A hydropower perspective on flexibility demand and grid frequency control Linn Saarinen

A hydropower perspective on flexibility demand and grid frequency control Linn Saarinen
Linn Saarinen • A hydropower perspective on flexibility demand and grid frequency control 2014
Kph, Trycksaksbolaget AB, Uppsala 2014
A hydropower perspective
on flexibility demand and
grid frequency control
Linn Saarinen
A hydropower perspective
on flexibility demand and
grid frequency control
Linn Saarinen
UURIE 339-14L
ISSN 0349-8352
Division of Electricity
Department of Engineering Sciences
Uppsala, December 2014
Abstract
The production and consumption of electricity on the power grid has to balance
at all times. Slow balancing, over days and weeks, is governed by the electricity
market and carried out through production planning. Fast balancing, within the
operational hour, is carried out by hydropower plants operating in frequency
control mode. The need of balancing power is expected to increase as more varying renewable energy production is connected to the grid, and the deregulated
electricity market presents a challenge to the frequency control of the grid.
The first part of this thesis suggests a method to quantify the need for balancing or energy storage induced by varying renewable energy sources. It is found
that for high shares of wind and solar power in the system, the energy storage
need over a two-week horizon is almost 20% of the production.
The second and third part of the thesis focus on frequency control. In the second part, measurements from three Swedish hydropower plants are compared
with the behaviour expected from commonly used power system analysis hydropower models. It is found that backlash in the guide vane and runner regulating
mechanisms has a large impact on the frequency control performance of the
plants.
In the third part of the thesis, the parameters of the primary frequency control
in the Nordic grid are optimised with respect to performance, robustness and
actuator work. It is found that retuning of the controller parameters can improve
the performance and robustness, with a reasonable increase of the actuator work.
A floating deadband in the controller is also discussed as a means to improve
performance without increasing the actuator work.
© Linn Saarinen Uppsala, 2014
List of papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
III
L. Saarinen, N. Dahlbäck and U. Lundin. "Power system flexibility
need induced by wind and solar power intermittency on time scales of
1-14 days." Submitted to Renewable Energy, July 2014.
L. Saarinen, P. Norrlund and U. Lundin. "Field measurements and
system identification of three frequency controlling hydropower
plants." Submitted to IEEE Transactions on Energy Conversion,
September 2014.
L. Saarinen and U. Lundin. "Robust primary frequency control in a
system dominated by hydropower." Submitted to Control Engineering
Practice, November 2014.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1
Previous research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.1 Frequency control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.2 Hydropower plant dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2
Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Generator and grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Hydropower turbine and waterways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
System identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
PID control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Quantifying balancing need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Rampage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 Rampage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4
Hydropower plant dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Summarised results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Primary frequency control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1
The Nordic power grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2.1 Disturbance suppression - Gwu y (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.2 Model uncertainty - T (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.3 Sensitivity - S(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.4 Control signal restriction - Gwu (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2.5 Optimisation cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2.6 Evaluation of controller performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.1 Optimisation with reduced uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.2 Summarised results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
15
15
16
18
18
20
28
28
30
30
31
6
Conclusions
................................................................................................
45
7
Future work
................................................................................................
47
8
Summary of papers
....................................................................................
48
9
Acknowledgements
...................................................................................
50
10 Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
References
........................................................................................................
53
Abbreviations
Symbol
D
Ep
Ep0, Ep-läge
f
FCR-D
FCR-N
FRR-a
FRR-m
Ki
Kp
LFC
M
n
P
PID
PV
r
Sk
Tf
Ti
Tp
Tw
Ty
TSO
u
u pos
up
VRE
w
wu
WP
y
Y
Yc
Ymv
Ypos
Unit
Hz/MW
Hz
s
pu
s
MW
pu
s
s
s
s
s
pu
pu
pu
pu
%
%
%
%
Description
Load damping constant, load frequency dependency
Droop, inverse static gain of controller
Frequency controller parameter settings
Grid frequency
Frequency containment reserve, disturbed operation
Frequency containment reserve, normal operation
Frequency restoration reserve, automatic
Frequency restoration reserve, manual
Integral gain of controller
Proportional gain of controller
Load frequency control, secondary control
System inertia
Measurement disturbance
Power
Proportional, integral, derivative controller
Photovoltaics
Reference signal (grid frequency)
Energy storage need over the time horizon k
Input filter time constant
Feedback time constant of controller
Time constant of P-model
Water time constant
Servo time constant
Transmission system operator
Control signal, guide vane opening
Guide vane position (after backlash)
Grid input signal, power output from plant
Varying renewable energy
Output disturbance, grid frequency disturbance
Input disturbance, load disturbance
Wind power
Output signal, grid frequency
Guide vane opening
Guide vane opening control signal
Guide vane opening measured value
Guide vane opening position
7
1. Introduction
At all times, the production and consumption of electricity on the power grid
have to be in balance. A small amount of energy is stored as kinetic energy
in the rotation of the large machines that are directly connected (without inverter) to the grid, and any mismatch of production and consumption will lead
to acceleration or deceleration of the whole system and thereby adding to or
borrowing from the rotational energy of the system. The rotational speed of
the machines, and thereby the electrical frequency of the grid, has to be kept
close to its nominal value (50 Hz in the Nordic grid). Grid frequency deviations are especially harmful to thermal power plants, and if the grid frequency
drops too low, they will disconnect in order to protect the machinery. If this
happens, the grid frequency will start to drop even faster due to the increased
mismatch between production and consumption, and eventually there will be
a blackout. The objective of grid frequency control is to make sure that this
never happens.
In the Nordic power system, grid frequency control is carried out mostly
by hydropower plants. Hydropower can be regulated faster and with smaller
loss of efficiency than most other power sources, and since there is plenty of
hydropower in the Nordic system (especially in Norway and Sweden), it takes
on most of the regulation task. However, there is an increasing concern among
hydropower owners that the frequency control wears the turbines down. There
have been several premature failures of Kaplan turbine runners in recent years,
and increased grid frequency control activity is believed to be one of the reasons.
From the power system point of view, there are also challenges for the grid
frequency control. The quality of the Nordic grid frequency, measured in minutes outside the normal operation band 49.9-50.1 Hz, has been gradually deteriorated during the last decades. There are several possible explanations
for this, for example: The deregulation of the electricity market has led to
more activity on the production side (starts and stops and load changes), especially around the hour shifts, since electricity is bought and sold in blocks of
one hour; the production with varying renewable energy sources such as wind
power, which need to be balanced, has increased; the amplitude of very low
frequency oscillations of the grid frequency (period of 40-90 s) has increased;
the operation of the grid frequency control might have changed due to the
deregulation of the market, as the production companies gradually adjust to
the new market situation.
The aim of this thesis is to start in the system we have today, with a diverse
fleet of hydropower plants regulating the grid frequency with relatively simple
9
PI controllers, and find improvements that are easily implemented and beneficial for both the plants and the grid. One of the basic principles in process
control is to "try simple things first", because the greater part of the possible
improvement of the control of a process can often be achieved by good tuning
of simple controllers. This thesis aims at this low hanging fruit, and the results are meant to form a baseline to which more advanced control strategies
can be compared. This strategy also applies to the modelling approach of the
thesis. The principle is to use models that are good enough for the application
- not too simplified and not unnecessarily complex. One part of the thesis is
focused on comparing the models to measurements on real plants, in order to
find out what needs to be included in the model to actually describe the reality
well enough.
The aim of the thesis is also to make an outlook on the future needs for
balancing and frequency control. The challenge is primarily to balance the
production from an increasing amount of varying renewable energy sources
in the system. Since previous research has focused on the need for balancing
over the intra-day or seasonal time horizon, the research on future needs in
this thesis focuses on the 1-2 week time horizon.
A short note on the word frequency: In this thesis, two different frequency
concepts are used: First; the grid frequency is the frequency of the voltages and
currents of the grid, which are tightly connected to the speed of the machines.
The Nordic grid frequency has a nominal value of 50 Hz, and it is the quantity
one attempts to control with (grid) frequency control. Secondly, there is the
frequency content of any signal (for exampel the frequency content of the
grid frequency signal), derived from the Fourier transform of the signal. This
is a means to describe the dynamic characteristics of the signal, that is, if it
changes slowly or quickly in time. In this thesis, the word "frequency" will
generally be used for the concept related to the Fourier transform, and "grid
frequency" will be used for the concept of grid frequency. The exception to
this principle is the widely established notion "frequency control", which will
be used instead of the somewhat cumbersome "grid frequency control".
1.1 Previous research
In recent years, there has been an upswing in research on power system balancing and control. One reason for this is the increase of varying renewable
energy (VRE) production, and the outlook on a future system dominated by
renewables.
The flexibility need of the power system can be divided into four categories
which are related to different time scales: Planning, correction and backup,
rampage and power system stability [1]. Increased VRE production has a
potential impact on all four time scales.
10
The planning time horizon stretches from one or several years (use of hydropower reservoirs, fuel purchase for thermal and nuclear power) to dayahead unit commitment and dispatch. In the Nordic power system, the electricity is sold on long-term fixed price contracts and on the NordPool spot
market. Both the load and the power generated by naturally varying sources
like wind, solar, wave and tidal power vary with the time of the day and the
weather. Some part of the variations are predictable, and some are stochastic.
Day-ahead weather forecasts can be reasonably reliable, but the exact timing
of events, like the passing of a weather front, are often uncertain. For longer
time horizons like 5 or 10 days, weather prognoses are often very uncertain.
With more varying power producers in the system, the role of dispatchable
power plants is transformed from base load production to backup or balancing power. This means more part load operation and more starts and stops.
For thermal power this is associated with lower efficiency, higher emissions,
shorter lifetime due to thermal stress and lower incomes due to fewer equivalent full load hours per year. Ummels et al. [2] has analysed the dispatch
pattern for thermal power in a system with high integration of wind power,
showing an increased need for thermal plants to operate on minimal load, especially during the nights. For hydropower, increased regulation (primary control) and frequent starts and stops means lower efficiency and shorter lifetime
due to wear. There is also a cost for low utilisation of the installed power.
Heide et al. [3, 4] and Rasmussen et al. [5] has analysed the yearly storage
and balancing need in an European power system with 100% VRE, the optimal
mix of wind and solar power and the relation between storage need and VRE
overcapacity. In another Danish study, the occurrence and length of periods
with very high and very low net loads that can be expected in the Danish power
system in 2025 was analysed [6]. The result showed that periods of negative
net load for 2-7 days would be common, while longer periods of high load
would occur less often.
Seasonal variations for wind power has also been studied, for example by
Tande et al. [7], showing that for Norway, wind power production co-varies
with the load (higher in the winter and lower in the summer), but that over
several years, there is a correlation between dry years (low inflow to the hydropower reservoirs) and a years with less wind.
The correction and backup time horizon stretches from day-ahead to hourahead in the Nordic power system. Corrections to the spot market result, for
example due to errors in the prediction of production or load, can be bought
and sold on the NordPool balancing market, ELBAS. The need of balancing
power due to varying wind power production over the correction and backup
time horizon has been thoroughly researched. In general, the net load (load
minus wind power) prediction error is analysed instead of the prediction error
for wind power only. One method of analysis is to decompose the net load
prediction error into periodic components of different frequency, by using the
Fourier transform. This information can be used for sizing of energy storage
11
[8]. Several other methods for storage sizing and storage operations can be
found in the literature [9–12]. Different sizes of storages have been suggested,
ranging from 2 hours of wind power peak production and a power capacity
of 22% for a Portuguese pumped hydro storage [10] to 30-90 minutes and
10% power capacity for the Hungarian power system [13] and 1-3 hours and
25% power capacity for a Danish wind farm, using a dynamic sizing method
[14]. For the Nordic countries, the increased reserve requirement at 10% wind
energy penetration level has been estimated to 1.5-4% of installed wind power
capacity [15].
Rampage can be seen as a special requirement within the correction and
backup time horizon. When a weather front passes over a wind farm, the production will change with a certain ramp speed and amplitude, and some other
production will need to make a corresponding but opposite ramp to balance
the power system. These ramps can, at least partly, be predicted, but the exact
time when the ramp will occur is usually hard to predict. The speed and amplitude of such ramps compared to the ramping capacity of the dispatchable
production are of great interest. Rampage of wind power is discussed further
by Sorensen [16].
The power system stability time horizon stretches from momentary to intrahour balancing of power and load on the grid. The inertia of the large generators connected to the grid functions as a small energy storage that momentarily balances production and load. On the second to hour time horizon, automatic and manual frequency control handles the balancing. Traditionally,
VRE has not contributed with inertia or frequency control reserves, although
it is technically possible to do so at least for wind power plants [17]. For photovoltaics (PV), fast cycling energy storage to firm the power output has been
suggested [18].
Energy storage can be used for balancing over all the above mentioned time
horizons. The utilization of an energy storage is limited by three factors: Volume (energy), flow (power) and transmission capacity. The relation between
the volume and the flow determines for how long time the storage can maintain maximum output or input. Pumped hydro storages are usually designed
with storage volumes lasting one day, while the Nordic hydropower has reservoirs big enough for seasonal storage. Some rivers, for example Luleälven in
Sweden, are equipped with several large turbines in parallel of which some
are only operated during peak loads. However, even though there is enough
volume and flow, the output can still be limited by the transmission capacity.
1.1.1 Frequency control
Power grid frequency control has been an active research area for over 50
years, and the literature on the topic is extensive. There are several surveys that
provide overviews [19, 20]. Primary frequency control was mostly studied in
12
the 70’s and 80’s, using for example the root locus method [21–23], but since
the 90’s the research has mainly been directed towards secondary frequency
control (often called load frequency control, LFC), and all sorts of control
design methods have been proposed.
Robust control theory and design is well suited for power system frequency
control, since the power system is a large, complex and time varying system.
LFC controllers based on H2 , H∞ and other robust methods have been designed
and showed good performance also in presence of generation rate constraints,
backlash and other nonlinearities [24–26], which are known to in many cases
deteriorate controller performance [27]. A drawback with the classical robust
control design is that it results in high order controllers which may be difficult
to implement. One way to get around this problem is to use model reduction or to design a PID controller that imitates the behaviour of the high order
controller. It has been shown that a PID controller designed this way for hydropower LFC can come very close to the performance and robust stability of
a H∞ -controller [28].
PID controllers and their variants (PI, PI with droop, lead-lag) are the most
commonly used controllers, and there is a continued research interest for LFC
PID design and tuning. PID design methods using for example frequency
domain specifications [28], quantitative feedback theory [29], internal model
control [30], constant M-circles in the Nichols chart [31], optimal output feedback [32], and Differential Evolution [27] have been suggested for LFC applications. In recent years, many have focused on the robustness of the PI(D)
controllers, for example by using specifications in the Nyquist diagram [33],
Kharitonov’s theorem [34] or structured singular values [35].
1.1.2 Hydropower plant dynamics
Hydropower plants are highly non-linear and complex systems, and much research effort has been put into modelling of hydropower plants with different
levels of detail over the years. The amount of detail needed in the model depends on what type of phenomenon one wishes to study. To be able to study
fast hydraulic transients, for example due to load rejection, models including
the waterhammer effect with compressible water, elastic penstock and nonlinear turbine equations are needed [36, 37]. On the electrical side, there are
oscillation phenomena in the 1 Hz-range, which are modelled with the swing
equation and a generator model [38]. Dynamics from surge tanks [39], shared
waterways [40] and the runner angle regulation of Kaplan turbines [41, 42]
may also be of importance. To be able to study more than one operational
point, one needs to include the nonlinear (steady state) relation between guide
vane opening and power [43–45]. For frequency control studies, when small
signal amplitudes are studied and the interesting dynamics are in the range
13
below 0.1 Hz, a linear model of the water acceleration in the penstock is often
considered enough [38].
The dynamic response of a hydropower plant can be measured and defined in different ways. Jones et al have suggested that step response, ramp
response and random signal response should be used to benchmark the frequency control ability of hydropower plants [46]. Others have been more interested in testing the response to load rejection [43], [47] or the frequency
response [47], [48], [49]. Step response tests are recommended by IEEE [50]
to define the governor dynamics, and are used by for example the Swedish
TSO in the specifications of the primary frequency control products, FCR-N
and FCR-D. In Paper II, it will be argued that grid frequency step response is
not the best way to test the performance of the frequency controller of a plant.
1.2 Outline of this thesis
Chapter 2 briefly presents the theoretical foundations of the research carried
out in this thesis. Chapter 3 suggests a method to assess the flexibility need
induced by varying renewable power, and summarises the results of Paper I.
In Chapter 4, the validity of the linear hydropower plant model for frequency
control studies is discussed in the light of measurement results from three
Swedish hydropower plants. This chapter connects to Paper II. Chapter 5 and
Paper III proposes a method to optimise the primary frequency control of the
Nordic power grid within the framework of the controller structure implemented today, taking robustness, actuator work and performance into account.
In Chapter 6, the conclusions of the thesis are summarized and discussed, and
Chapter 7 outlines the work of the PhD thesis to come.
14
2. Theory
In this chapter, the theoretical foundations of this thesis are presented briefly.
First, a model of the power grid, valid for small deviations from the operating
point and for frequencies below 0.2 Hz, is derived. Then, a corresponding
small signal analysis model of a hydropower plant is derived. These parts are
based on Kundur [38]. After that, the basic concepts of system identification
and robust control theory are presented. These parts are based on Ljung and
Glad [51], [52].
2.1 Generator and grid
The equation of motion for a synchronous generator is
J
d ωm
= Tm − Te
dt
(2.1)
where J [kgm2 ] is the moment of inertia of the turbine and generator, ωm
[rad/s] is the mechanical rotational speed, Tm [Nm] is the driving mechanical torque and Te [Nm] is the braking electrical torque. Defining the inertia
constant H [s] as the kinetic energy at rated speed divided by the power base
H=
2
1 J ω0m
2 VAbase
(2.2)
and acknowledging that the torque base is Tbase = VAbase /ω0m and that the
per unit electrical angular velocity, ω̄r , is the same as the per unit mechanical
angular velocity, ω̄m , the per unit equation of motion of the machine is
2H
ω̄r
= T̄m − T̄e ,
dt
(2.3)
where the superscript bar is used to denote the normalised values. Defining a
new constant M = 2H (in the following, M will be referred to as the inertia of
the system), and taking the Laplace transform of this equation gives
ω̄r =
T̄m − T̄e
.
Ms
(2.4)
The relation between the power and frequency of a generator is
P = ωr T,
(2.5)
15
where P [VA] is the electrical power and T [Nm] is the torque. Small deviations from the initial values (denoted with subscript 0) can be expressed
by
(2.6)
P0 + ∆P = (ω0 + ∆ωr )(T0 + ∆T )
which can be approximated by
∆P = ω0 ∆T + T0 ωr
(2.7)
since the term ∆ωr ∆T is small. The torque on the machine is the difference
between the driving mechanical torque Tm and the braking electrical torque Te ,
giving the equation
∆Pm − ∆Pe = ω0 (∆Tm − ∆Te ) + (Tm0 − Te0 )∆ωr .
(2.8)
∆P̄m − ∆P̄e = ∆T̄m − ∆T̄e ,
(2.9)
The last term is equal to zero, since the mechanical torque and the electrical
torque are equal in steady state. The initial speed ω0 = 1 if it is expressed in
per unit. This means that
and the relation between power and frequency is the same as the relation between torque and speed, and (2.4) can be rewritten as
∆ω̄r =
∆P̄m − ∆P̄e
.
Ms
(2.10)
∆P̄m − ∆P̄L
.
Ms + D
(2.12)
The change in braking electrical power is the sum of the load change,∆PL
[VA], and the frequency dependency of the load, expressed with the loaddamping constant, D,
(2.11)
∆P̄e = ∆P̄L + Dω̄r
which inserted in (2.10) gives
∆ω̄r =
This equation describes the dynamics of one machine connected to a load. It is
also a model of the whole power system, with all connected machines lumped
into one. The crucial assumption made here is that the whole power system has
the same grid frequency. This is a good approximation in the low-frequency
band where the frequency control operates, but not on higher frequencies [38].
2.2 Hydropower turbine and waterways
The velocity U [m/s] of the water in the penstock of a hydropower plant is
√
(2.13)
U = KuY H,
16
with guide vane opening Y [%], head H [m] and the proportionality constant
Ku . Assuming small deviations, ∆, from the initial values, the change of the
water velocity can be expressed as
∆U =
∂U
∂U
∆H +
∆Y.
∂H
∂Y
(2.14)
Inserting the partial derivatives and normalising the signals by their initial
(steady state) values lead to the equation
1
∆Ū = ∆H̄ + ∆Ȳ .
2
(2.15)
The mechanical power Pm [VA] of the turbine is given by
(2.16)
Pm = K p HU
with the proportionality constant K p .
Again assuming small deviations from the initial values, the change of the
normalised power is
(2.17)
∆P¯m = ∆H̄ + ∆Ū.
With the expression of ∆H̄ in (2.15), the power is
∆P¯m = 3∆Ū − 2∆Ȳ .
(2.18)
When the head over the turbine changes, Newton’s second law of motion
gives the following equation for the acceleration of the water in the penstock:
ρ LA
d∆U
= −Aρ g∆H
dt
(2.19)
Here ρ [kg/m3 ] is the density of the water, L [m] and A [m2 ] is the length
and area of the conduit, g [m/s2 ] is the acceleration due to gravity and t [s] is
the time. The factor ρ LA is the mass of the water in the conduit and ρ g∆H is
the incremental change in pressure at the turbine. The equation on normalised
form becomes (by dividing both sides by Aρ gH0)
Tw
with
d∆Ū
= −∆H̄ = 2 (∆Ȳ − ∆Ū) ,
dt
Tw =
LU0
.
gH0
(2.20)
(2.21)
Taking the Laplace transform of (2.20) and solving for ∆Ū gives
∆Ū =
1
1 + 21 Tw s
∆Ȳ .
(2.22)
17
Finally, substituting with ∆Ū from (2.18) gives
�
1 − Tw s
∆P̄m ∆Ȳ =
1 + 21 Tw s
(2.23)
This transfer function describes how the mechanical power of the turbine responds to changes in the guide vane opening of an ideal, lossless hydraulic
turbine [38]. The water time constant, Tw , varies with the load.
2.3 System identification
In system identification or empirical modelling, measurements of the input
and output signals of a system are used to estimate dynamical models of the
system. A linear, discrete time system can be modelled by
A(q)y(t) = B(q)u(t)
(2.24)
where u(t) is the input signal, y(t) is the output signal, and A and B are polynomials in the time shift operator q, with parameters a1 , ..., ana and b1 , ..., bnb
and with a time delay of nk samples. Stacking all the time shifted input and
output signals and the parameters in

 

a1
−y(t − 1)

 a2 

−y(t − 2)

 . 

..

 . 

.

 . 


 

−y(t − na)

ana 

(2.25)
θ =   and ϕ (t) = 
,
u(t
−
nk)
b

 1


 

 b2 
 u(t − nk − 1) 

 . 

..

 .. 

.
u(t − nk − nb + 1)
bnb
the model can be written
ŷ(t) = θ T ϕ (t).
(2.26)
A least squares estimation of the parameter vector θ can be calculated using for example the ARX (Auto Regression eXtra input) method. In many
cases when the model structure is not on the form in (2.24), a linear regression
problem on the form 2.26 can still be formulated through construction of a
different ϕ vector that could contain any combinations of y(t) and u(t), for
example u2 (t − 1) or y(t − 1)u(t − 2) [51].
2.4 Robust control
A generalised closed loop system is depicted in Figure 2.1, with the system
G(s) and the controller F(s), consisting of one feedforward link Fr (s) and
18
one feedback link Fy (s). In many cases, for example the controllers studied
in this thesis, Fr = Fy = F. The input signals to the system are the reference
signal, r(t), which is 50 Hz in our case (or zero if only the grid frequency
deviation is considered); input disturbances, wu (t), which in our case is the
mismatch of electricity consumption and production on the power grid; output disturbance w(t), which in our case is grid frequency disturbances; and
measurement noise, n(t). The control signal is called u(t), and is in our case
the power output of the plants that operate in frequency control mode; and the
physical quantity we wish to control is called y(t), and is in our case the grid
frequency. The closed loop system can be expressed as
Y = (I + GFy )−1 GFr R + (I + GFy )−1W − (I + GFy )−1 GFy N+
(I + GFy )−1 GWu , (2.27)
where I is the identity matrix and capital letter denote the Laplace transform of
the signals. For simplicity, the argument of the transfer functions and signals
are omitted. The equation (2.27) can also be expressed as
Y = Gc R + SW − T N + GSuWu
(2.28)
where Gc (s) is the closed loop system from r to y; S(s) is the sensitivity function or the transfer function from w to y; T (s) is the complementary sensitivity
function or the transfer function from n to y; and Su (s) is the input sensitivity
function or the transfer function from wu to u.
The control signal can be expressed as
U = (I + Fy G)−1Fr R − (I + Fy G)−1 Fy (W + N) + (I + Fy G)−1Wu =
= Gru R + Gwu (W + N) + SuWu (2.29)
with the transfer function Gru (s) from r to u and the transfer function Gwu (s)
from w to u.
In the context of control theory, the concept of robustness means insensitivity to model errors and disturbances. In some applications, the robustness
of the controller is very important, for example due to large or unknown disturbances or uncertain or time varying system dynamics. Several methods to
design robust controllers have been developed. Two of the most well known
methods are called H2 and H∞ controller design. Both these design methods
aims at minimising the transfer functions S(s), T (s) and Gwu (s) of the system,
which describe the system sensitivity to model errors and disturbances. It is
not possible to make all these transfer functions small on all frequencies, so
part of the design method is to choose weighting functions in the frequency
domain, in order to prioritise in what frequency band each of the transfer functions should be pushed down, and in what frequency band it is allowed to be
larger. The H2 design methods minimises the total energy of the weighted
transfer functions (the H2 -norm), while the H∞ design minimises the highest
peaks of the weighted transfer functions (the H∞ -norm) [52].
19
w
wu
r
Fr
+
+
+
u
G
y
+
+
-Fy
+
+
n
Figure 2.1. A generalised closed loop system.
2.5 PID control
Proportional-integral-derivative (PID) controllers are commonly used for control of industrial processes. In many cases, only the PI part is used, and the
derivative part is set to zero. The most basic form of the PID controller is
Ki
(2.30)
U(s) = K p + + Kd s E(s)
s
where the control error e(t) = r(t) − y(t), K p is the proportional gain of the
controller, Ki is the integration gain and Kd is the derivative gain. The inverse
of Ki is often called the integration time, Ti , of the controller. However, in
this thesis, Ti is used to denote the feedback time constant of the PI controller
with droop which is implemented in the hydropower plants of Vattenfall, so
Ti �= 1/Ki . The droop limits the static gain of the controller (a PI controller has
infinite static gain, due to the integration).
The derivative part of a PID controller is very sensitive to high-frequency
disturbances. In general, it is therefore necessary to filter e(t) before it enters
the controller, or at least to filter the signal going to the derivative block.
20
3. Quantifying balancing need
As described in Section 1.1, the balancing need of the power system can be divided into four categories which are related to different time scales: Planning,
correction and backup, rampage and stable power system [1]. In the Nordic
power system, hydropower is the main resource for balancing on all four time
scales. In Sweden, the power capacity of the hydropower was increased in the
1980’s to be able to handle diurnal as well as seasonal load variations. The
large reservoirs, located at the river sources, make it possible to save large
amounts of energy from the spring and summer to the winter. This storage
capacity and power capacity has a potential to balance also production from
VRE. The question of how much VRE the Swedish hydropower can balance
has caused a heated debate among researchers in Sweden [53]. This thesis will
approach the question from another angle, and suggest a method to quantify
the storage need induced by VRE over time horizons of 1-14 days, attempting to fill the gap between previous research on intra-day storage need and
seasonal storage need.
3.1 Method
The power production and load of the power system has to be equal. When
some of the power is produced by non-dispatchable sources (VRE), the dispatchable power needs to be equal to the net load, that is
PDISP = PNL = PGL − PVRE ,
(3.1)
where PDISP is the dispatchable power production, PNL is the net load, PGL is
the gross load (power consumption) and PVRE is the varying renewable power
production. Without VRE in the system, the balancing need corresponds to
the variations of the consumption or gross load. The balancing need induced
by VRE can then be defined as the difference between the balancing need of
the net load and the balancing need of the gross load.
The balancing need can be quantified by an energy storage measure. The
basic idea of the energy storage measure is illustrated in Figure 3.1, with data
from one week. In the top figure, the gross load and the net load during one
week are plotted, both with their mean value subtracted. To balance this load
with an energy storage, energy has to be put into storage whenever P > 0, and
taken out of storage whenever P < 0. In the bottom figure, the accumulated
21
Normalized gross and net load
1
GL
NL
P [pu]
0.5
0
−0.5
−1
0
20
20
40
60
80
100
120
140
Time [h]
Accumulated normalized gross and net load
GL
NL
15
Σ P [pu]
160
10
5
0
−5
0
20
40
60
80
100
Time [h]
120
140
160
Figure 3.1. Illustration of the energy storage measure. Gross load (GL) and net load
(NL) with mean values subtracted (top) and accumulated gross and net load (bottom)
over one week. The difference between the largest peak and valley of the accumulated
curve is the energy storage need of this week.
sum of the gross and the net load are plotted. The accumulated sum of the
net load increases from t=1 to t=45, since there is surplus energy, meaning
that the energy volume in the (imagined) storage is increasing. After t=45,
the normalised production falls below zero and the accumulated sum starts to
decrease - the stored energy volume is reduced. After a few more peaks and
valleys, the curve ends up on zero, since the average power is zero (the mean
value has been subtracted). Looking at the entirety of the curve, the maximal
storage volume needed can be determined as the difference between the highest peak and the deepest valley, 16 pu. For the gross load, the accumulated
sum starts downwards and eventually rises to 8 pu. This means that a certain
energy level in the storage is needed at the beginning, and the total needed
storage size is 9pu − (−1pu) = 10pu. The storage need induced by the VRE
production for this week is 16pu − 10pu = 6pu. The same calculations can
be made for all one-week periods of the year, and also for other time horizons
than one week. The histogram of the storage of all periods with a certain hori22
zon during one year constitutes a statistical measure of the size of the needed
energy storage. The method is described with equations in the following.
The energy storage need of the net and gross load respectively are calculated
according to (3.2), with energy storage requirement S, time horizon k and load
P.
j
t+k
P(n) −
P(t
+
i)
−
∑
∑
j ε [t,t+k] i=0
n=t k
j t+k
P(n) min ∑ P(t + i) − ∑
j ε [t,t+k] i=0
n=t k
Sk (t) = max
(3.2)
This means that for each hour (t=1:8784), a vector with the length of the
time horizon k is formed [t:t+k]. The mean value of the vector is subtracted
from the vector, and then the accumulated sum of the vector is calculated.
The needed storage volume is considered to be the difference between the
maximum and the minimum of the accumulated sum. The same operation is
carried out for each hour and for each time horizon.
The cumulative distribution function F of the storage need can then be calculated as the probability of the storage need Sk being smaller than or equal to
a certain value sk :
(3.3)
F(sk ) = Pr {Sk ≤ sk }
The cumulative distribution function can be used as a measure of the firmness level. To reach a firmness level of 90%, a storage need sk is needed such
that 0.9 = F(sk ). The storage need at a 90% firmness level can be interpreted
as the storage volume that is large enough to cover 90% of all occasions.
The above described method is applied to load data and wind and solar
power production data from Germany the year 2012 (see further details in
Paper I).
3.1.1 Rampage
The speed and duration of VRE production changes is also of interest, since
it will put demands on the rampage capability of the balancing power plants.
The rampage of VRE production can be calculated as the differenc between
the production at the start and end of data samples with the same length as the
investigated time horizon
P(t) − P(t − k)
(3.4)
k
with ramp speed Ṗ, time horizon k and VRE power production P.
If a production forecast, P̂, is available, part of the rampage will be known
in advance, and part of the rampage will be unpredicted (due to errors in the
Ṗk (t) =
23
forecast). The ramp speed of prediction error, ėk , that is the unpredicted production, can be calculated as
P(t) − P̂(t) − P(t − k) − P̂(t − k)
.
(3.5)
ėk (t) =
k
Wind power production data from the wind park Horn’s Rev (2009), from
the entire Danish wind power (2009) and the production from photovoltaics in
Germany (2011) are analysed with respect to ramp speeds. The ramp speeds
are calculated with a moving window the size of the time horizon, but only the
largest not overlapping ramp speeds are selected to form the results.
3.2 Results
The results on storage are presented in detail in Paper I, and will only be
summarised here. The results on rampage are not included in the paper and
will therefore be presented in detail.
3.2.1 Storage
The storage need induced by VRE increases with the time horizon and with the
VRE share of the total energy production (Figure 2-4 in Paper I). The storage
need normalised with the time horizon is approximately 20% for WP, 5% for
PV and 10% for a combination of WP and PV if the energy share of VRE is
20% and the time horizon is 1-2 weeks (Table 2 in Paper I). This means that
the required storage volume for a 20% share of VRE on two-week horizon is
14 · 0.1 = 1.4 times the average two-week VRE energy production. For higher
shares of VRE, the storage need is approximately 30% for WP, 12% for PV
and 18% for the combination of PW and WP, over time horizons of 7-50 days
(Figure 5 in Paper I).
Photovoltaics has a positive impact on the power system when it comes
to a one day horizon and small energy shares. With the 2012 PV share in
Germany, the storage need over the one-day horizon is decreased. However,
with higher shares of photovoltaics, the positive correlation of solar power and
load is overshadowed by the solar variation as such, resulting in larger storage
needs, especially on shorter time horizons.
Combining WP with PV mitigates the storage need since their variability
have different patterns. While an integration of 80% WP would increase the
storage need 6 times over a two week horizon, a combination of PV and wind
increases the storage need 4 times.
The European Union has a goal that 20% of the energy supply should come
from renewable sources in 2020. Wind power is expected to grow to 140-210
GW installed power in 2020 [54], with as much as 100 GW in the North Sea
24
region. Assuming a capacity factor of 0.3, the 100 GW wind power in the
North Sea region would induce a storage need of 2.2 TWh over a 14-days
horizon if the share of wind power in the receiving areas is 20%, and even
more if lack of transmission makes the local share of wind power higher than
that. This can be compared to the size of the entire storage capacity of the
European pumped storage, which is 2.5 TWh [54]. On the other hand, seasonal
reservoirs of the conventional Nordic hydropower holds 121 TWh [55], that at
least partially could be used for balancing of VRE. With enough transmission
capacity, this might become an important task for Nordic hydropower in the
future.
3.2.2 Rampage
The worst case ramp speeds of the analysed data are presented in Table 3.1,
3.3 and 3.4. Naturally, fast ramps occur more often in the production of Horns
Rev than in the more geographically distributed entire Danish wind power or
the German photovoltaic production. During 2009, there are 10 occurrences
when the production of Horns Rev falls with 80% of the installed power (128
MW) in one hour (Table 3.1). Ramps of shorter duration are even steeper, for
example there was 10 occurrences of 15-minute ramps of more than 250%/h,
that is 100 MW in 15 minutes. The ramp speed of the prediction error (Table 3.2 has equal or even higher values than the production ramp speed. The
high ramp speed of the prediction error may be explained by errors in the timing of production changes. If a change in production comes earlier or later
than predicted, this will give a large prediction error ramp speed.
Table 3.1. Horns rev ramp speed in % of installed capacity per hour. Observe that
when the duration of the ramp is short (e.g. 5 minutes), the ramp speed can be much
higher than 100% per hour.
Duration of ramp
5 min
15 min
30 min
1 hour
2 hours
6 hours
Ramp speed [%/h]
1 occurrence 5 occurrences
588 / -772
502 / -430
311 / -305
268 / -272
175 / -177
160 / -161
93 / -95
85 / -85
49 / -48
46 / -43
16 / -16
15 / -16
10 occurrences
422 / -401
256 / -237
146 / -136
82 / -80
40 / -40
15 / -15
For the entire Danish wind power (Table 3.3), there are 10 occurrences of
ramps of approximately 15% per hour, and the fastest 1-hour ramps are 62%
increase and 56% decrease. These ramp speeds are moderate compared to the
ramp speeds of Horn’s rev, but still considerable. Power forecast data was not
available for the Danish wind power.
25
Table 3.2. Horns rev prediction error, ramp speed in % of installed capacity per hour.
Duration of ramp
5 min
15 min
30 min
1 hour
2 hours
6 hours
Ramp speed [%/h]
1 occurrence 5 occurrences
591 / -770
487 / -434
316 / -301
267 / -273
186 / -174
161 / -162
113 / -95
86 / -89
69 / -60
48 / -48
22 / -22
20 / -19
10 occurrences
424 / -400
255 / -237
145 / -144
83 / -82
44 / -44
17 / -16
Table 3.3. Danish wind power ramp speed in % of installed capacity per hour.
Duration of ramp
1 hour
2 hours
6 hours
Ramp speed [%/h]
1 occurrence 5 occurrences
62 / -56
18 / -15
33 / -30
16 / -13
13 / -14
10 / -9
10 occurrences
16 / -13
13 / -12
9 / -8
German solar power has smaller rampage extreme values than the Danish
wind power (fastest 1-hour ramp is 29% up and 34% down), but ramps of 2530%/h with a duration of 1-2 hours occur more often. The fast ramps of solar
power are caused by sunrise and sunset as well as weather changes (cloudiness), and therefore occur more frequently than the wind power ramps which
are caused by changes in the wind speed. It can also be noted from Table 3.4
that the downward ramps are generally steeper than the upward ramps, which
might be explained by cloud formation during the day leading to increasing
cloudiness during the afternoons.
The prediction error ramp speeds are slightly lower than the production
ramp speeds for PV, especially on one hour or longer time horizons. The
sunrise and sunset ought to be well known factors, improving the predictions,
but the weather has a great impact on how high the midday peak will be.
Table 3.4. German PV production ramp speed in % of installed capacity per hour.
Duration of ramp
15 min
30 min
1 hour
2 hours
6 hours
26
Ramp speed [%/h]
1 occurrence 5 occurrences
49 / -38
37 / -33
33 / -36
29 / -30
29 / -34
28 / -29
27 / -31
26 / -27
16 / -17
16 / -16
10 occurrences
34 / -30
28 / -29
26 / -28
24 / -25
16 / -16
Table 3.5. German PV prediction error, ramp speed in % of installed capacity per
hour.
Duration of ramp Ramp speed [%/h]
1 occurrence 5 occurrences 10 occurrences
15 min
50 / -52
43 / -48
41 / -46
30 min
31 / -31
24 / -28
21 / -26
1 hour
16 / -29
14 / -26
13 / -25
2 hours
12 / -29
11 / -25
10 / -24
6 hours
8 / -16
6 / -16
4 / -16
27
4. Hydropower plant dynamics
The linear model of the hydropower plant derived in Chapter 2.2 is often used
for power system analysis. As discussed in Chapter 1.1, hydropower plants
can be modelled with much more detail, but due to the non-linearities of the
system, the models quickly become very complex. This lead to the question of
how big the error of the linear model is, what type of error it is and which part
of the complex hydropower plant system that should be added to the model
to improve the results the most. All this will naturally depend on what type
of disturbance or what type of operation one wishes to study, and also on the
properties of the individual plant. In this chapter, and in Paper II, measurements on three Swedish hydroplants are analysed in order to find in what way
these plants deviate from the linear model in the specific case of frequency
control operation.
4.1 Method
The dynamic behaviour of hydropower plants operating in frequency control
mode was investigated through experiments on three hydropower plants in
Sweden (Plant A, B and C). The plants were selected as representatives of the
diverse Swedish hydropower fleet. Plant A has one diagonal turbine which is
similar to a Kaplan turbine in the sense that the runner blades are adjustable.
Plant B has three Francis turbines with separate intakes and a common tail
race tunnel. Plant C has three Francis turbines with a common free surface tail
race tunnel. More information about the plants can be found in Paper II.
A block diagram of the studied hydropower plants is given in Figure 4.1.
During the experiments, a signal representing the grid frequency deviation ∆ f
was created by a signal generator and given as input signal to the governor
instead of the real grid frequency. The guide vane opening setpoint could
be changed manually in a step-wise manner. The guide vane control signal,
guide vane measured value and the output power were measured as well as the
pressure before and after the turbine (which cannot be seen in Figure 4.1).
Three types of experiments were carried out on each plant: Grid frequency
deviation sine wave with different frequency and amplitude, grid frequency
deviation step changes and production setpoint step changes. All types were
carried out at different loads and if possible, with different controller settings
("Ep-läge").
28
Turbine
Governor
f
Yc
Y mv
Y pos
P
Y sp
Figure 4.1. Model of governor and plant. Input signals are the frequency deviation
∆ f (created by a signal generator) and guide vane opening setpoint Ysp (manually
controlled). Output signals are the governor control signal Yc , the guide vane measured
value Ymv (used as feedback signal) and the output power P.
Backlash was observed in the runner regulating mechanism of the diagonal
turbine, and in the guide vane regulating mechanism of particularly the Francis
turbine equipped with a guide vane regulating ring (Plant C).
As can be seen in Figure 4.1, two intermediary signals, Yc and Ymv , were
measured in addition to the input signals ∆ f and the output signal P. Therefore it was natural to divide the model into three parts: The controller, which
structure and parameters were already known; the servo Gs (s) from Yc to Ymv
and the rest of the plant, Gt (s), from Ymv to P, including turbine, waterways
and generator. The servo of a hydropower plant is usually modelled by a first
order lag filter [38]. Based on visual inspection of the data a time delay, Tdel ,
was also included in the servo model,
Gs (s) =
1
e−sTdel .
Ty s + 1
(4.1)
The main dynamics of the third part of the model, Gt (s), is the acceleration
of water in the penstock due to a change in the guide vane opening. A model
of this was derived in Chapter 2.2 (2.23). This equation describes the system
from guide vane opening to mechanical power. The dynamics of the generator
is fast in comparison and can be omitted. Including a proportionality constant,
K, to account for the steady state relation between guide vane opening and
electrical power, the model from Ymv to P at the load Y0 is
−Y0 Tw s + 1
.
(4.2)
0.5Y0 Tw s + 1
This model, in the following called the PZ-model of Gt (s), has a pole in the
left half plane and a zero in the right half plane. The pole corresponds to the
lag time constant of the model, meaning how fast it responds to any change
Gt (s) = K
29
in the input signal. The zero in the left half plane corresponds to the model
having non-minimum phase, which means that the response to any change of
the input signal Ymv leads to an initial response of the output signal P that is
in the opposite direction compared to the final value of the response. This
behaviour could be clearly seen in the step response data from Plant B and C,
but not at all in the step response of Plant A. Therefore, a similar model but
without the non-minimum phase behaviour (that is, without a zero in the right
half plane) was also used as a comparison
Gt (s) = K
1
.
Tp s + 1
(4.3)
This model will in the following be called the P-model of Gt (s).
Having defined the model structure of Gt (s), it is straight forward to estimate the parameters K, Tw and Tp from the measured data using system identification methods. However, these models cannot adequately describe the
backlash of the runner and guide vane regulating mechanisms that were observed in the data, since backlash is a non-linear (amplitude dependent) phenomenon, and the models are linear. A convenient way to include the backlash
in the identification process is to model the backlash separately and pre-treat
the data by running it thorough the backlash model (creating the guide vane
position signal Ypos ) before doing the system identification (from Ypos to P).
The size of the backlash cannot be directly identified with this method, but
if the pre-treatment and identification are repeated for different sizes of backlash, the backlash size can be determined indirectly as the one that gives the
best fit to data in the estimation step.
The parameters of the model Gt (s) were estimated from the measured data
using the System Identification Toolbox in Matlab [56]. The estimation was
carried out for both the PZ- and P-model and with different sizes of backlash.
The parameters were estimated separately for each experiment (frequency sinusoid, frequency step, setpoint step and different points of operation).
4.2 Results
The main results are presented in detail in Paper II. Some additional results
will be presented here, together with a summary of the main results.
4.2.1 Summarised results
In normal operation, the actuator movements demanded by the frequency controller are small, especially when the disturbances have periods of one minute
or less. This means that even a small backlash will have an impact on the
response of the plant. The backlash cuts off some of the amplitude of the response, and also adds to the negative phase shift. For Plant A, the phase shift
30
from guide vane opening to power for a disturbance with a 60 s period was
−40◦ and for Plant C it was −17◦ in Ep2 and −42◦ in Ep0 (see Figure 8-10 in
Paper II). These numbers can be compared to the phase shift of Plant B, −13◦ ,
which is closer to the value expected from the linear models.
The backlash of the guide vane regulating mechanism was much smaller
in the plants equipped with individual guide vane servos (Plant A and B) and
larger in the plant equipped with guide vane regulating ring (Plant C).
The backlash of Plant C was well described by a backlash between the
guide vane opening feedback, Ymv and the turbine model Gt (s). The backlash
of Plant A was not situated in the guide vane but in the runner regulating
mechanism. Still, the impact on the output power could be reasonably well
approximated with a guide vane backlash situated before Gt (s).
The diagonal turbine of Plant A did not show any non-minimum phase behaviour, and was better described by a first order lag without zero (P-model)
than by the standard non-minimum phase model (PZ-model), see Figure 11 in
Paper II. This can be partly explained by the slow regulation of this turbine,
but there might also be other explanations.
The measured incremental gain from guide vane opening to power correspond well with tabulated values based on index-testing of the turbines (Figure 5-7 in Paper II). The incremental gain is highly depending on the point of
operation, as have been showed in previous research [45]. This fact deserves to
be stressed, since the praxis among many hydropower plant owners is to use
the linear approximation K ≈ Prated /Ymax to calculate the frequency control
reserves. This method is heavily biased. For example, for Plant C, the incremental gain (and thereby the "reglerstyrka") is 30% lower than Prated /Ymax at
80% load and 40% higher than Prated /Ymax at 65% load.
The simplest way to decrease the impact of backlash on the frequency control would be to increase the steady state gain of the plants participating in
frequency control (and correspondingly decrease the number of plants participating at a given time). The normal control actions of the participating plants
would then be larger compared to the size of the backlash, and therefore the
response of the plants would be closer to the response of the linear models.
That is, there would be less reduction in gain from ∆ f to ∆P and less negative
phase shift.
4.2.2 Backlash
It may sometimes be useful to have a linear approximation of non-linear backlash of hydropower plants. In this section, two different linearisations of the
backlash are compared. The first one is a first order system, characterised by
a time constant TBL and a gain KBL ,
GBL1 =
KBL
.
TBL s + 1
(4.4)
31
The second approximation is a time delay TBL and a gain KBL ,
GBL2 = KBL e−sTBL .
(4.5)
The dynamic response of a backlash is highly dependent on the amplitude
of the input signal. To map the behaviour, a ±0.05% guide vane opening
backlash, modelled by the Matlab Simulink backlash block, is simulated with
sinusoidal input signals of different amplitudes and frequencies, and the gain
and phase of the response is calculated with the fast Fourier transform. The result is plotted in a bode-like diagram in Figure 4.2. Two models are simulated:
The Vattenfall frequency controller, with parameter setting Ep0, connected to
a backlash block, and the backlash block alone. In the case when the input
signal first passes through the controller, the amplitude of the input signal to
the backlash will vary depending on the frequency of the input signal. This
means that the amplitude dependence of the backlash will be mirrored in the
frequency domain (the backlash will reduce the gain and phase more on higher
frequencies than it does for constant amplitude input signals). Note that even
though the input signal passes through the controller, the gain and phase responses plotted in Figure 4.2 are caused only by the backlash. The gain and
phase response of the controller and backlash would be even steeper.
Now, comparing the backlash gain and phase with the two approximations,
using TBL = 1 and KBL = 0.9, it can be seen from Figure 4.3 that the two
approximations give the same result for frequencies lower than 0.02 Hz. For
frequencies higher than 0.2 Hz, it can be seen that although the second approximation (time delay and gain) is closer to the backlash behaviour for a constant
amplitude input signal, the first approximation (pole and gain) is closer to the
backlash when the input signal amplitude is affected by the controller dynamics.
32
From Ymv to Ypos
Gain [abs]
1
0.5
0
−3
10
−2
10
−1
Frequency [Hz]
10
0
10
Phase [°]
0
−50
−100
With controller, ampl=0.1 Hz
Without controller, ampl=0.1%
Without controller, ampl=1%
−3
10
−2
10
−1
Frequency [Hz]
10
0
10
Figure 4.2. Gain and phase of the response of a ±0.05% guide vane opening backlash
to sinusoidal input signal. Simulation of a system with the Vattenfall controller, with
parameter setting Ep0, and backlash, compared to simulation of only the backlash.
33
From Ymv to Ypos
Gain [abs]
1
0.5
0
Backlash
GBL1
GBL2
−3
10
−2
10
−1
Frequency [Hz]
10
0
10
Phase [°]
0
−50
−100
−3
10
−2
10
−1
Frequency [Hz]
10
0
10
Figure 4.3. Approximations of backlash (pole and gain approximation marked with
squares, time delay and gain approximation marked with diamonds), compared to a
non-linear ±0.05% guide vane opening backlash (marked with stars). A system with
the Vattenfall controller, with parameter setting Ep0, and three different backlash models are simulated with sinusoidal ∆ f input signal with amplitude 0.1 Hz and varying
frequency.
34
5. Primary frequency control
In this chapter and in Paper III, a method to optimise the tuning of the primary
frequency control in the Nordic grid is suggested. The method is based on
ideas from robust control theory (Chapter 2.4) and the PID design methods
of Åström and Hägglund [57]. The general idea is to optimise the parameters
of the currently used PI controller with droop with respect to certain demands
specified in frequency domain on performance, robustness and actuator work.
The main contribution of this work to the research field is the discussion on
how to set up these demands, and what the main trade-offs are. The power
grid is complex and time varying, and there are great uncertainties in the system models and parameters. In this work, an attempt is made to take these
uncertainties into account in practice.
In this chapter, and in Paper III, a control theory oriented notation is used
instead of the physical quantity oriented notation in the other chapters. The
notation is explained in Table 5.1. In this chapter, the signals are treated in pu,
while in Chapter 4, they were treated in their physical units.
5.1 The Nordic power grid
The Nordic power system is dominated by hydropower, and frequency control
is almost entirely carried out by hydropower plants. The transmission system
operators (TSO) buy frequency control products from the plant owners. There
are two products for primary frequency control (or frequency containment reserve): FCR-N for normal operation (49.9 < f < 50.1 Hz) and FCR-D for
disturbed operation ( f < 49.9 Hz). The technical specifications for the two
reserves are defined in different ways in the different Nordic countries. In
Sweden, the guideline is that FCR-N reserves should be committed to 63%
within 60 seconds, while FCR-D reserves should be committed to 50% within
5 seconds. In addition to the primary frequency control, there is a manually
operated secondary control (FRR-m), and since 2013, an automatic secondary
control (LFC, FRR-a) has been tested and is currently in operation during a
few hours per day. In this thesis, the primary frequency control for normal
operation, FCR-N, is analysed.
The Nordic power grid can be seen as the single-input single-output (SISO)
system, depicted in Figure 5.1. The system is lumped into one rotating mass
and one hydropower plant controlling the frequency of the system. Electromechanical oscillations between one generator and the grid and inter-area
35
Governor
+
-
+
-
Grid
Plant
Ki
s
+
+
1
T y s +1
K
-Tw s +1
0.5 Tw s +1
+
+
w
1
Ms+D
+
+
y
Figure 5.1. Model of the Nordic power system, with all power plants lumped into one.
In the model used for controller design, the two grey backlash blocks are omitted. In
the model used for evaluating control performance, they are included. Signals and
parameters are explained in Table 5.1 and Table 5.3.
oscillations are not modelled explicitly, but handled as grid frequency disturbances w. This approximation is reasonable since the unmodelled dynamics
are present in a higher frequency band than the dynamics of the grid frequency
control [38]. The main disturbance of the system is load disturbance wu . Load
disturbance can arise both from changes in the load and changes in the power
production. The sign of the load disturbance is chosen so that if more power
is added to the system (increase of generation, decrease of load), the sign is
positive. The signals of the system and their abbreviations are explained in
Table 5.1. All signals are scaled to per unit with the bases given in Table 5.2.
The power base is not chosen as the typical total load of the system, but instead
as the total power of the plants participating in FCR-N, if they all operate with
10% droop (E p = 0.1). This base is chosen to simplify comparison with one
plant operating with the Vattenfall controller setting Ep0.
Table 5.1. Definitions of the signals in Figure 5.1. The corresponding notation of
Chapter 4 is also specified. Note that in this chapter, the signals are treated in pu,
while in Chapter 4, they were treated in their physical units.
Signal
r
y
u
u pos
up
wu
w
n
Ch.4
∆f
∆Ymv
∆Ypos
∆P
Physical variable
Grid frequency deviation reference (=0)
Grid frequency deviation from 50 Hz
Plant control signal, guide vane opening dev.
Plant guide vane opening position dev.
Plant output power deviation
Load disturbance (Positive if power is added)
Grid frequency disturbance
Measurement disturbance
Physical unit
Hz
Hz
%
%
MW
MW
Hz
Hz
The governor is modelled as a first order low-pass input signal filter, a PI
controller with droop and a first order lag representing the hydraulic guide
vane servo
K p s + Ki
.
(5.1)
C(s) =
2
(T f s + 1)(Tys + (E p K p + 1)s + E pKi
36
Table 5.2. Per unit bases used in this chapter.
Type
Pbase
fbase
Ybase
Value
37650 MW
50 Hz
100%
Applies to signal
u p , wu
r, y, w, n
u, u pos
This controller structure is implemented in the hydropower plants owned by
Vattenfall Vattenkraft AB, the largest hydropower owner in Sweden. Although
the controller structure varies among the generating companies, their functionality can in most cases be translated into this structure. It should be noticed
that this controller uses guide vane opening feedback and not power feedback,
and that this is the most common case in Nordic hydropower plants.
A backlash or floating deadband is also included in the governor model.
Sometimes a floating deadband is included in the controller in order to decrease small movements and wear on the turbine and actuator. In most Nordic
hydropower plants, no such deadband is implemented.
The hydropower plant and power grid are modelled with a standard linear
model [38], but including a guide vane opening backlash in the hydropower
plant and an uncertain gain, K, from guide vane opening to output power, motivated by the use of guide vane opening feedback in the controller. The gain
of a plant varies with the operational point, and in practice the total gain is
calculated in advance, in order to commit the right number of plants to frequency control during each operational hour. There is however a considerable
uncertainty in the total gain. As was showed in Paper II, the power output can
be affected by backlash in the guide vane regulating mechanism. Therefore a
guide vane opening backlash is included in the model.
The system parameters are listed in Table 5.3 with a nominal, minimum and
maximum value. The values for inertia, M, and frequency dependence of the
load, D, are based on personally communicated estimations by the Swedish
TSO. The water time constant, Tw , is based on typical values for hydropower
plants owned by Vattenfall. The servo time constant, Ty , is based on the measurements described in Paper II, and the maximal value is set rather high in
order to implicitly include some of the effect of backlash in the uncertainty of
the linear part of the model. The standard setting of the controller parameters
used in Vattenfall plants, called "Ep0", is used as nominal controller parameters.
5.2 Method
A linear model (Figure 5.1 but with the shaded backlash blocks omitted) is
used for controller design, and a non-linear model (all of Figure 5.1) is used
for evaluation of the performance of the optimised controllers by simulation.
37
Table 5.3. System and controller parameters presented with nominal, minimal and
maximal values in per unit (and in physical units in parenthesis).
Parameter
Nominal
System parameters
M
13 (250)
D
0.5 (360)
Tw
1.5
K
1 (37.65)
0.2
Ty
Min
Max
Unit
8 (150)
0.4 (300)
0.5
0.9 (34)
0.1
24 (450)
1 (750)
2
1.1 (41)
1
s (GWs)
pu/pu (MW/Hz)
s
pu/pu (GW/%)
s
Controller parameters (Nominal values, "Ep0")
Kp
1
1/6
Ki
Ep
0.1
1
Tf
1/E p
Ti = 1/(E p Ki )
10 (7530)
60
pu
s−1
pu
s
pu/pu (MW/Hz)
s
The linear model used for controller design can be characterised by the
transfer functions from the input signals n, wu and w to the output signals u,
u p and y. The approach of robust controller design taken here is to make a
weighted minimisation of these transfer functions in frequency domain [52].
The Matlab Robust Control Toolbox [58] is used to define the system, set
up weighting functions and optimise the controller parameters. The weighting
functions are expressed as soft and hard tuning goals, which are translated into
normalised functions fi (x) (soft) and gi (x) (hard) by the software which then
solves the minimisation problem
Minimise max fi (x) subject to max g j (x) < 1,
i
j
xmin < x < xmax
(5.2)
with
fi (x) =� Wi (s)−1Gi (s, x) �∞
gi (x) =� Wi (s) Gi (s, x) �∞ ,
−1
(5.3)
(5.4)
where the user-defined optimisation objectives Wi (s) are the maximal gains of
the closed loop transfer functions Gi (s, x) with the parameter values x.
The tuning goals are described in subsections 5.2.1-5.2.4 and plotted in
Figure 2 in Paper III.
The following optimisation constraints on parameters were used:K p > 0,
Ti > 0, E p > 0.1. The constraint on E p means that the total amount of FCR-N
in the system cannot be increased.
38
5.2.1 Disturbance suppression - Gwu y (s)
The main goal of the controller is to attenuate load disturbances wu , which
means that the transfer function Gwu y from load disturbance wu to grid frequency y should be minimised. First of all, small steady state error is required.
Today, the Nordic TSO:s aim at a fixed amount (753 MW) of regulating power
participating in primary frequency control in the grid frequency band 49.950.1 Hz. This corresponds to a steady state gain from wu to y slightly below
0.1 pu/pu. The different TSO:s have slightly different demands on activation time. In Sweden, the reserve should be activated to 63% after 60 seconds. To keep the low-frequency performance of today, a low frequency tuning goal of 0.1 pu/pu maximal gain (hard from 0 to 0.000167 Hz and soft from
0.000167 Hz to 0.00167 Hz) for Gwuy is used in the optimisation. Further, it is
desired to minimise the peak gain of Gwu y . This is expressed as a soft tuning
goal � Gwu y (iω ) �∞ < 0.5 pu/pu.
5.2.2 Model uncertainty - T (s)
The transfer function from measurement noise n to output signal y is called
the complementary sensitivity function T (s). The system robustness to model
errors is connected to T (s) [52]. Given a true system G0 = (I + ∆G )G with
the model G and the model error ∆G , where G and G0 has the same number of
unstable poles, the true system is stable if
|T (iω )| <
1
.
|∆G (iω )|
(5.5)
The model uncertainties given in Table 5.3 can be expressed as a set of
transfer functions ∆Gi (iω ). The minimum of 1/∆Gi (iω ) is used as a hard
tuning goal for maximal gain of T (iω ).
5.2.3 Sensitivity - S(s)
The transfer function from grid frequency disturbance w to grid frequency y is
called the sensitivity function S(s), and describes how a relative model error is
transformed to an error on the output signal [52]. The peak of the sensitivity
function is also a result of the distance from the Nyquist curve of the open
loop system to the point (-1,j0), and determines how damped or oscillatory the
step response of the system is. The hard tuning goal � S(iω ) �∞ < 1.7 pu/pu
is used in the optimisation. This value is based on based on visual inspection
of the system step response. Further discussion of sensitivity margins can be
found in the literature [57].
39
5.2.4 Control signal restriction - Gwu (s)
There are two reasons to limit the high frequency content of the control signal.
First of all, there are unmodelled electromechanical dynamics in the high frequency band (>0.2 Hz) [59] which will act as a disturbance w on the system.
Limiting the transfer function Gwu , from w to u, makes sure that the controller
mostly will ignore these disturbances. Secondly, control actions cause wear
on the turbines and actuators. The wear is believed to be correlated to the
travelled distance of the actuator and the direction changes. This also makes
restriction of the high frequency gain of Gwu desirable. Based on this reasoning, a high frequency tuning goal of 1 pu/pu maximal gain (soft from 0.2 Hz
to 0.5 Hz and hard from 0.5 Hz to infinity) for Gwu is used in the optimisation.
This will also indirectly limit the gain from measurement noise n and load
disturbance wu to u.
5.2.5 Optimisation cases
Three optimisation cases are chosen, in order to put some light on the most
important trade-offs in the design:
C1) All the tuning goals described in the previous section are applied.
C2) The Gwu tuning goals are not applied. This case is constructed to illustrate the trade-off between performance and disturbance sensitivity of the
control signal.
C3) The Gwu y low-frequency gain tuning goal is increased from 0.1 pu/pu
to 0.2 pu/pu. This case is constructed to illustrate the trade-off between lowfrequency and mid-range frequency disturbance suppression.
The controllers C1-C3 are evaluated in frequency domain in Figure 2 in
Paper III.
Further, the controller C1 is re-optimised for a system with one system parameter at the time changed to its extreme value. The purpose is to describe
how each system parameter influences the optimal controller tuning. The results are presented in Figure 5 and Table 6 in Paper III.
Finally, the system is expanded to include three hydropower plants with
different characteristics, instead of one lumped plant. The controller parameters of each plant is optimised in this system, and compared to the parameters
optimised for the one-plant system. The purpose is to check the feasibility
of controller optimisation on the simplified one-plant system. The results are
presented in Figure 6 and Table 7 in Paper III.
5.2.6 Evaluation of controller performance
The performance of the optimised controllers can also be evaluated by simulation of the non-linear system in Figure 5.1, including backlash. In the literature, the controller performance is most often evaluated by simulation of a
40
load step disturbance [30, 32, 50, 60]. This type of evaluation is reasonable if
the purpose of the controller primarily is to handle larger disturbances, such
as loss of a power line or a production unit. However, the purpose of the frequency control analysed in this thesis, FCR-N, is to handle normal operation
without large disturbances, and typical load variations would be a better input
signal in evaluation of its performance. Unfortunately, there is no continuous measurement of the total load of the grid available. The grid frequency is
measured, but any calculation of the load disturbance from the grid frequency
presupposes a model of the system, which is uncertain. Therefore, to imitate normal operation, the system model is simulated with white noise with
standard deviation 0.0035 pu as load disturbance input signal. The standard
deviation of the white noise is chosen to give a frequency standard deviation
with the Ep0-controller that is the same order of magnitude as the real grid
frequency standard deviation.
The performance of the controllers is quantified with the standard deviation
of the grid frequency, the travelled distance of the actuator (corresponding to
the signal u) and the guide vanes (corresponding to the signal u pos ), and the
number of direction changes of the actuator and the guide vanes. The result is
presented in Table 5 and Figure 3 in Paper III.
5.3 Results
The main results are presented in detail in Paper III. Some additional results
will be presented here, together with a summary of the main results.
5.3.1 Optimisation with reduced uncertainty
The robustness demands on the complementary sensitivity function T (s) of
the system has great impact on the controller optimisation. In the analysis in
Paper III, the tuning goal for T (s) is kept the same in all optimisation cases,
in order to get consistent comparisons, for example to the three-plant system.
However, one might also ask what happens to the optimisation result if the
uncertainties of the system parameters are decreased (for example by more
measurements or further analysis of already available data).
To assess the impact of model uncertainty on the optimisation, the controller is optimised for systems where the uncertainty of one system parameter
at the time is reduced. If for example the system inertia could be measured
or estimated with good accuracy, it may be possible to differentiate between
low-inertia and high-inertia periods and to reduce uncertainty. In Table 5.4,
the system parameters are changed to the mid point of the downward and upward original uncertainty interval, and the uncertainty interval is reduced to
be only the downward, only the upward or half the interval. Looking at the
result for optimisation with the nominal parameter value (M = 13, D = 0.5,
41
Tw = 1.5, K = 1) but decreased uncertainty interval, compared to the optimisation with nominal system parameters and the original uncertainty interval
(first line in the table), it can be seen that the uncertainty pushes the controller
in a conservative direction. For example, for M = 13, with the original uncertainty interval K p = 2 and Ti = 56, while for M = 13 with reduced uncertainty
interval, K p = 2.3 and Ti = 49. However, also with the reduced uncertainty
intervals, the optimisation leads to a conservative controller for the low inertia system and a more aggressive controller for the high inertia system. The
other requirements (limited S(s) and Gwu (s)) are still limiting the achievable
performance.
Table 5.4. Controller parameters optimised for different values of the system parameters and reduced uncertainty.
Varied parameter
Nominal
Uncertainty interval
Optimised parameters
Kp Ti [s] E p T f [s]
2
56 0.1
0.83
M =10
M =13
M =19
8 - 13
10.5 - 18.5
13 - 24
2
2.3
2.8
63
49
36
0.1
0.1
0.1
0.78
1.1
1.4
Tw =1
Tw =1.5
Tw =1.75
0.5 - 1.5
1 - 1.75
1.5 - 2
2.5
2.3
2.2
37
49
57
0.1
0.1
0.1
1.3
1.1
0.89
D =0.45
D =0.5
D =0.75
K =0.95
K =1
K =1.05
0.4 - 0.5
0.45 - 0.75
0.5 - 1
0.9 - 1
0.95 - 1.05
1 - 1.1
5.3.2 Summarised results
2.2
2.1
2.3
2.1
2.2
2.1
54
53
42
41
52
56
0.1
0.1
0.1
0.1
0.1
0.1
0.93
0.91
1.2
0.36
0.96
0.98
All the optimisation cases results in controllers with larger proportional part,
K p , than the Ep0-controller. It is also clear that the optimal tuning of the controller is highly dependent on the inertia of the system. With a high inertia,
the primary control can be more aggressively tuned (higher K p and shorter
feedback time constant Ti ), and with a low inertia the tuning needs to be less
aggressive. Since the inertia of the grid varies with the type and number of
synchronised power plants, it could be beneficial to use a gain scheduling approach with different parameter settings depending on the current inertia of
the grid. The inertia depends on the number of synchronised generators and
their type and size, which means that the TSO has an approximate knowledge of how this parameter varies over time. To produce an accurate, on-line
approximation of the inertia, however, might be challenging.
42
The optimisation shows that from the point of view of the grid, the primary
control should be tuned more conservatively if the water time constants of the
plants are long. Ideally, the whole system should be optimised globally, but
this is impractical due to the vast number of plants that participate in primary
frequency control. Comparing the result from tuning one plant at the time
under the assumption that it alone provides primary control for the system
does gives similar results as a global optimisation of all plants. Since different
plants participate in frequency control at different times, it is also reasonable
to use a parameter setting that can work well independently of the other plants.
Taking that into account, it is reasonable to optimise each plant separately.
The conclusions about how the controllers should be tuned for different
system parameters are highly dependent on the demands on the sensitivity
functions T (s) and S(s) and on Gwu (limiting the high frequency content of the
control signal). If performance was prioritised over robustness and disturbance
insensitivity, the conclusion might be that a faster controller is needed if the
inertia is low or the water time constant is high.
There are some trade-offs to consider. First of all, demanding a small steady
state and low-frequency grid frequency error limits the possibility to achieve
good performance in the mid-frequency range, at least if it is combined with
demands on low disturbance sensitivity of the control signal on high frequencies. The demand on low-frequency disturbance suppression is a somewhat
open question, since there are other types of control that might take on this
task. Maybe the normal reserve (FCR-N) could focus on mid-frequency range
disturbance suppression, and secondary control (FRR-A, LFC) could handle
the persistent load deviations. Such a solution needs to be further studied before anything can be concluded about its feasibility.
Demands on low disturbance sensitivity of the control signal in the high
frequency range greatly limits the performance. The demands are set conservatively in this thesis, since the high frequency behaviour of the power grid
is not included in the model. It is clear, however, that the performance can
be improved if higher sensitivity can be allowed, that is, if it can be showed
that there is no risk for unwanted interaction with inter-area oscillations and
other phenomena in frequency range above 0.2 Hz. The draw-back is that it
would also most likely increase the wear on the turbines and actuators of the
controlling plants.
There is a clear trade-off between actuator work and performance. Simulation of a white noise load disturbance showed that while the grid frequency
deviation decreased with the optimised controller, the actuator work increased.
However, with a small floating deadband in the controller, the actuator work
could be considerably decreased while keeping some of the improved performance. Since there is already some backlash in the actuator of most plants,
allowing a small backlash in the controller seems reasonable. However, there
is always a risk in introducing non-linearities in the system. No limit cycle
oscillation due to backlash can be seen in the simulations of the non-linear
43
system, but the margins to limit cycle oscillations could be further analysed
with for example the describing function method.
It can be concluded that the grid frequency quality can be much improved,
especially in the 40-90 s band, by retuning of the currently used primary frequency controllers. The robustness of the system can be improved at the same
time.
44
6. Conclusions
The aim of this thesis has been to develop research results that are practically applicable for the Nordic grid transmission operators and hydropower
industry. I will therefore venture to draw some conclusions and make some
recommendations, that I hope can be useful:
The proportional part of the controller is important. Increasing the proportional part of the PI controller improves the timing (or phase shift) of the
response. A good balance between the proportional and integral part of the
controller is crucial for a good controller performance. Today, the proportional part is set lower than what is optimal, at least in the investigated Vattenfall plants. If the droop (E p ) of the plant is changed, the other controller
parameters (K p and Ki ) also have to be changed if the same dynamic performance should be achieved (so that one plant operating with low droop is
interchangeable with two or several plants operating with higher droop).
Avoid high droop (low static gain or "reglerstyrka"). Hydropower plants
can be expected to have some backlash in the guide vane regulating mechanism (and in the case of Kaplan turbines, in the runner regulating mechanism).
If the static gain of the controller is low (typically 20%/Hz), the dynamic response to disturbances with periods of approximately one minute will be very
small, which means that even a small backlash (typically 0.1-0.3% will have a
considerable impact on the amplitude and phase of the response. With a higher
static gain, the dynamic gain will also be higher and the backlash will have
less impact on the response. Therefore it is preferable to distribute the control
task on fewer plants, operating with lower droop, compared to distributing the
control task on many plants operating with higher droop.
The quota Prated /Ymax is not a good approximation of the incremental
gain. The incremental gain of a plant in frequency control mode, measured in
MW/Hz, is highly dependent on the operating point of the plant. The approximation that the power is proportional to the guide vane opening can lead to
considerable over- or under-estimation of the static gain ("reglerstyrka"). This
has been shown before [45], but deserves to be stressed since this approximation is often used by the hydropower plant owners. Instead, the incremental
gain can be calculated from data collected from index tests of the turbine.
The results of Paper II shows that for the investigated plants, the incremental
gain calculated from index test data corresponded well to the incremental gain
measured at frequency control tests of the plants.
45
Sinusoidal input signals are recommended for frequency control testing. Experiments with a sinusoidal grid frequency input signal of varying
frequencies gives relevant information of the backlash of the system as well as
the linear dynamics. The impact from surge on the plant response can also be
investigated through such measurements. Grid frequency step response tests,
which are recommended by IEEE [50], give some information about the dynamics of the plant, but mostly in the low-frequency range. It is important to
note that amplitude dependent dynamics (such as backlash) might be important for the normal behaviour of the plant, and therefore it is important to make
tests with both small and larger amplitudes.
There is a trade-off between the performance of the frequency control in the low-frequency range and the mid-frequency range. Demands
on high disturbance suppression at low frequencies limit the performance at
mid-range frequencies, at least if combined with demands on limited high frequency control signals and limited sensitivity functions.
There is a trade-off between actuator work and frequency control performance. Limiting the high frequency content of the control signal affects
the mid-range disturbance suppression.
Low system inertia calls for a slower controller. Given the demands on
on robustness and sensitivity discussed in Paper III, a conservative tuning of
the primary control is better for a system with low inertia, while a more aggressive tuning is preferable for a system with high inertia. However, the
disturbance suppression is not as high for the low inertia system with the conservative controller tuning. If high mid-range disturbance suppression is prioritized over robustness and disturbance insensitivity, one might come to the
opposite conclusion, that low inertia calls for a faster controller.
Slow plants call for slower controllers. From a system perspective, a conservative tuning of the primary control is suited for plants with long water time
constant. Just like the conclusion about inertia above, this conclusion depends
on the demands on system robustness and disturbance sensitivity discussed in
Paper III.
Balancing VRE production over weeks and months may become more
challenging than the intra-day and intra-hour balancing. With high shares
of VRE in the energy mix, the energy storage need is approximately 20% times
the time horizon (for a mix of wind power and photovoltaics). To manage
long periods of low or high VRE production, very large storage volumes are
needed. The seasonal reservoirs of the Nordic hydropower constitutes such
large energy storages, and could play an important role in balancing the future
power grid, over short and long time horizons. Flexible operation of thermal
power plants, like gas turbines, or demand side management may also be a
part of the solution.
46
7. Future work
The research of this thesis has been focused on primary frequency control. The
next step is to include the secondary frequency control (LFC) in the analysis.
One central topic is the interaction between primary control and LFC. There is
a concern that the division into two different control schemes leads to double
work, so that each disturbance is balanced twice, first by the primary control
and then by the LFC which attempts to restore the primary control reserves.
Can the two control schemes be coordinated or in some other way adjusted so
that double work is avoided? And how should each control system be tuned to
achieve the best overall performance?
I would also like to address the more general question of advantages and
drawbacks with different types of LFC controllers. What can be gained from
using additional input signals to the LFC controller, such as scheduled changes
in the load on HVDC cables or other known or predicted load or production
changes? What can be gained from more advanced LFC controllers in general,
for example model predictive control, MPC?
In the research on primary frequency control, I have so far used a simplified
model of the power grid, in which electromechanical oscillations are omitted.
In the continuation of my research, I would like to include more power system
dynamics in the model, at least the major inter-area oscillations of the Nordic
power system. With a widened frequency range of the model, it would be
interesting to benchmark a controller with derivative part (PID or lead-lag)
with the PI controller suggested in Paper III. What is the gain and what is
the cost with a more aggressive frequency control? With the strict demands on
disturbance sensitivity used in Paper III, a derivative part does not significantly
improve the controller performance. However, if these demands are softened,
the result might be somewhat different.
Another related topic is the question of how harmful different types of control actions are to the turbine and actuator, and how to design control strategies
that minimises wear. In my research so far, the number of direction changes of
the actuator and guide vanes and the total distance travelled by the actuator and
guide vanes have been used as measures of the wear. With a more extensive
model of the hydropower plant or with results from other studies, it might be
possible to improve the measures of wear and to use them explicitly in control
design. This work will be done in cooperation with other researchers.
47
8. Summary of papers
Paper I
Power system flexibility need induced by wind and solar power intermittency on time scales of 1-14 days. This paper describes a method to assess the
needed production flexibility to adapt the power system to the production from
varying renewable energy sources such as wind power and photovoltaics over
time horizons of 1-14 days. Load and production data from the German power
system is used to quantify the flexibility need in terms of power and energy
storage requirement due to higher shares of renewable energy (20-80%).
I performed the analysis and wrote the paper. N. Dahlbäck came up with
the idea of the method.
The paper was submitted to Renewable Energy in July 2014.
Paper II
Field measurements and system identification of three frequency controlling hydropower plants. The dynamic behaviour of hydropower plants participating in primary frequency control is investigated in this paper through
frequency response, step response and setpoint change tests on three Swedish
hydropower plants. Grey-box system identification is used to estimate the parameters of simple linear models suitable for power system analysis and the
major shortcomings of the linear models are discussed.
I performed the major part of the analysis and wrote the paper. P. Norrlund
contributed with suggestions to the analysis, quantification of the backlash,
theoretical calculations (based on drawings) of the water time constants and
surge periods and calculation of the polynomial that is called "tabulated incremental gain" in the paper. The field measurements were planned and performed by P. Norrlund and Gothia Power, with some smaller contribution from
me.
The paper was submitted to IEEE Transactions on Energy Conversion in
September 2014.
48
Paper III
Robust primary frequency control in a system dominated by hydropower.
In this paper, the parameters of the primary frequency controller currently used
in the Nordic power system are optimised using a robust control approach.
The trade-off between performance, actuator work and robustness is analysed
in frequency domain and time domain, and the sensitivity to disturbances and
model errors is discussed.
I performed the analysis and wrote the paper.
The paper was submitted to Control Engineering Practice in November
2014.
49
9. Acknowledgements
The research presented in this thesis was carried out as a part of "Swedish
Hydropower Centre - SVC". SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University
of Technology, KTH Royal Institute of Technology, Chalmers University of
Technology and Uppsala University.
Participating companies and industry associations are: Alstom Hydro Sweden, Andritz Hydro, E.ON Vattenkraft Sverige, Falu Energi & Vatten, Fortum
Generation, Holmen Energi, Jämtkraft, Jönköping Energi, Karlstads Energi,
Mälarenergi, Norconsult, Skellefteå Kraft, Sollefteåforsens, Statkraft Sverige,
Sweco Energuide, Sweco Infrastructure, SveMin, Umeå Energi, Vattenfall Research and Development, Vattenfall Vattenkraft, Voith Hydro, WSP Sverige
and ÅF Industry.
I would like to express my gratitude to some persons who have been important to this licentiate project. First of all my supervisors at Uppsala Universtity: Urban Lundin, Per Norrlund and Bengt Carlsson, thank you for for
your support and for valuable discussions. Thanks to colleagues at Vattenfall: Niklas Dahlbäck and Erik Spiegelberg, for sharing your knowledge and
ideas. Jonas Funkquist and Katarina Boman, for teaching me everything I
know about modelling and control (prior to this project) and for being such
good mentors. Johan Bladh, for opening the path for me into the area of hydropower and frequency control in the first place, and for your support ever
since. Karin Ifwer, Jonas Persson, Reinhard Kaisinger, Roger Hugosson, Harald Boman and the operators at Voullerim DC for taking an interest in this
project and for helping me along. To Evert Angeholm and his colleagues
at Gothia Power for measurements and data. To the SVC reference group,
for supporting this project with your knowledge. Thanks also to my fellow
hydropower PhD students at Uppsala University: Birger Marcusson, Weijia
Yang, José Pérez, Jonas Nøland and Mattias Wallin, for sharing the everyday
ups and downs. And last but not least, to my girlfriend Tove Solander, for love
and support and for making sure that I do not get too preoccupied by work.
50
10. Svensk sammanfattning
Produktionen och konsumtionen av elektricitet måste alltid vara i balans på
elnätet. En liten energimängd finns lagrad som rotationsenergi i de tunga maskiner som är direktkopplade till elnätet, framför allt vattenkraftens och kärnkraftens generatorer och turbiner, men den motsvarar bara några sekunders
förbrukning. Om produktionen är lägre än konsumtionen så lånas energi från
den roterande massan, vilket innebär att maskinerna roterar långsammare och
långsammare. Denna rotationshastighet motsvarar också frekvensen på elnätets spänning och ström, vilket innebär att den elektriska frekvensen i hela
nätet sjunker. Vattenkraftverken mäter frekvensen och ökar sin produktion om
frekvensen sjunker, eller minskar sin produktion om frekvensen stiger. Detta
kallas frekvensreglering. Alltför låg frekvens kan medföra haverier i termiska
kraftverk, vars långa axlar kan hamna i resonans och skjuvas av. Frekvensregleringens mål är att hålla frekvensen nära 50 Hz, det vill säga se till att
produktionen balanserar konsumtionen, även vid stora störningar. Frekvenskvaliteten på det Nordiska elnätet har försämrats under de senaste 20 åren.
Det kan finnas många olika förklaringar till detta, exempelvis avregleringen
av elmarknaden, ökningen av varierande förnybar produktion och en gradvis
förändring av hur frekvensregleringen körs. Man har också lagt märke till en
svävning i nätfrekvensen med periodtiden 40-90 sekunder, vars amplitud tycks
öka. Det finns också farhågor från vattenkraftsägarna om att slitaget på vattenkraftturbinerna har ökat, kanske på grund av en ökad frekvensreglering.
Den här avhandlingen består av tre delstudier. Den första delstudien beskriver hur behovet av balansering över långa tidshorisonter (1-2 veckor) påverkas
av en ökad andel sol- och vindkraft. De andra två delarna handlar om balansering inom drifttimmen, närmare bestämt frekvensreglering.
Delstudien om balanseringsbehov utgår från ett energilagringsperspektiv.
Behovet av energilagring definieras här som integralen av nettolastens avvikelse från genomsnittet. Produktions- och lastdata från det tyska elnätet skalas
upp och används för att beräkna energilagringsbehovet över olika tidshorisonter och för olika mängd sol- och vindkraft (varierande förnybar energi,VRE) i
systemet. Resultaten visar att med 20 energiprocent VRE så behöver man lagra ungefär 10% av den typiska tvåveckorsproduktionen för att klara balansen
över tvåveckorshorisonten. Om VRE dominerar energiproduktionen så närmar
man sig lagringsbehovet 20%. Långa perioder med svaga vindar och lite sol
kan bli en utmaning för kraftsystemet i framtiden, eftersom stora energimängder krävs för att klara balansering över långa tidhorisonter.
51
Delstudierna om frekvensreglering tar avstamp i systemet som det ser ut
idag, med avsikt att försöka hitta enkla förbättringsåtgärder som kan ge betydande resultat. En gyllene regel inom reglertekniken är att testa enkla lösningar först, eftersom det har visat sig att rätt intrimmade enkla regulatorer ofta
kan ge nog så bra resultat som mer avancerade regulatorer i många typer av
system.
Genom experiment på tre svenska vattenkraftverk och analys av mätresultaten undersöks hur bra enkla, linjära modeller kan beskriva dynamiken i ett
vattenkraftverk, och vilka som är de viktigaste avvikelserna som på något sätt
bör inkluderas i modellen. Resultaten visar att glapp i ledskene- och löphjulsregleringen har stor betydelse för kraftverkens frekvensregleringsarbete. Glappet gör att små regleråtgärder får större fasvridning och mindre amplitud än
förväntat, vilket kan vara negativt för elnätets frekvenshållning. Mätningarna tydliggör också att reglerstyrkan från en anläggning är starkt beroende av
driftpunkten, eftersom sambandet mellan uteffekt och pådrag inte är linjärt.
Om systemoperatörerna ska få en bra uppfattning om hur mycket reglerstyrka
som verkligen finns tillgänglig vid en viss tidpunkt så krävs det att kraftbolagen bygger sina beräkningar av reglerstyrkan på den aktuella punkten på
turbinkurvan.
En jämförelse mellan den information om anläggningarnas dynamiska beteende som kan erhållas från olika typer av experiment (nätfrekvenssteg, nätfrekvenssinusvåg och börvärdesändring) görs också. Sinusformad nätfrekvenssignal med olika periodtider visar sig ge mest information om glappet och om
systemets dynamik för övrigt, och kan därför rekommenderas.
Informationen om anläggningarnas dynamik används sedan för att optimera primärregleringens reglerparametrar. Mål för robusthet, störningskänslighet
och prestanda sätts upp i frekvensdomän och reglerparametrarna optimeras i
ett system där alla frekvensreglerande vattenkraftverk klumpas ihop till ett.
Resultaten visar att undertryckningen av störningar med periodtider kring 4090 sekunder kan förbättras så att den kvarvarande amplituden nästan halveras
jämfört med dagens regulator (Vattenfalls Ep0-inställingar används som jämförelse). Den viktigaste åtgärden för att åstadkomma detta är att öka proportionaldelen i turbinregulatorn, det vill säga öka regulatorns omedelbara svar på
frekvensavvikelser. Det optimala parametervalet beror också på systemets parametrar, vars värden i viss mån är osäkra. Om elnätets rotationströghet är låg
är det mer lämpligt med en långsammare regulator, förutsatt att de uppställda
kraven för känslighet och robusthet ska uppfyllas. För ett kraftverk med en
längre vattentidskontant är det också lämpligt med en långsammare regulator,
medan en snabbare regulator kan väljas för ett system med hög rotationströghet och för kraftverk med korta vattentidskonstanter.
52
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57
Linn Saarinen • A hydropower perspective on flexibility demand and grid frequency control 2014
Kph, Trycksaksbolaget AB, Uppsala 2014
A hydropower perspective
on flexibility demand and
grid frequency control
Linn Saarinen
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