Evaluation of Power Oscillation Damping in the Nordic Grid Using HVDC

IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2020

Evaluation of Power Oscillation Damping in the Nordic Grid Using HVDC

JOHAN HANSSON

Abstract

With a rapid transition towards renewable energy sources and increasing demand for electricity, we are facing major changes in the electricity grid.

With increased congestion and reduction of inertia the grid is becoming more sensitive to rotor angle stability issues, such as the stability of inter- area modes or power oscillation damping (POD). This thesis shows that the use of local measurements for POD improvement could lead to re- duced transient rotor angle stability. Depending on fault location it is shown that reduction of the first swing is not always feasible, and de- pending on the controller the first swing could be amplified and in the worst case lead to a power blackout due to the tripping of generators and transmission lines. The reason for the behavior is, among other things, rooted in the fundamental limitations of using local measurements, where the estimate of the inter-area mode detects the wrong signs of the distur- bance. Nonlinear simulations are performed in the nordic32 bus Simulink model in order to validate the results.

Sammanfattning

Med en snabb ¨overg˚ang till f¨ornybara energik¨allor och ¨okad efterfr˚agan p˚a el st˚ar vi inf¨or stora f¨or¨andringar i eln¨atet. Med ¨okad belastning och minskad r¨orelsem¨angd blir n¨atet mer k¨ansligt f¨or problem relaterade till rotorvinkelstabilitet, s˚a som stabiliteten av interareapendlingar eller kraf- toscillationsd¨ampning (POD). Denna avhandling visar att anv¨andningen av lokal m¨atning f¨or POD-f¨orb¨attring kan leda till minskad transient ro- torvinkelstabilitet. Beroende p˚a var felet intr¨affar visas det att d¨ampning av den f¨orsta sv¨angningen inte alltid ¨ar m¨ojlig, och beroende p˚a regu- latorndesign kan den f¨orsta sv¨angningen f¨orst¨arkas och i v¨arsta fall leda till en str¨omavbrott p˚a grund av brotkopplade generatorer eller transmis- sionsledningar. Detta fenomen ¨ar bland annat sprunget ur de fundamen- tala begr¨ansningarna i anv¨andnignen av lokal m¨atning, d¨ar estimeringen av interareapendlingen initialt estimerar fel tecken p˚a st¨orningen. Icke- linj¨ara simuleringar utf¨ors i nordic32 Simulink-modellen f¨or att validera resultaten.

Acknowledgments

For your invaluable help, motivation, feedback and never ending support, I want to thank my supervisor Joakim Bj¨ork. Thank you Karl Henrik Johansson for the role of examiner at the division of decision and control systems at KTH. Furthermore I would like to send my thanks to sup- porting friends and family, my mother Brita who encouraged me to study as well as my fellow students. Last but not least, Karin, you and your inexhaustible source of optimism and love means the world to me.

Johan Hansson Stockholm, November 2020

Contents

1 Background 7

1.1 Problem Definition . . . 7

1.2 Method . . . 8

1.3 Report Structure . . . 8

1.4 The Nordic Grid . . . 8

1.5 Simulink model . . . 9

2 Theory 11 2.1 Dynamic system . . . 11

2.2 Classic model of a synchronous generator . . . 12

2.3 Two-machine system . . . 13

2.3.1 Two-machine example - Definition of a simple two-machine system . . . 13

2.4 Inter-area oscillations . . . 14

2.5 Power system stability . . . 15

2.5.1 Frequency stability . . . 15

2.5.2 Voltage stability . . . 16

2.5.3 Rotor angle stability . . . 16

2.5.4 N-1 criterion . . . 17

2.6 Modal transformation . . . 17

2.6.1 Two-machine example - Modal transformation . . . 18

2.7 Stabilizing controllers . . . 18

2.7.1 Lead-lag design . . . 18

2.7.2 H2-controller . . . 20

2.8 Fundamental sensor feedback limitations . . . 21

2.8.1 Filtering sensitivity functions . . . 22

2.8.2 Filtering limitations using local measurements . . . 23

2.8.3 Two-machine example - Filtering limitations . . . 23

2.8.4 Feedback control limitations using local measurements . . 27

2.8.5 Two-machine example - Feedback control limitations . . . 28

2.9 Two-machine example - Perturbed system . . . 30

3 Simulation study 31 3.1 Case 1 - Load decrease in the north region . . . 31

3.2 Case 2 - Load increase in the south region . . . 33

3.3 Case 3 - Severe disturbance in perturbed system . . . 33

4 Results 33 4.1 Case 1 - Load decrease in the north region . . . 33

4.2 Case 2 - Load increase in the south region . . . 35

4.3 Case 3 - Severe disturbance, perturbed system . . . 36

5 Conclusion 37

6 Future work 38

References 40

A Linearized system 42

1 Background

Melting ice, rising water levels, extreme weather, famine and a global climate refugee crisis are some of the predicted consequences if global emissions con- tinues to increase [1]. Reduction of fossil fuels and green house gasses is hence a must, for which electricity will play a mayor role. The Nordic grid, as grids all over the world, are seeing a great expansion in renewable power production such as wind and solar power. But renewables does not only come with good features. Intermittent production leads to production uncertainty and reduced inertia and its damping properties results in impaired dynamic stability, and increased sensitivity to interference.

In this thesis we address the issue of rotor angle stability in connection with inter-area modes. Inter-area modes are low frequency oscillations which are a product of the electro-mechanical coupling between large groups of plants or generators. The resulting power oscillation does not only take up transmission capacity in the grid but can at worse lead to tripping generators and black outs.

Conventional local power system stabilizers (PSS) are implemented to curb neg- ative effects and oscillations in the grid but in order to achieve effective power oscillation damping (POD), coordination with local PSSs in a wide area may be necessary, if at all feasible [2]. However, when a huge amount of sensor data from a wide area is used, issues regarding both robustness due to delay and noise as well as general security arises.

Another technique to attenuate oscillations involves high voltage DC (HVDC) links, these are widely used to connect asynchronous grids and to balance loads over large distances. HVDC has advantages over high voltage AC transmission when it comes to submarine or underground transmission [3], but have also been proven efficient to suppress oscillations [4]. This is due to the fact that the HVDC link is conveniently controllable [5], with high efficiency and precise control over active power [2]. It has been found that local measurements for POD have good properties regarding observability and robustness but that an undesirable effect, first swing instability may occur [6]. The fundamental lim- itations and performance issues of POD due to local sensor measurements are studied in [6]. Here it is investigated on a small scale system. Using the findings in [6], the following thesis focuses on the effect of performing POD with local measurements on the Nordic grid using HVDC. We will look at a hypothetical HVDC connection, which may symbolize the already existing HVDC link ”Fen- noSkan 2” between Dannebo in Sweden and Rauma in Finland, and use it to perform POD.

1.1 Problem Definition

In this thesis we study the use of local frequency measurements to suppress inter-area oscillations in the Nordic electricity grid. Using the HVDC intercon- nection ”FennoSkan” for POD, the effect of geographical location of fault and

measurement is examined from the perspective of power system stability. In the study an attempt is made to answer the problem statement: ”What is the con- sequences of using the HVDC link FennoSkan for POD with local measurements in the Nordic grid?”.

1.2 Method

In order to analyze and determine the limitations of POD using local mea- surements in the Nordic power grid, simulation experiments are performed.

This enables to study the effects under controlled forms which also results in transparency and easy scalability or modification/development. Simulations and computations are conducted in the software Matlab and Simulink by Math- Works.

1.3 Report Structure

The first part of this thesis gives a brief overview of the Nordic grid and the Simulink model used, it provides an introduction for the reader as well as shows the tools used for the project. The second part contains relevant theory to the current problem, it provides the reader with the theoretical foundations and a simple example of what we in this essay will call ”sensor feedback limitations”.

A less complex two machine power system is used throughout this part to ex- emplify the theory. The third part reveals the fault cases used for simulation, for which results are presented in part 4. Part 5 summarizes the conclusions by analyzing the results and it is followed by some suggestions for future work in part 6 .

1.4 The Nordic Grid

The synchronous inter-Nordic system Fig. 1 consists of the Swedish, Norwegian, Finish as well as the east part of the Danish power grid (Denmark 2). It also involves the asynchronous interconnections to nearby countries, and is in itself a subsystem to the European power grid. The Nordic power grid has an approx- imate demand of 400 TWh annually, and the demand is increasing. Table 1 is to illustrate the division of energy between the countries. The growing demand of energy has led to an investment plan by the Nordic TSOs of 15 billion euro until 2028 [7].

The most important interconnections, such as the HVDC submarine cables from Sweden to Finland (FennoSkan), Sweden to Denmark (KontiSkan) and Germany (Baltic Cable) etc. are listed in Table 2. Some of which are currently under construction. Their primary task is to strengthen the grid and allow import and export of power. With large transmission capacity, HVDC links are indeed important components in the grid. A sudden commutation problem or other loss of transmission capacity of the link is therefore an important dimensional

Table 1: Production and consumption in the Nordic grid 2019 [8].

Country Production [TWh] Consumption [TWh]

Sweden 163 136

Norway 133 133

Denmark 2 8 13

Finland 64 83

Total 368 365

error to take into account.

The electrical topography of the Nordic grid can be divided into a north and a south part. The northern region is dominated by hydro generators whereas the southern region has a larger share of nuclear production as well as power consumption in the form of both industries and HVDC export. Hence, the power in the Nordic grid flows from north to south.

Table 2: Interconnections in the Nordic grid.

Cable Inter-connection Power rating [MW]

KontiSkan 1 SE-DE 250

KontiSkan 2 SE-DE 300

Baltic Cable SE-GER 600

SwePol SE-POL 600

Nord Balt SE-LIT 700

FennoSkan 1 SE-FIN 500

FennoSkan 2 SE-FIN 800

NordLink NO-DE 1400

NorNed NO-NE 700

Skagerrak 1 NO-DE 250

Skagerrak 2 NO-DE 250

Skagerrak 3 NO-DE 440

Skagerrak 4 NO-DE 700

1.5 Simulink model

The simulink model nordic 32, based on Long term dynamics. Phase II. Final Report (Cigre) [9], will be used to represent the Nordic grid. The model consists of 20 machines which is a slight modified version of the original Nordic32A as the twin machines at bus 4047 and 4063 are combined into single equivalents.

This is assumed not to affect the overall characteristics of the grid. Frequency dependent loads and line impedances are modeled according to [9].

Det svenska transmissonsnätet för el består av ca 17 000 km kraftledningar, drygt 200 transformator- och kopplingsstationer samt utlandsförbindelser med både växel- och likström.

TRANSMISSIONSNÄTET FÖR EL 2020

Ofoten

Røssåga

SVERIGE

Luleå

Petäjäskoski Keminmaa

Nea

Umeå FINLAND

NORGE

Hasle Oslo

Stockholm Forsmark

Rauma

Helsingfors Tallinn Sundsvall

Göteborg

Ringhals

DANMARK

Oskarshamn

ESTLAND

LETTLAND Riga

Vilnius Klaipeda LITAUEN Karlshamn

Slupsk Malmö

Köpenhamn

Rostock Güstrow Lübeck Eemshaven

Flensburg Wilster Likström (HVDC) Utlandsförbindelse med lägre spänning än 220 kV Förberedelse/entreprenadfas Vattenkraftstation Värmekraftstation Vindkraftpark Transformator/kopplingsstation 400 kV-ledning 275 kV-ledning 220 kV-ledning

Figure 1: The Nordic power system 2020 (map courtesy of Svenska kraftn¨at).

2 Theory

In this part theory that is relevant for this thesis is presented. Common knowl- edge and rigorous derivations are left out to retain the reader’s interest and focus on the essence. For the interested, omitted details can be found in the references.

2.1 Dynamic system

Consider the linear dynamic system (For linearization of nonlinear system see Appendix A):

˙

x(t) = Ax(t) + Bu(t),

y(t) = Cx(t) + Du(t). (1)

Here x(t) denotes the state vector and u(t) and y(t) is the input and output vectors. The matrices A, B are the state and input matrices and C, D are the output and direct feed through matrices respectively. The state at time t is given by:

x(t) = e^A(t−t0)x(t0) + Z t

t0

e^{A(t−τ )}Bu(τ )dτ.

Here x(t0) defines the initial state at time t = t0 and u(t) denotes the input.

The transfer function G(s) of (1) is given by

G(s) = C(sI − A)⁻¹B + D, (2)

and the eigenvalues of the system (1) are determined by the characteristic equa- tion [10]

det(λI − A) = 0.

A system is said to be asymptotically stable if A does not have any open right half plane (ORHP) eigenvalues. That is, all eigenvalues have a negative real part.

Right and left eigenvectors vi and wi satisfies

Avi= λivi, i = 1 . . . n,

w_i^TA = λ_iw^T_i i = 1 . . . n, (3) respectively.

The eigenvectors arranged into matrices becomes

V =v_i, . . . v_n, , W =





 w^T₁

... w^T_n





, where V , W fulfills

AV = V Λ,

W A = ΛW. (4)

Where Λ = diag(λi) ∈ C^n×n, i = 1 . . . n.

An eigenvector can be scaled and still be an eigenvector, hence by normalizing V and W then

V⁻¹= W. (5)

Combining (4) and (5) yields

W AV = Λ. (6)

2.2 Classic model of a synchronous generator

The physical structure of a generator is complex, and modeling of generators can be done in many ways. The detail level used depends on the purpose of the study. For this work, the second-order classical machine model [11] shown in Figure 2 will be used.

E_q⁰∠δ

X_d⁰

Uk∠θ Iq→

+

− Figure 2: Classical model of a synchronous generator.

The electrical components of the classic model Figure 2 is a constant voltage E_q⁰ behind a transient reactance X_d⁰ as well as the terminal voltage Uk. The relation between E_q⁰∠δ and U^k∠θ is shown in a phasor diagram in Figure 3

I_q

E_q

Ut

jIqX_d⁰ δθ

θ

Figure 3: Phasor diagram the classical generator model.

2.3 Two-machine system

HV DC

G1 G2

1 3 2

jX₁ jX₂

Figure 4: Two machine power system model, with parallel HVDC connection.

Consider the two-machine system in Figure 4 to be a representation of the Nordic grid. Here the two synchronous generators represents north and south area of the grid, connected via three buses and two transmission lines. The generators are modeled as classic machine models from subsection 2.2. The dynamics of generator k, k = 1,2, is given by the second order differential equation (7).

˙δ_k= ω_k,

Mkω˙k= Pm,k− Pload,k

| {z }

∆Pk

−E_qk⁰ Uk

X_dk⁰ sin(δk− θ)

| {z }

P_ek

−Dkωk.

(7)

Here ωk and δk denotes the deviation from the nominal frequency and the volt- age phase angle of the k-th machine. The voltage phase angle at the buses is θ and M is the machine inertia, and Uk can either be the voltage at the machine terminal or an adjacent network bus. Higher-order dynamics, such as damper windings, voltage regulators etc. are simplified into a single damping constant Dk. The difference in power ∆Pk represents the active power injections from generators or HVDC, and P_ek denotes the transmitted active power from adja- cent buses.

2.3.1 Two-machine example - Definition of a simple two-machine system

We model a two machine network using the model parameters presented in Table 3. Linearizing the system around this working point gives a model on the form as (31) in Appendix A, and this system has the state vector x =δ₁ δ₂ ω₁ ω₂. The input matrix B is representing active power injec- tion at bus1, bus2 followed by active power injection at the HVDC connection.

The outputs represent the phase angle measurements and bus 1,2 and 3. The

matrices A, B, C and D for the linearized system is found in 8, and the system has four stable poles listed in Table 4.

Table 3: Parameter values for a two-machine representation of the dominant north-south inter-area mode in the N32 model.

Parameter Value

X₁, X₂ 0.5

M1 4.6479

M2 6.4323

D1, D2 102

Data value [p.u]

Load bus 1 34.8

Load bus 2 74.6

Production G₂ 44.4 Power flow G₁ to G₂ 30.2

A =









0 0 1 0

0 0 0 1

−4.27 4.27 −0.07 0

5.59 −5.59 0 −0.05







 , B =









0 0 0

0 0 0

0.21 0 0.11

0 0.16 −0.01







 ,

C =





0.76 0.24 0 0 0.39 0.61 0 0 0.57 0.43 0 0



, and D =





0.005 0.002 0.004 0.003 0.005 −0.003 0.004 0.003 0.001



.

(8)

Table 4: Poles of the linearized system.

-0.03 ± 3.14i -0.00 + 0.00i -0.06 + 0.00i

2.4 Inter-area oscillations

Inter-area oscillations or inter-area modes are low frequency oscillations between groups of generators which occurs when large groups of generators are intercon- nected with relatively weak tie lines. The mode is a product of the interaction between the mechanical parts in the power generators through the electrical part of the system [12]. The modes are typically in the frequency range 0.1 to 0.8 Hz but the nature of the inter-area mode is highly determined by the grid components and their structure [13]. This can for instance be the mechanical structure of the plants such as inertia, the electrical impedance’s of the trans- mission lines as well as local excitation systems such as PSS and AVR. Inter-area

modes are indeed not the only and possibly the worst stability issue, but have historically had major consequences such as black outs. One famous example is the North American black out which occurred the 10th of August 1996 caused by insufficient damping of an inter-area mode [14].

2.5 Power system stability

From the definition in [15] power system stability is defined as: ”Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact.”

Power system stability might be threatened by a various number of problems, and even though the outcome of all these problems might be a destabilized system bundling them all together will be very complex. Instead Kundur et al. in [15] proposes a classification of dynamical power system stability. This classification separates three different stability definitions:

• Frequency stability

• Voltage stability

• Rotor angle stability

Power system stability

Voltage stability Frequency

stability

Rotor angle stability

Small-signal analysis

Transient stability Figure 5: Stability definitions.

2.5.1 Frequency stability

This refers to maintaining the system frequency as well as the ability to recover after a load disturbance. The frequency stability can easily be illustrated by the aggregated swing equation (9). The equation is derived from Newton’s second

law applied on a synchronous generator [11], but can also be extended to a group of generators.

M ˙ω = Pm− Pload, (9)

where M denotes the combined inertia, ω the overall average frequency, Pm

denotes the total mechanical input power and P_load the total demands of the connected loads. One can observe from (9) that a disturbance in the load results in angular acceleration, hence must be compensated by mechanical power in order to maintain desired frequency.

2.5.2 Voltage stability

In each bus the voltage must be kept at reasonable level regarding the distur- bances. It is usually the case for voltage instability that disturbed loads tries to restore power and hence increases stress in the grid by consuming more re- active power. This in turn reduces the voltage further. An example of this is the motor slip on an induction motor trying to compensate for a heavier load [15]. Voltage instability are also heavily affected by the inductive characteristics of the transmission lines as it limits the ability to retain an acceptable voltage level.

2.5.3 Rotor angle stability

The rotor angle or the load angle is the angle between the induced electromotive force (EMF) and terminal voltage of an alternator. During normal operations of a generator, both stator magnetic field and rotor magnetic field rotates with the same speed. However between the two magnetic fields there will be a slight separation in phase angle, depending on the power output of the generator.

Increased load angle results in increased power, but this is true only up to a certain limit. If the load angle exceeds this limit the power decreases which in turn could jeopardize the synchronism in the grid. For small load deviations oscillations might occur. The rotor angle stability is divided into two sub- stability categories regarding the size of disturbance.

Transient stability The power systems ability to maintain synchronism when subject to a major disturbance is called transient stability. This type of distur- bances could potentially be disconnected generators or short circuited transmis- sion lines. The system responds to this by large deviation of rotor angle relative to pre-fault operation point and it is influenced by the nonlinear power-angle relationship. Such large disturbances could potentially lead to separation in the system, also known as first swing instability.

Small-signal stability Small-signal stability compare to transient stability refers to stability under a small disturbance. Here the deviations in rotor angle is small enough so that an analysis is not noticeably affected by non-linear behavior. This means that the performance of a linearized model is sufficient

which facilitates analysis and calculations noticeably. There are two forms of small-signal instability both which are dependent on the initial operating state of the system:

• rotor oscillations, as a result of insufficient damping torque

• rotor angle increase through aperiodic mode, due to absence of adequate synchronizing torque

2.5.4 N-1 criterion

The N-1 criterion is a well-established way of analyzing robustness of a power grid. It is a measure telling that the grid should be operational even under the circumstances of loosing one unit. A unit could for instance be a generator, interconnection or transmission line. Moreover, it states that the power system should withstand individual ”normal” faults in the 400 and 220 kV grids with- out any interruption regarding power production and power consumption, as well as not leading to any secondary failures.

2.6 Modal transformation

For the analysis of oscillatory power system modes, it is convenient to transform the system to its modal form. First consider a linearized systems state vector such as (31) in Appendix A. By change of basis using the eigenvectors (4), we can form

x = V ξ. (10)

Here ξ becomes the new state vector and the state variables are throughout the entire paper deviations from the linearization point. Applying this transforma- tion on (31) yields

V ˙ξ = AV ξ + Bu,

y = CV ξ + Du. (11)

And by using (4), (5) and (6) the transformed system becomes ξ = Λξ(t) + ¯˙ Bu(t),

y(t) = ¯Cξ(t) + Du(t). (12)

Where ¯B = W B, ¯C = CV and Λ = W AV are the same as presented in Sec- tion 2.1.

Trough the systems eigenvalues the information about the modes frequency and damping ratio is obtained:

Mode: λi= σi± jωi, Frequency: f_i= ω_i[rad/s], Damping ratio: ζi= − σi

|λi|.

(13)

2.6.1 Two-machine example - Modal transformation

Using the two machine model derived in subsection 2.3 and following the modal transformation in subsection 2.6, we see that the system has a poorly attenuated mode at 3.14 [rad/s].

Table 5: Modal analysis of two machine system

λ f [rad/s] ζ

-1.17e-15 1.17e-15 1

-6.30e-02 6.30e-02 1

-3.02e-02 ± 3.14i 3.14 0.0096

2.7 Stabilizing controllers

As this thesis focus on the fundamental limitations rather than optimal control solutions, only two different controllers are used for evaluation and comparison.

One industry standard lead-lag controller and one H2-optimal controller. Both implemented with a washout filter, described in the following section.

2.7.1 Lead-lag design

KP SS T_ws 1+T_ws

1+T₁s 1+T₂s

1+T₃s 1+T₄s

Uin Upss

Figure 6: Block diagram of PSS-controller.

The industry standard controller for power systems is a lead-lag compensator, often called power system stabilizer or PSS. The PSS standard layout can be seen in Figure 6 and it consists of a proportional controller, a high pass filter or

”washout filter”, followed by one or two phase compensators.

The washout filter’s purpose is to intercept any contribution from steady-state deviation of the input signal. It has zero static gain filtering out low frequencies acting on the input, and it is tuned in order not to intervene with the modal frequency Ω. As can be seen in Figure 7 the filter does not amplify nor does it have any significant effect on the phase at the angular frequency of the mode and above.

Figure 7: Bode diagram of the washout filter with Ω = 5 rad/s.

The phase compensators is shifting the phase in order to achieve positive damp- ing, and has the form

1 + T1s 1 + T₂s.

Assume we have a linearized system on modal form (12) where we want to control the mode λ1, then the phase compensator is tuned as follows [16]: First we compute the residue of the mode, that is

R¯₁= Cv₁w₁B,

where v1 and w1 corresponds to the right and left eigenvector for the desired mode as described in (3). The residue ¯R is the product of the controllability and observability of the targeted mode.

The residue is then used to calculate the angle φ defined as φ = 180^◦− arg( ¯R1).

Let nf be the number of phase compensators, then nf = 1, if 0^◦< |φ| ≤ 60^◦, nf = 2, if 60^◦< |φ| ≤ 120^◦.

if 120^◦ < |φ| ≤ 180^◦ then set φ = −arg( ¯Ri), KP SS is set to be −KP SS and nf = 1. Then T1-T4can be calculated by

α =

1 + sin_n^φ

f

1 − sin_n^φ

f

, T = 1

ωp

√α, T₁= αT and T₂= T.

For

n_f = 2 =⇒ T₃= T₁, T₄= T₂, nf = 1 =⇒ T3= T4,

here ωp = |λ1|. Lastly the gain KP SS is chosen according to the desired atten- uation but also with respect to how the eigenvalues changes. A root locus can be used to detect what gain produces the highest damping of the desired mode as well as a stable system.

2.7.2 H₂-controller

The H2-controller seeks to synthesize the controller Fy which solves the opti- mization problem

minF_y kP k₂.

Here P is the desired transfer function to be minimized and k·k₂ refers to the H₂ norm [10] defined for a MIMO system as:

kGk₂= 1 2π

Z ∞

−∞

tr(G(iω)^HG(iω))dω

1/2

.

Here G^H denotes the complex conjugate transpose of G and tr() is the trace operator. We form the extended system Figure 8

F u P

y

d z

Figure 8: Block diagram of the extended system.

With P being an input output matrix with the structure (14), see [2] for details.

P =





G_zd 0 G_zu

0 0 I

G_yd I G_yu



. (14)

Here as an explanation: Gyddenotes the transfer function (2), from disturbance d to measurement output y. In order to tune the regulator, weights are added on each transfer function by post- and pre-multiplication of the extended system with weight matrices

P_weighted= W_outputP W_input.

In this work, we will only consider diagonal weights. The larger weight rela- tive to the others the higher the input or output will be penalized during the optimization.

2.8 Fundamental sensor feedback limitations

Given a simple closed looped system, the transfer functions from inputs and measurement noise to the output is commonly known as the sensitivity function S, and the complementary sensitivity function T respectively. Given the feed- back controller u = −Ky, where u denotes the controller input, K the controller and y the output, S and T can be written as: S = (1 + G_yuK)⁻¹and T = 1 − S.

Assuming the system is internally stable, meaning no open right half plane (ORHP) poles or zeros of the plant are cancelled by the controller, the following results holds for S and T : Let pi and qi denote any ORHP poles and zeros of Gyu then:

S(pi) = 0, T (pi) = 1,

S(qi) = 1, T (qi) = 0, (15)

and S + T = 1, for all frequencies. With or without cancellation of poles or ze- ros, disturbance rejection and noise suppression conflict each other hence cannot coincide. Under the circumstances that disturbances and noise act in different frequency ranges this trade off problem is dealt with by shaping the controller accordingly and achieving sufficient performance.

Another trade off is the so called ”waterbed effect” derived from Bode’s integral.

Assuming the loop-gain G_yuK is proper i.e. the degree of the numerator does not exceed the degree of the denominator. If S(∞) = S_∞6= 0 and the loop gain has ORHP poles located at s = np then by the Cauchy integral theorem

Z ∞ 0

ln    

S(jω) S_∞

dω = π 2 lim

s→∞sS(s) − S∞

S_∞ + π

np

X

i=1

pi. (16)

Moreover, if the loop gain is strictly proper then S∞= 1. And if the relative degree (pole excess) ≥ 2, then the right side limit of (16) tends to zero, and (16) is reduced to

Z ∞ 0

ln |S(jω)|dω = π

np

X

i=1

pi. (17)

This shows that disturbance attenuation at some frequencies always comes at the expense of an equally large region of amplification at others, as well as the addition from any ORHP poles in Gyu[10].

2.8.1 Filtering sensitivity functions

u G_yu

Gzu

G_zd G_yd

F Fu

−Kz

d

+ +

˜ z z

ˆ z

−

+ + y +

n

Figure 9: General control configuration.

The performance variable z which is desired to be controlled is not necessarily the same as the output y. To get an overview of the problem we model it as a general control configuration Figure 9 [17]. Here the sensor controller feedback y = −Ky where K = (1 + KzF Fu)⁻¹KzF . The aim is here to design the sensor controller feedback K in order to achieve acceptable disturbance attenuation on the performance variable z. The closed loop transfer function from d to z is

Tzd = G_zd− GzuK(1 + G_yuK)⁻¹G_yd. (18) The General control problem in subsection 2.7 can be divided into two separate problems, control and filtering. If we assume that all unobservable states of the system are stable and that the error estimate of the performance variable is both unbiased and bounded. Then we can define the filtering sensitivity function as P = (G_zd− F G_yd)G⁻¹_zd and M = F G_ydG⁻¹_zd. (19) This is under the conditions that G_ydhas no unobservable unstable states, G_zd is right invertible and the filter F is a stable filter. For clarity, the estimate ˆ

z is unbiased if ˜z = z − ˆz is completely decoupled from the input u meaning that whenever the disturbance and measurement noise is zero for all inputs and initial states, the estimation error ˜z decays asymptotically to zero. A transfer function is said to be right invertible if it has at least as many input signals as signals to be estimated.

The filtering sensitivity function P represents the relative effect of the distur- bance on the estimation error, and the complementary sensitivity function M represents the relative effect of the disturbance d acting on the estimate ˆz [17].

Like S and T we have the relation that P(s) + M(s) = 1 for any s ∈ C but not at poles in P or M. And by defining p as an ORHP pole of G_zd and ξ is an ORHP zero of G_ydwhich do no coincide with a zero in G_zd then we have

P(pi) = 0 M(pi) = 1

P(ξi) = 1 M(ξi) = 0 (20)

assuming F is a bounded error estimator.

Again like the case for the sensitivity function S we can apply bodes integral on P assuming P is proper as well as assuming F to be the bounded error estimate.

Let % denote the ORHP zeros of P, and ς denote the ORHP zeros of Gzd but under the condition that F (ς)Gyd(ς) 6= 0 for any same zero. If P(∞) = P∞6= 0 then we have

Z ∞ 0

ln    

P(jω) P∞

dω = π 2 lim

s→∞sP(s) − P_∞ P∞

+ π

n_%

X

i=1

%_i− π

nς

X

i=1

ς_i. (21) Similar to the ”water-bed” constraint (17) this shows that it is impossible to achieve reduction of the estimation error over all frequencies at the same time as ORHP zeros of Gzd are present.

2.8.2 Filtering limitations using local measurements

When dealing with for instance unwanted inter-area oscillations the ability to measure relative frequency difference and in an ideal way capture the mode would be optimal from a control point of view. But external measurements are not typically available hence we have to use local frequency measurements.

2.8.3 Two-machine example - Filtering limitations

Consider the same two machine system as in subsection 2.3, ideally we would like to capture the generators relative frequency difference z = ω1− ω2. But as z is not available we try to estimate z using the voltage phase angle at bus 1 as measurement input y = θ₁, and estimator F , as seen in Figure 10.

The disturbances d₁ and d₂ are active power perturbations acting directly on generator 1 and 2 respectively. For this set up the input output mapping is given by:

Gzd1 Gzd2

Gyd1 Gyd2



= G₀

 0.21s(s + 0.05) −0.16s(s + 0.07) 0.16(s²+ 0.05s + 7.37) 0.04(s²+ 0.07s + 17.74)

 (22)

ω1

F

V ∠θ1 V ∠θ3 V ∠θ2

ω2

HV DC

G₁ G₂

1 3 2

u = P_dc

d1= ∆P1 d2= ∆P2

y = θ1

ˆ z

Figure 10: Two machine power system model, with signals and estimator.

where G0= s(s+0.06)(s²¹+0.06s+9.86). Since y 6= z we want to design an estimator ˆ

z = F y so that ˆz ≈ z. Let P1 and P2 be the filtering sensitivity functions (19) coupled to the disturbance d₁ and d₂respectively. Then it is only possible to achieve |P_i(jω)| < 1, i = 1, 2 for frequencies in the range ω ∈ (|q₁|, |q2|), where q₁= −0.03 ± j2.71 and q₂= −0.04 ± j4.21 being the complex conjugated zeros of G_yd1 and G_yd2 respectively. This is due to the fact that ˆz must have the same sign as z, i.e., M_i > 0. Using (22) the complementary sensitivity functions become

M1= F Gyd1G⁻¹_zd1= F0.76(s²+ |q1|²) s²(s + 0.05) , M₂= −F G_yd2G⁻¹_zd2= −F0.24(s²+ |q₂|²)

s²(s + 0.07) ,

(23)

and obviously Mi > 0 only for s = jω ∈ (|q1|, |q2|), and the zeros derived there- fore impose filter limitations.

From the bode diagram of the complementary sensitivity functions Figure 11, it can be seen that the two complementary sensitivity functions has the same sign for the interval ω ∈ (|q1|, |q2|). Hence, the estimate of z will be good no matter the origin of the disturbance. However, in the interval ω > |q2| we see that the complementary sensitivity function M₁> 0 has opposite sign compared to M₂. This means that a disturbance acting on d₂will initially be estimated with the wrong sign. The same problem occurs for ω < |q₁|.

To further illustrate the filtering limitations, two different estimators are de- signed to the system in (22). The block diagram of the set up is shown in Figure 12. Neglecting the measurement noise n an estimator at first sight could be

Figure 11: Bode diagram, Visualization of local measurement limitations.

G_zd

Gyd F

d

n

+ +

z +

y

ˆ z

−

¯ z

Figure 12: Filter design.

F = G⁻¹_ydG_zd, (24)

but as seen from (22) G⁻¹_ydG_zd is not a proper estimator, hence not stable.

Instead a simple estimator consisting of two derivatives and a complex conju- gated pole approximately at the modal frequency, followed by an H₂ optimal estimator are designed.

The simple estimator for (22) is designed to be

10 s²

s²+ 2s + 10.86, (25)

and the H2 optimal estimator is tuned to penalize the performance variable z and the disturbance d. The resulting estimators ability to imitate z for the simple estimator and the H2optimal estimator when the system is subjected to a disturbance is presented in Figure 13 and Figure 14.

(a) Disturbance input 1 (b) Disturbance input 2

Figure 13: Estimation of inter-area mode using a simple estimator.

(a) Disturbance input 1 (b) Disturbance input 2

Figure 14: Estimation of inter-area mode using a H2optimal estimator.

It can be concluded from Figure 13 that both observers suffers from the local sensor limitation. When a disturbance is acting on input 2, both observers initially estimates the wrong sign of the mode. This validates what was seen in Figure 11 where the phase differs by 180 degrees between M1 and M2. 2.8.4 Feedback control limitations using local measurements

The purpose of a feedback controller is to create a closed loop system such as Tzd < Gzd which is the same as reducing the amplitude of the disturbance in closed loop system. Multiplying with G⁻¹_zd we can form the disturbance response ratio [18].

|Rzd| =

1 − GzuK(1 + GyuK)⁻¹GydG⁻¹_zd

< 1. (26) If we denote Rzd1 and Rzd2which represents the disturbance response ratio for disturbance input d₁ and d₂then we would like to design a feedback controller u = −ky so that

|Rzdj| < 1, j = 1, 2.

Similar to the previously describe filtering limitations however, this is not pos- sible for all frequencies [6]. If we assume that the closed loop system is stable and the performance variable z = ω1− ω2, then according to Bode’s integral

Z ∞ 0

ln |Rzdj(jω)|dω = π

nγ

X

i=0

γi≥ 0. (27)

Here γi are the ORHP zeros of Rdzj This means we have the same water bed effect as presented in (16). Disturbance attenuation achieved in the interval ω ∈ (|q1|, |q2|) using the feedback controller u = −ky imposes that

sup

ω

max(|R_zd1|, |R_zd2|) > 1, for

(ω < |q1|

ω > |q2| . (28) The feedback controller Fu in Figure 9 can be chosen to decouple the control input u from the estimation error by selecting a controller such that Gzu = F (Gyu+ Fu). Then by using the substitutions Gyu= Gyu+ Fu and K = KzF we can rewrite (26) as

|Rzd| =

1 − (1 + G_zuK_z)⁻¹G_zuK_zMj

< 1, (29)

and by letting K_z → ∞ we see that Rzdj → Pj in (19). It should however be noted that the ORHP zeros in R_zdj do not need to be the exact same zeros as for P_j.

2.8.5 Two-machine example - Feedback control limitations

Synthesizing PSS- and H₂-controllers as described in subsection 2.7, the aim is now to achieve disturbance rejection. Plotting the disturbance response ratio (26) for both controllers Figure 15 we see that disturbance rejection is achieved for the modal frequency at about 3.14 rad/s. However both controllers show the limitation of the zeros, here marked in the figure as dashed vertical lines. It is clear that disturbance rejection for disturbances acting on disturbance input 1 is not achieved above q2 or likewise not possible for disturbances acting on disturbance input 2 below q1.

(a) PSS controller (b) H2-controller

Figure 15: Disturbance response ratio using a simple controller and a H2- controller.

A disturbance unit step is performed for both controllers, in Figure 16 the rel- ative frequency difference for the closed loop system Tzd and open loop system Gzd can be compared.

Shown in Figure 16 both controllers achieves attenuation of the mode as de- sired, except for the first swing caused by disturbance input 2. A close up of the first swing shown in Figure 17, shows how the controller amplifies the first swing disturbance response when the disturbance acts on input 2. This is linked to what was shown in Figure 13 b and Figure 14 b where the observer is ini- tially estimating the wrong sign, hence amplification is inevitable. The resulting dominant modes of the close loop systems are listed in Table 6.

Table 6 shows that the H₂-controller increases the absolute value of the real part, but still keeps the mode at the original frequency. The PSS controller also increases the absolute value of the real part but slightly shifts the modes frequency.

(a) PSS controller (b) H2-controller

Figure 16: Relative frequency difference from local phase angle measurement following step disturbance.

(a) PSS controller (b) H2-controller

Figure 17: Close up of the relative frequency difference from local phase angle measurement following a step disturbance in input 2.

Controller λ f [rad/s] ζ

PSS -0.02 ± 2.98i 2.99 0.073 -2.92 ± 4.81i 5.63 0.510 H2 -0.08 ± 3.14i 3.14 0.086 -0.40 ± 2.84i 2.87 0.140

Table 6: Resulting dominant modes of the closed loop systems.

2.9 Two-machine example - Perturbed system

Adding a perturbation to the system in the form of a 15% load increase in the south area yields a system which mode frequency is lowered to 2.97 rad/s compared to the original system. Using the same PSS and H₂-controller as in subsubsection 2.8.5 and generating step responses, this shows how the con- trollers performs to a more congested system.

(a) PSS controller (b) H2-controller

Figure 18: Relative frequency difference from local phase angle measurement following step disturbance in a perturbed system.

It can be noted in Figure 18 that the controllers achieves less damping compare to the original system in Figure 16. In Figure 19 still the overshoot occurs but without any significant difference. Since the controllers are based on the model, perturbations changing the system could potentially reduce the performance of the controllers. We see however in this example that the impact of a load increase is not causing any immediate problems.

(a) PSS controller (b) H2-controller

Figure 19: Close up of the relative frequency difference from local phase angle measurement following step disturbance in a perturbed system.

3 Simulation study

In order to examine oscillatory phenomenons and fundamental limitations in the Nordic power grid, a simulation study is performed. The potential sources of error such as load losses, tripping of generators, and tripping transmission lines are considered in order to detect relevant cases based in the N-1 criterion in subsubsection 2.5.4. The countless number of possible faults are impossible to cover in this report, hence limited to the most distinctive cases. Once the fault cases are established, simulations with multiple regulators and measure- ment signals are performed. The model used is presented in Figure 20 as a schematic.

Analysis of power production, consumption and power flow in the Nordic grid leads to the following assumptions

• Severe load losses can be assumed more likely to occur in the south region of the grid due to locations of industries and interconnections. These faults could be thought of as faulty HVDC interconnection or failure in an industry. In such case the magnitude of load loss lies in the spectrum up to 1400 MW, as can be seen in Table 2.

• Loosing a generator can occur in both north and south region with poten- tially larger production losses in the south region.

• Faults on HVDC connections used either for import or export of power could lead to increased or decreased power in the region of the connection.

This can also be seen as an increase or decrease in loads.

We consider control of the FennoSkan HVDC link. The link is modeled as two controllable loads at each HVDC terminal. Reactive or resistive losses as well as potential time delays in the HVDC connection are assumed relatively small, hence neglected.

Based on the previous analysis and assumptions three different test cases are constructed. We consider faults that increase the load angle.

3.1 Case 1 - Load decrease in the north region

A failing HVDC connection such as an exporting east link results in a 1 second pulse of a 500 MW load decrease in the north bus 4072 marked as a red flash in Figure 20. This increases the already high amount of power in the north area.

4071 4011

4072 4012

1011 1013

1012 1014

1021

1022

2032 2031

4022 4021

4031 4032

4041

4042

4043 4046 4044

1043 1044

1041

1045 1042

4045 4061

4062

4063

4047

4051

HVDC

North South

Figure 20: Modified Nordic 32 bus system.

3.2 Case 2 - Load increase in the south region

A failing HVDC in the southern region of the Nordic grid leads to a 1 second disturbance pulse. This is a 500 MW power dropout corresponding to a link importing power from an interconnected neighbor grid. The disturbance occurs at bus 4051 marked as a red flash in Figure 20.

3.3 Case 3 - Severe disturbance in perturbed system

A severe disturbance of 1650 MW power occurs in the north (bus 4072). The system is now under heavier congestion and inertia in the south part of the system has been reduced due to expanding use of solar power. Controllers are designed for the nominal model used in Case 1 and 2.

The reason that pulses are used instead of constant load steps in both cases is partly that a failing HVDC can be reset and connection reestablished, but also due to the fact that a constant step implies that other dynamic properties of the system gets triggered which are not a part of this study.

4 Results

For the following simulations the generator located at the north end of the Fen- noSkan HVDC link (bus 4072) is called the North generator and the generator at the south end of the HVDC link (bus 4044) is referred to as the South gen- erator. Phase measurements for plotting/evaluation are taken on load buses where disturbances are acting, that is bus 4072 (North) and bus 4051 (south).

These are the two buses that have a strong participation in the inter-area mode.

Two H2-controllers and two PSS controllers are synthesized for each separate output measurement with one moderate version and one more aggressive one.

4.1 Case 1 - Load decrease in the north region

As seen in Figure 21 the inter-area oscillation caused by the disturbance pulse is being attenuated by all controllers synthesized with the local measurement of the generator speed located at the same North bus. The relative phase between bus 4072 and 4051 is suppressed compared to the open loop performance both when it comes to the first swing and the overall damping.

If then with the same controllers, a disturbance occurs in the south (bus 4051) Figure 22, it is shown how the controllers attenuates the inter-area mode but increases the phase difference in the first swing.

(a) Relative phase difference between bus 4072 and 4051

(b) Close up of first swing

Figure 21: Relative phase subject to a pulse disturbance in the north, with measurement in the north.

(a) Relative phase difference between bus 4072 and 4051

(b) Close up of first swing

Figure 22: Relative phase subject to a pulse disturbance in the south, with measurement in the north.

4.2 Case 2 - Load increase in the south region

In Figure 23 the relative phase difference subject to a load increase at the south bus (4051) is shown, measurement output is at the south bus. All controllers are attenuating the mode and suppressing the first swing seen in the right figure.

(a) Relative phase difference between bus 4072 and 4051

(b) Close up of first swing

Figure 23: Relative phase subject to a pulse disturbance in the south, with measurement in the south.

If the disturbance instead occurs at the north bus 4072 we get the following behaviour, seen in Figure 24

(a) Relative phase difference between bus 4072 and 4051

(b) Close up of first swing

Figure 24: Relative phase subject to a pulse disturbance in the north, with measurement in the south.

It is shown in Figure 24, that all controllers attenuates the inter-area mode, but with the risk of increasing the phase difference in the first swing. It seems to be a matter of tuning as the moderate H2-controller more or less leaves the first swing unaltered, when the other increases it.

Taking a closer look at this example where the disturbance occurs in the south region. We plot the control signal for the controllers with different measurement

locations. The result of the H2-controller is shown in Figure 25, and the PSS is shown in Figure 26.

(a) Measurement in south (b) Measurement in north

Figure 25: Relative phase and control signal subject to a pulse disturbance in south using H2 controller.

(a) Measurement in south (b) Measurement in north

Figure 26: Relative phase and control signal subject to a pulse disturbance in the south using PSS controller.

Both Figure 25 and Figure 26 a) shows that when disturbance and measurement occurs in the same geographic part of the grid the control signal initially acts correctly to attenuate the disturbance. It is however shown in Figure 25 and Figure 26 b) that when the disturbance and measurement are at different geo- graphic locations. The control signal instead enhances the disturbance. As the control signal initially should be positive, that is injecting power to compensate for the disturbance loss, the negative peak decreases the power further.

4.3 Case 3 - Severe disturbance, perturbed system

The system is perturbed by increased generation in the north generators as well as decreased inertia in some of the south generators. This represents a congested

system with more wind and solar power.

A severe disturbance in the north region occurs as a one second pulse with an amplitude of 1650 MW. This is increasing the power in the northern region hence causing an even more congested system. The aggressive H2 controller from the previous cases are used, both with measurement signal in the north as well as in the south. Results are presented in Figure 27.

(a) Pulse disturbance in the north (b) Closeup of the first swing

Figure 27: Relative phase subject to a severe pulse disturbance in north with the perturbed system.

Shown in Figure 27 the regulator which measurement is taken at the same bus as the disturbance suppresses phase difference between the north and south generators. However, when measurement is taken at the south end of the system, the controller (as shown in previous examples) causes the first swing to increase the phase difference. In this case, when the system is heavily congested and with lower inertia the controller causes the system to separate. This is clearly shown in Figure 27 b) as the red graph diverges in the first swing.

5 Conclusion

Using FennoSkan for damping inter-area modes using local measurements has been shown possible, yet must come with a trade off. Arbitrary disturbance at- tenuation will not be possible in all locations due to the local sensor limitation, and this is linked to the observer initially estimating the wrong sign as has been shown. It is however possible to design a controller that achieves arbitrary large POD improvement. Designing the controller is a huge topic itself and it has been shown that a somewhat aggressive controller could lead to instability dur- ing the first swing. The design is by no means a simple task, especially since the electricity grid is an extremely complex system. With countless faults that can occur it will be very cumbersome to find an optimal control for all possible situ- ations. Not only must the controller be optimized for the disturbances, but also robust against model perturbations as the grid expands and changes. This was

shown for the heavily congested grid with a model perturbation in case 3, the issue with the first swing became hazardous with tripping generators as a result.

The model used for the Nordic grid is shown to be quite robust and can handle some major disturbances without failing to recover. It clearly meets the N-1 criterion, but it should however be mentioned that the models simplifications might have altered the system’s behaviour. Neglecting weak bonds when for ex- ample grouping multiple generators into one, might have made the model more robust to load disturbances compare to the true grid. It should also be noted that safety features neglected in the model can in the real grid cause tripping generators for minor disturbances compared to what was shown in this report.

The negative impact of the local measurement limitations seems however not to be a serious problem unless under extreme conditions as been shown. That includes heavy congestion, multiple coinciding disturbances as well as high gain POD controllers.

In order to achieve the best oscillation damping with a local measurement, it is obvious that a measurement should be taken at the disturbance source. How- ever, with only one measurement source one must choose where to measure.

The north area of the Nordic grid is dominated by production, and a failure here is then likely to decrease the exporting power from the region. This is shown in the simulations to be a minor problem to transient stability. In the south however, a loss of a importing HVDC link or loss of generator causes the phase angle difference to increase, forcing the power flow from north to south to further increase. This increases the stress in the network, and then having a control system that amplifies the negative effect is undesirable. Therefore, it is recommended that measurements take place in the southern area to maximize the benefit of damping using local measurement.

Multiple sensor or even wide area measurements would most likely circumvent the problem of the observer generating the wrong sign but local measurement is significantly cheaper and easier to implement than wide area measurement techniques. Not only are the amount of sensors an expense but also the com- munication and safety precautions needed for a more complex system, not to mention the communication time delay. Even though better performance is achieved with wide area measurements, the robustness and economic benefit of using the local measurement compensates for poorer performance.

6 Future work

For future research it would be of great interest to explore some of the following mentioned tracks.

External measurement - Throughout this thesis the external measurement was limited to the rotor speed of given generator, this is under the assumption

that generator speed measurements are available. It would be very interesting to explore different types of measurement sources such as bus voltages or active power flow. These measurements are already available for transmission system operators as they are of great concern regarding stability and balance in the power grid. But how they capture the inter-area modes and their limitations would be interesting to dig further into.

HVDC dynamics - In order to focus on the theory and fundamental limi- tations of the local measurement technique, none of the HVDC dynamics are treated in this thesis. This could for instance be the ramp up / ramp down time, as well as reactive power and power capacity limitations. Adding these to the model used, the behaviour of the system would most likely be different and less responsive compared to what has been shown. Since HVDC connections in a congested grid are more likely able to lower the power rather than increase it, this would be an interesting property to explore. That would mean that com- pared to what was shown in this thesis, the HVDC connection would initially be at an equilibrium with maxed out power transmission and thus limited to a decrease in power.

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