (4.6)
Where B = AX−1AT is known as the susceptance matrix. Considering equation (4.4b)
and (4.6), PF can be expressed as:
PF = (X−1ATB−1)P = HP (4.7)
H is an (m × (n − 1)) matrix and is referred to as the matrix of shift factors, power
transfer distribution factors (PTDFs) or simply distribution factors. Hli is the amount
by which the active power flow over the link l varies with a change in the injection at
node i [12].
33
4.4 Optimal Power Flow Chapter 4
In the DC power flow model, the flow on any network element can be represented as a
linear function of the injections at the nodes of the network [12].
4.3 Economic Dispatch (ED)
This work supposes that the electric energy system is managed by a central operator
with full information on the technical and economic data of the generators, the loads,
and the network. Then, the Economic Dispatch (ED) problem seeks to find a solution
for the problem of allocating the total demand among the generators in order to have
the minimum production cost. The ED problem has as objective function the minimiza-
tion of the production cost with respect to constraints such as energy balance and the
operational limits of the generating units. There are different methods to solve the ED
problem such as the Lambda Search Technique (for rather simple cost functions), the
Gradient Search and Newton’s method [49]. In an ED problem, the generator outputs
are the only adjustable variables.
4.4 Optimal Power Flow (OPF)
After allocation of the total demand among the generators so that the production cost
is minimized by the ED, there is a need to include the transmission losses as well as
power flow constraints in the optimization problem. When the complete transmission
system model is included in ED, the problem is called an Optimal Power Flow (OPF)
[49]. The ED covers the generation limits, while the OPF includes many more of the
power system limits such as limits on the generator reactive power (in an AC Optimal
Power Flow), limits on the voltage magnitude or flows over transmission lines.
The mathematical formulation of the OPF introduces decision variables (such as gener-
ator voltage, LTC transformer tap position, load shedding, reactive injection for static
VAR compensators), state variables (that describe the response of the system to changes
in the control variables) and parameters (including known characteristic of the system
and assumed constant parameters such as network topology, network parameters, gen-
erator cost functions).
34
4.5 Introducing PFC Constraints Chapter 4
A compact form of the OPF problem is:
Minimize f(x, u) (4.8a)
s.t. G(x, u) = 0 (4.8b)
H(x, u) ≤ 0 (4.8c)
u ≤ u ≤ u (4.8d)
where u is the set of decision variables, such as generator active power, generator bus
voltage; x is the set of state variables, e.g. load bus voltage magnitude and angle; f(x, u)
represents the objective function, e.g. power generation cost or power transmission
losses; H(x, u) is the set of operational constraints, e.g. currents and voltage limits,
branch power flow limits; G(x, u) is the set of power flow equations and u and u are the
physical limits of the decision variables [19].
There exist challenges for solving the OPF due to the big size of problem and (in case
of an AC OPF) non-linearities or non-convexity of the problem. The General algebraic
Modeling System (GAMS) is a modeling software that is designed for modeling linear,
nonlinear or mixed integer optimization problems [18].
4.5 Introducing PFC Constraints
A classic DC OPF with objective function of minimizing the cost of generating power
and the cost of providing capacity for reserves is given below:
Minimize∑
i∈Iei(Pi) +
∑i∈I,i 6=l
si(Ri) (4.9a)
s.t.∑
i∈IPi =
∑j∈J
dj (4.9b)
− Lk ≤∑
i∈I,j∈JHik(Pi − dj) ≤ Lk, k ∈ K (4.9c)
0 ≤ Ri ≤ Rmaxi , i ∈ I, i 6= l (4.9d)
Pi +Ri ≤ Pmaxi , i ∈ I (4.9e)
Pi, Ri ≥ 0, i ∈ I (4.9f)
35
4.5 Introducing PFC Constraints Chapter 4
where
i ∈ I set of generators;j ∈ J set of loads;k ∈ K set of transmission lines;i = l largest system generator;Pi dispatched power of generator i (MW);ei(Pi) generator i energy cost function ($/h);Ri reserve of generator i (MW);si(Ri) generator i reserve cost function ($/h);Pmaxi total capacity of generator i (MW);Rmaxi maximum reserve of generator i (MW);dj load j (MW);Lk line k thermal limit (MW);Hik element (i,k) of PTDF matrix H.
Constraint (4.9b) represents the DC power flow power balance equations. Constraint
(4.9c) means that the line flow cannot exceed the line capacity limits. Finally, Con-
straints (4.9d), (4.9e) and (4.9f) set the reserve and generation limits.
Ri is the “Primary Reserve (PR)” of generator i which is the capacity of the generators
that is reserved for governors to apply PFC [14]. The PFC adequacy, depends as ex-
plained before on the inertia and the ramping capability of the governors. The outage
of the largest generator in the grid is considered for determining the PR requirement.
Therefore adding a governor response constraint guaranteing PFC adequacy in equation
(4.9a) is needed. The proof of these new expressions comes from [27] and are explained
as follow. Referring to Chapter 2, the swing equation (2.33) is given here again as:
df (t)
dt=
1
MH(Pm(t)− Pe (t)) (4.10)
Assume that at the time t = 0, a power plant with size Pl stops working and becomes
out of order. Next, the mechanical power Pm decreases with Pl, while the electrical load
power Pe stays constant (Figure 4.1), because the load does not change.
It can be seen from Figure 4.1 that the frequency first decreases as Pm−Pe = −Pl < 0.
When the frequency drop exceeds the dead-band value fdb, the governor starts working
and starts increasing the mechanical power from the prime movers. The frequency
decreases till tNadir, and then the frequency stabilizes when Pm −Pe = 0. After that, it
will increase because Pm − Pe > 0. Finally the frequency reaches the steady state value
36
4.5 Introducing PFC Constraints Chapter 4
C
fdb
tss
fss
Figure 4.1: Governor operation and frequency behavior after a power plant outage [27]
of fss at tss.
Equation (4.10) can be integrated over time between t = 0 and t = tNadir:∫ tNadir
0
df(t)
dtdt = fNadir − f0,
=1
MH
∫ tNadir
0(Pm(t)− Pe(t))dt
(4.11)
The first instants after a contingency are only governed by the inertial response:
td =MH
Plfdb (4.12)
In Figure 4.1, td is the time at which the frequency drop exceeds the dead-band fdb.
If the dead-band frequency fdb is reached, the governor starts to work. The governor
increases the mechanical power Pm with a ramp rate C:
C =∆Pm∆t
(4.13)
37
4.5 Introducing PFC Constraints Chapter 4
The integral on the right hand side of equation (4.11) can be developed as:
fNadir − f0 =1
MH(
∫ td
0(−Pl)dt)
+1
MH(
∫ tNadir−td
0(−Pl + C(t− td))dt)
=−1
MHPltd
+1
MH(
∫ tNadir−td
0(−Pl + Ct)dt)
=−1
MHPltd
+1
MH
(−Pl(tNadir − td) +
C(tNadir − td)2
2
)=−1
MHPltd +
1
MH(−Pl
2(tNadir − td))
=−1
MH(Pltd +
P 2l
2C)
(4.14)
Equation (4.14) expresses the relation between fNadir, inertia MH , the power loss Pl,
the governor dead-time td and the ramp rate C.
According to equation (2.36), for having PFC adequacy fNadir must be larger than fmin,
and thus:
fNadir = f0 −1
MH(Pltd +
P 2L
2C) ≥ fmin
−1
MH(Pltd +
P 2l
2C) ≥ fmin − f0
Pltd +P 2l
2C≤MH(f0 − fmin)
P 2l
2C≤MH(f0 − fmin)− Pltd
2C
P 2l
≥ 1
MH(f0 − fmin)− Pltd
C ≥12P
2l
MH(f0 − fmin)− Pltd= Cmin
(4.15)
Now substitute equation (4.12) into (4.15), Cmin can be written as:
Cmin =12P
2l
MH(f0 − fmin − fdb)(4.16)
Equation (4.16) defines the overall system governor ramp rate. Likewise, the PFC
constraint on individual units depends on the ramp rate ci of the individual plant. The
ci is described as the fastest possible mechanical power output change of the machine i.
38
4.5 Introducing PFC Constraints Chapter 4
The value of ci is calculated from so called stress test : observations of a large contingency
during low governor capability conditions (Figure 4.2).
td tNadir
PiNadir
Figure 4.2: Generator ramping capability ci [27]
[27] defines two necessary and sufficient conditions for satisfying equation (2.36): First,
the sum of the reserves (except i = l, the unit which experiences the outage) should be
larger than the largest possible loss Pl:
∑i∈,i 6=l
Ri ≥ Pl (4.17)
The second condition is that Ri should be delivered before tNadir [27]; therefore, Ri must
be smaller than PNadiri . PNadiri is the power delivered by unit i at tNadir:
Ri ≤ PNadiri (4.18)
From Figure 4.2, PNadiri = ci(tNadir − tdb), thus:
Ri ≤ ci(tNadir − tdb) (4.19)
According to Figure 4.1, tNadir − tdb is equal to PlCNadir
. However the CNadir should
be replaced by Cmin due to the fact that with Cmin the worst condition is examined,
as the frequency Nadir is deeper when the system ramp rate C is smaller. Therefore,
equation 4.19 becomes:
Ri ≤ ciPlCmin
(4.20)
Now substitute equation (4.16) into (4.20):
Ri ≤ ciPl12P 2l
MH(f0−fmin−fdb)
(4.21)
39
4.5 Introducing PFC Constraints Chapter 4
Yielding,
Ri ≤ ci2MH(f0 − fmin − fdb)
Pl(4.22)
Equation (4.22) and (4.17) can be added to the set of equations (4.9a). Then, the OPF
also includes a PFC adequacy constraint:
Minimize∑
i∈Iei(Pi) +
∑i∈I,i 6=l
si(Ri) (4.23a)
s.t.∑
i∈IPi =
∑j∈J
dj (4.23b)∑i∈,i 6=l
Ri ≥ Pl (4.23c)
Ri ≤ 2ciMH(f0 − fmin − fdb)
Pl, i ∈ I, i 6= l (4.23d)
− Lk ≤∑
i∈I,j∈JHik(Pi − dj) ≤ Lk, k ∈ K (4.23e)
0 ≤ Ri ≤ Ri, i ∈ I, i 6= l (4.23f)
Pi +Ri ≤ Pi, i ∈ I (4.23g)
Pi, Ri ≥ 0, i ∈ I (4.23h)
where
ci generator i governor ramp rate (MW/s);fmin minimum acceptable frequency (Hz);fdb system governor’s dead-band (Hz);f0 rated frequency (Hz).
Constraint (4.23b) is the power balance equation. Constraint (4.23c) and (4.23d) belong
to the constraints to ensure PFC adequacy in the grid. Constraint (4.23e) states that
the line flow capacity limits should be respected. Hik is an (m × (n − 1)) matrix of a
network with n nodes and m transmission lines and is referred to as the Power Transfer
Distribution Factor matrix. Hik is the amount by which the flows over the link k varies
with a change in the injection at node i [12]. Finally, constraints (4.23f), (4.23g) and
(4.23h) set the reserve and generation capacity limits, respectively.
40
Chapter 5
Case Study
5.1 Introduction
This Chapter presents a case study which aims to examine how different levels of wind
integration have an influence on frequency stability. First in Section 5.2, the Nordic 32-A
test system, a model of the Swedish grid, will be presented. Then, in Section 5.3, the
modification to the original Nordic 32-A test system which are necessary to have ability
of performing dynamic simulation in PSS/E will be explained.
5.2 Nordic 32-A Test System
The CIGRE Nordic 32-A test system is used as a model of the Swedish transmission
grid in this work [50]. As the single line diagram of the grid shows in Figure 5.1, the
Nordic 32-A test system includes four main areas. The northern area is characterized
by the presence of a lot of hydro generation and by a low load. The central area,
on the contrary, has a high demand for electrical energy and contains mostly thermal
power plants. The south-western area contains multiple thermal units and has a low
load. Finally, the external area is connected to the northern area and has a mixture of
different types of generation and load [40].
The total installed capacity of Nordic 32-A is 17.5 GW. However, the real Swedish grid
has the total installed capacity of 33.5 GW [51]. Thus, the Nordic 32-A test system
is considered a one half scaled-down version of the real Swedish grid in this work [52].
41
5.2 Nordic 32-A Test System Chapter 5
Central
North
External
SouthWest
Figure 5.1: The single-line diagram of Nordic 32-A test system
The model consists of 32 main buses, 51 transmission lines, 13 transformers and 22
generators, 13 of which are hydro units, whereas the rest are thermal generators. The
following dynamic models are used in the original Nordic 32-A test system [50]:
• GENSAL: represents a salient pole generator and is used for all hydro power units.
Figure 5.2 shows the block diagram of GENSAL generator.
• GENROU: is a model of a synchronous generator with a cylindrical round rotor
and represents the generators of the thermal power units. Figure 5.3 shows the
42
5.3 Modification of the Nordic 32-A Test System Chapter 5
Figure 5.2: The block diagram of GENSAL generator [53]
block diagram of GENROU generator.
Figure 5.3: Block diagram of GENROU generator [53]
• SEXS: represents the excitation system’s dynamic model and is used for all types
of synchronous generators. Figure 5.4 describes the control diagram of SEXS.
Figure 5.4: The control diagram for the Simplified Excitation System [53]
• SATB2A: is the stabilizer model, dampening the oscillations in the electrical output
power, Figure 5.5 shows the dynamic control model for STAB2A.
• HYGOV: represents the governor for hydro plants (no governor for thermal units).
The block diagram for the HYGOV governor is illustrated in Figure 5.6.
43
5.3 Modification of the Nordic 32-A Test System Chapter 5
Figure 5.5: The dynamic control model for STAB2A [53]
Figure 5.6: The block diagram for the HYGOV governor [53]
5.3 Modification of the Nordic 32-A Test System
The simulations carried out in this work were performed using the power system si-
mulation tool PSS/E 33.4. All the procedure for developing Nordic 32-A Test System
in PSS/E are provided in Appendix A. The Nordic 32-A grid is modeled based on the
parameters given in [50], but some modifications are made in order to be able to perform
a dynamic simulation using the model. The changes are kindly provided by Prof. Tuan
A. Le from Chalmers University. They are as the following (there are no changes in the
bus data):
With respect to the generator data:
• The active generation PG of the generator attached to bus 1012 is 600 MW in [50],
but is 400 MW in the modified test grid.
• The minimum allowed reactive power of the generator attached to bus 1012 is
-80 MVAr in [50], but is -200 MVA in the modified test grid.
44
5.3 Modification of the Nordic 32-A Test System Chapter 5
• The minimum allowed reactive power of the generator attached to bus 1013 is
-50 MVAr in [50], but is -200 MVAr in the modified test grid.
• The minimum allowed reactive power of the generator attached to bus 1014 is
-100 MVAr in [50], but is -200 MVAr in the modified test grid.
• The active generation of the generator attached to bus 4012 is 600 MW in [50],
but is 500 MW in the modified test grid.
• In [50], there is only one machine attached to bus 4051 with PG = 600 MW, but
in the modified test grid, there are 2, with PG = 600 and 400 MW.
• In [50], there is only one machine attached to bus 4051 with Qmax = 350 MVAr,
but in the modified test grid, there are 2, with Qmax = 350 and 350 MVAr.
• In [50], there is only one machine connected to bus 4051 with a scheduled voltage
of 1.02 pu, but in the modified test grid, there are 2, scheduled voltages 1.0 and
1.0 pu.
• In [50], there is only one machine attached to bus 4051 with a machine power base
of 700 MVA, but in the modified test grid, there are 2, each having a power base
of 700 MVA.
• All scheduled voltages are set to 1 pu in the modified test grid.
With respect to the branch data:
• The long-term line flow rating of line 41-4041 is 770 MVA but is 750 MVA in [50].
• The long-term line flow rating of line 43-4043 is 1430 MVA but is 1500 MVA in
[50].
• The long-term line flow rating of line 51-4051 is 1430 MVA but is 1500 MVA in
[50].
• The long-term line flow rating of line 61-4061 is 770 MVA but is 750 MVA in [50].
• There is an extra branch, between buses 4061 and 4062, not present in [50].
• The line charging parameter B is completely different from in [50].
45
Chapter 6
Method and Simulation Results
6.1 Introduction
This Chapter explains the method for studying the impact of wind integration on the
adequate operation of PFC, considering two metrics of PFC adequacy: the ROCOF and
the frequency Nadir. The following part of this Chapter discusses the second objective
of this work which is the cost analysis of the impact of wind integration on the adequacy
of operation of the Primary Frequency Control. A case study using the Nordic 32-A
test system, introduced in Chapter 5, is presented in this Chapter. In Section 6.2,
the parameters of the grid will be identified with an ad-hoc method. This requires
the automation of PSS/E. Next in that Section, wind turbines are integrated in the
identified Nordic 32-A test system and the effect of the wind integration on the ROCOF
and frequency Nadir, is investigated. Section 6.3 contains the second objective of the
work; investigating the economic impact of wind integration on the PFC, considering
the ramp rate capabilities for all the governors in the grid. Finally, an OPF is presented
which includes constraints ensuring the adequacy of PFC operation. At the end of each
Section, an overview of the simulation results and a summary are provided.
6.2 Impact of Wind Integration on PFC Adequacy
This Section explains the method for studying the technical impact of wind integration
on the adequacy of PFC operation. The Nordic 32-A system is taken as an example
46
6.2 Impact of Wind Integration on PFC Adequacy Chapter 6
to illustrate the method. The Center-Of-Inertia (COI) frequency should be calculated.
In order to acquire the COI frequency, PSS/E needs to be automated. This is done by
Python and the psspy module. Next, an ad-hoc method to identify the grid parameters
using actual data of a contingency occurring in the NORDEL system, will be presented.
Wind turbines of type GE 1.5 MW are subsequently integrated in the model. In the
next step, the effect of different levels of wind power generation on the ROCOF and
frequency Nadir of the COI frequency of the grid are investigated. Finally, the amount
of wind capacity that leads to an inadequate primary frequency control is calculated.
6.2.1 Center-of-Inertia (COI) frequency
A starting point for analyzing the frequency stability in the Swedish grid is to study the
primary frequency control when there is a sudden load-generation imbalance. Figure 6.1
shows the frequency during three generation outage events in the Nordic system [54].
We will model the Nordic system with the Nordic 32-A grid, which is a model for
0 50 100 150 200 250 30049.5
49.6
49.7
49.8
49.9
50
50.1
50.2
Freq
uenc
y(H
z)
Time(seconds)
10-04-1609-08-1009-07-02
16/04/2010 (550 MW)
10/08/2009 (1100 MW)
02/07/2009 (800 MW)
Figure 6.1: Measured frequency in the Nordic system (NORDEL) after a suddentripping of 530, 800 and 1100 MW generation
mainly the Swedish grid. Since the installed capacity of the Nordic 32-A model is half
of that of the real Swedish grid, the machine on bus 1014 with a generation of 1100/2
=550 MW is considered for the contingency on 10/08/2009 (1100 MW). The question
that arises here is which frequency of the Nordic 32-A test system should be compared
with the real data? Bus frequencies can indeed be different from each other. The
answer is the center-of-inertia (COI) frequency of the grid. First of all, the definition of
47
6.2 Impact of Wind Integration on PFC Adequacy Chapter 6
the Center-Of-Inertia frequency, which was discussed in Chapter 2, is given here. The
“Center-Of-Inertia frequency” of the system is defined as:
∆f ≡
n∑i=1
Hi∆fi
n∑i=1
Hi
. (6.1)
According to equation (6.1), the speed/frequency of each machine after applying a con-
tingency, is needed for calculating the COI frequency. Analyzing and simulating the
speeds of all the machines in the system requires that we automatically perform the si-
mulation in PSS/E. For this purpose, PSS/E contains an embedded Python interpreter.
The PSS/E package includes the following Python extension modules, documented in
the PSS/E Application Program Interface (API) document [55]:
• psspy - provides access to the PSS/E API,
• dyntools - tools for processing channel output files,
• redirect - some tools to connect I/O streams between PSS/E and Python.
First, the Python interpreter should be initialized: