In the name of the

most merciful

Seminar on:

Pole Placement and Robust

Adjustment of Power Systems

Stabilizers(PSSs) through Linear

Matrix Inequalities(LMIs)

Presented by: Mahsa Rezaei (909238)

Supervised by: Dr. Maryam Dehghani 2

The damping of inter-area oscillations is an

important problem in electric power systems.

Inter-area oscillations provide restrictions on the

transferred power.

To enhance damping, power system stabilizers

(PSSs) have been adopted.

PSSs provide phase compensation as

supplementary signals into the excitation

systems of the generators.

3

How to control Power systems?

Generally power Systems are modeled as large scale

systems composed of a set of small interconnected

subsystems.

Complication of incorporation of many feedback loops

into the controller design for large scale interconnected

systems is a motivation for the development of

decentralized control.

On the other hand, Varying operating condition of

power systems and also the nonlinear nature of them,

limits the applications of conventional PSSs.

There is a need for controllers which are robust to

changes in the system operating conditions.

4

This work presents the application of LMIs to the optimal

adjustment of PSSs with pre-defined structure, much common in

the power industry.

Results of some tests show that gain and zeros adjustments are

sufficient to guarantee Power Systems robust stability and

performance with respect to various points of operation.

Making use of the flexible structure of LMI’s, the algorithm permits

one to choose in the compromise between low gain and high

damping factor.

The technique used here is the pole placement design together

with gain limitation, which places the poles of the closed loop

system in a specific region of the complex plane.

5

Outline:

Power System Modeling

Power System Stabilizers and the Closed loop

Structure

Pole placement through linear matrix

inequalities

Experimental results and discussion

More general case

6

Power System Modeling:

The power system electromechanical stability problem can be

represented by a set of differential equations together with a set of

algebraic equations to be solved simultaneously with each other:

𝑥 = 𝑓( 𝑥, 𝑧 )0 = 𝑔( 𝑥 , 𝑧 )

(1)

Where x is the state vector and z is a vector of algebraic variables.

Small disturbance stability analysis normally involves the linearization

of (1) around the system operating point.

(2)

7

∆ indicates a little variation of the variable.

The sub-matrix 𝑨𝟏𝟏 represents the dynamic description of the

system;

𝑨𝟏𝟐 represents the link between the dynamic of the system

and its algebraic variables;

𝑨𝟐𝟏 and 𝑨𝟐𝟐 represent the link between state and algebraic

variables;

𝑩𝟏 and 𝑩𝟐 relate the states and the dynamic of the system to

its inputs,

𝑪𝟏 and 𝑪𝟐 relate measurements to state variables. 8

Power System Stabilizer and the Close loop Structure:

The model described in (2) can be transformed in a more

appropriate form:

𝑥 = 𝐴. 𝑥 + 𝐵. 𝑢 (3)

𝑦 = 𝐶. 𝑥

The PSS structure is fixed, This fixed structure is given by:

(4)

In this scheme, we work with fixed poles and free zeros and gain;

so, we have to obtain the gain (𝐾𝐺) and the zeros (given by the

values of 𝑘2 and 𝑘1) of the controller.

The values chosen for the poles of the PSS are of practical

interest. 9

10

For the decentralized structure, the controllers transfer function

matrix will assume the following form:

(5)

n is the number of machines, and each 𝑲𝒊(𝒔) has the structure

given by (4).

Each PSS will be linked to a machine of the system. The control

structure given by (5) can be transformed in an state space form:

(6)

11

where 𝐴𝐶, 𝐵𝐶 , 𝐶𝐶 and 𝐷𝐶 are matrices that define the structure

of the controller given in (4), such that:

(7)

By using state space representation:

(8)

(9)

12

Applying the control structure (6) to the system described by (3),

we have the following description for the closed loop system:

(10)

By defining the matrices:

(11)

(12)

The modified system, which is equivalent to (10), is given by:

(13)

13

Where 𝑥𝑚= 𝑥 𝑥𝐶 𝑇

and the control law 𝑢𝑚 = 𝐾𝐶 . 𝑦 = 𝐾𝐶 . 𝐶𝑚. 𝑥𝑚

Using the state space description of the system and the

controller fixed matrices 𝐴𝐶 and 𝐶𝐶 , we can calculate

matrices 𝐴𝑚, 𝐵𝑚 and 𝐶𝑚.

Then, we must determine the static output feedback gain

matrix 𝑲𝑪 that places the closed loop poles of the system in

a determined region of the complex plane.

14

Pole Placement through Linear Matrix Inequalities:

A. LMI Regions

An LMI region is any subset D of the complex plane that can be

defined as

(14)

where L and M are real matrices such that 𝐿𝑇= L.

Important characteristics of LMI regions are:

1. Intersections of LMI regions are LMI regions;

2. A real matrix A is D-stable, that is, has all its eigenvalues in

the LMI region D if and only if a symmetric matrix Q exists

such that:

(15)

15

LMI regions of interest in control applications for pole

placement purposes are the following:

1. Half-plane Re(z) < – 𝛼 : L = 2.𝛼 and M = 1;

2. Disk centered at the origin with radius r:

3. Conic sector with apex at the origin and inner angle 2𝜃:

We will place the poles of the power system in a LMI region that

is the intersection of these three regions. It ensures a minimum

decay rate 𝛼, a minimum damping ratio 𝜉 = cos (𝜃), and a

maximum undamped natural frequency 𝜔𝑑 = 𝑟. sin 𝜃 for

the closed loop system.

16

B. Output feedback control and Pole Placement through LMI’s

Applying an output feedback control law 𝑢𝑚 = 𝐾𝐶 . 𝑦 to the

system (13), we get the following closed loop system:

(16)

Our objective is to place the poles of the closed loop system in a

region of the complex plane that is given by the intersection of

the three regions described. To do this, we have to apply (15) to

the closed loop system.

If we substitute the closed loop matrix 𝐴𝑐𝑙 into equation (15), the

term 𝐴𝑐𝑙.Q will not be convex, Because:

(17)

17

To solve this problem and transform (17) into a LMI, we

proceed the following change of variables:

(18)

Substituting (18) into (17), we have:

(19)

Thus, now we have a LMI problem. To recover the controller

gain matrix, we do the following:

(20)

The existence of the inverse in (20) is guaranteed if matrix 𝐶𝑚 is

full row rank. After getting matrix M, we calculate the gain

matrix 𝐾𝐶:

(21)

18

Using this change of variables, equation (15) can be applied to

the closed loop system (16) in order to place the poles of this

system in an adequate region of the complex plane:

(22)

Substituting the values of L and M for these three regions yields:

1. Half plane Re(z) < – 𝛼:

(23)

2. Disk with radius r:

(24)

3. Conic sector with inner angle 𝜃:

(25)

19

where * denotes symmetric term, and:

(26)

20

Summarizing the procedure:

1. Firstly, Define 𝐴𝑚, 𝐵𝑚 and 𝐶𝑚 from (11)

and the specifications of performance for the closed loop system

(that is, the values of 𝜃, 𝛼 and 𝑟);

2. solve the system of LMI’s given by (23), (24), (25) and Q > 0

in the variables Q and N;

21

3. Calculate matrix M, given by (20);

4. Calculate the static gain matrix 𝐾𝐶 = 𝑁.𝑀−1;

5. Recover the controller variables 𝐷𝐶 and 𝐵𝐶 , considering (12);

6. Calculate the transfer function of each decentralized controller,

considering (7), (8) and (9).

22

C. Constraining the Static Gain

As we are working with a static controller formulation, it is

interesting to bound the norm of the static gain matrix 𝐾𝐶 in

order to avoid infeasible values for the controller parameters.

To bound the Euclidean norm of the controller matrix

𝐾𝐶 = 𝑁.𝑀−1 (with M given by (20)), we do the following:

(27)

In other words, we are restricting at the same time the norms of

matrices N and 𝑀−1 . Rewriting (27), we have:

(28)

C. Constraining the Static Gain

As we are working with a static controller formulation, it is

interesting to bound the norm of the static gain matrix 𝐾𝐶 in

order to avoid infeasible values for the controller parameters.

To bound the Euclidean norm of the controller matrix

𝐾𝐶 = 𝑁.𝑀−1 (with M given by (20)), we do the following:

(27)

In other words, we are restricting at the same time the norms of

matrices N and 𝑀−1 . Rewriting (27), we have:

(28)

23

The values of 𝑲𝑵 and 𝑲𝑸 are project parameters and have

to be chosen carefully, in order to avoid large controller

parameter values and, at the same time, not restrict too

much the controller gain matrix.

24

D. The Robust Procedure

It is necessary to ensure that the power system will present good

performance even in case of variations in the operating point. To

deal with this problem, we will make use of the so called

polytopic models.

let the i-th power system model linearized around a specific

operating point be denoted by the triple ( 𝐴𝑖 , 𝐵𝑖 , 𝐶𝑖 ); considering

the model used here, matrices B and C do not vary with changes

in the operating points; so, the triple that defines the system will

be simply (𝐴𝑖 , B , C). A polytope is the set Ω defined below:

(29)

where n is the dimension of matrices 𝐴𝑖 and m is the number of

operating points. The matrices 𝐴𝑖 are called vertices of the

polytope.

25

To ensure that the poles of any closed loop system associated to a

matrix 𝐴 ∈ Ω will lie in the region of the complex plane described

previously, we have to solve m LMI’s jointly in the same variables

Q and N, that is:

(30)

26

for i = 1, 2, … , m , where m is the number of operating points

for the power system, and:

(31)

(32)

𝑨𝒊, i = 1, 2, … , m are the state transition matrices that define the

power system model, and these matrices represent nonlinear

models linearized around one specific operating point.

The procedure for recovering the controller gain matrix is the

same described before.

27

Experimental Results and Discussion: The studied case is the New England system, that consists of nine

synchronous machines with PSSs and one machine (number 39)

modeled as an infinite bus.

28

where 𝜟𝒙 composed by variables 𝜟𝝎, 𝜟𝜹 , 𝜟𝑬𝒒′ and 𝜟𝑽𝑭𝑫.

𝜟z is the algebraic variables vector, composed by variables

𝜟𝑰𝒅, 𝜟𝑰𝒒, 𝜟𝑽𝒓 and 𝜟𝑽𝒎. 𝜟y is the output vector (or measurements vector), composed by

measurement of variable 𝜟𝝎.

𝝎 is the synchronous machine rotor velocity, 𝑽𝒓 is the voltage

magnitude at the real axis, 𝑽𝒎 is the voltage magnitude at the

imaginary axis, 𝑽𝑭𝑫 is the field voltage, 𝑰𝒒and 𝑰𝒅 are the currents

at the machine coordinates, 𝜹 is the synchronous machine load

angle, 𝑬𝒒′ is the voltage magnitude of the synchronous machine

during the transitory state and ∆ indicates a little variation of the

considered variable. ∆u is the input vector (or control vector), given by variable

∆𝑽𝑹𝑬𝑭 (it represents a reference input voltage, that is present in

the automatic voltage regulator model).

29

Eight Operating Conditions that were used in this work:

30

The eigenvalues of the power system before control are presented in

figure 1. As can be seen, there are many unstable eigenvalues, as

well as low damped eigenvalues.

31

Considering the operating conditions given and the following

performance specifications:

32

33

34

35

Now Let us consider a more general case,

We state a Robust PSS design method in which by the

help of pre-defined structure of controller and the

three interesting regions for poles to be placed on,

could change to a LMI problem.

But What if we want to design a Robust free structure

controller whose poles are placed on a desired region?

We will investigate this problem through another paper

and try to solve it with ILMI.

36

The design of a PSS can be cast as a static output control synthesis

using a system augmentation technique. Thus, we deal only with

static output feedback problems. First, we discuss the design of a

static output controller that locates its closed-loop poles within a

specified region. We then describe its extension to simultaneous

stabilization of a finite collection of LTI plants.

Let us consider the LTI system:

𝑥 = 𝐴. 𝑥 + 𝐵. 𝑢 , 𝑦 = 𝐶. 𝑥 (33)

By pole-placement using static output feedback, we mean to address

the problem of searching for a static control law 𝑢 = 𝐾𝑦 such that

all the eigenvalues of the closed-loop matrix 𝐴 + 𝐵𝐾𝐶 are placed in

a specified complex-plane described by:

(34)

37

Lemma 1:

For the LTI plant (33), we can find a static control 𝑢 = 𝐾𝑦 that

places all its closed-loop poles in region (34) if and only if there

exist symmetric matrices W ∈ 𝑅(𝑚+𝑝)×(𝑚+𝑝) , W≥ 0 and P ∈𝑅𝑛×𝑛,

P ≻ 0 that satisfy the following LMI and rank condition,

(35)

(36)

If W satisfying (35) and (36) is obtained, K can be computed as

(37)

Assuming

(38)

38

Proof of lemma 1: Let 𝐴 ∈ 𝐶𝑛×𝑛 be a given complex matrix and let

𝐷 = 𝑠 ∈ 𝐶 ∶ 1𝑠

∗ 𝑝 𝑞𝑞∗ 𝑟

1𝑠< 0

denote a given open region of the complex plane, where Hermitian matrix

𝑝 𝑞𝑞∗ 𝑟 ∈ 𝐶2×2

has one strictly negative eigenvalue and one strictly positive eigenvalue,

and the star denotes transpose conjugate.

Theorem 1 (Lyapunov’s Inequality): Matrix A has all its eigenvalues

in region D if and only if there is a matrix 𝑃 = 𝑃∗ > 0 ∈𝐶𝑛×𝑛 such that

𝐼𝐴

∗ 𝑝𝑃 𝑞𝑃𝑞∗𝑃 𝑟𝑃

𝐼𝐴

< 0 (39)

Matrix inequality is referred to as Lyapunov’s inequality.

39

What we almost always work with from Lyapunov’s inequality is

in the case that D is the open left half-plane, i.e., p= r = 0, q = 1

and inequality (34) becomes 𝐴𝑇𝑃 + 𝑃𝐴 < 0.

Sometimes this synthesis is called D-stabilization. The following

lemma states a necessary and sufficient condition for the

existence of a pole-placement static output feedback controller.

40

According to Lyapunov stability theory , the existence of a pole-placement

static output feedback controller is equivalent to the existence of a positive

definite matrix P ∈ 𝑅𝑛×𝑛 such that the following inequality holds

(40)

for all x and u satisfying the static control law

(41)

where R ∈ 𝑅𝑚×𝑚 is any nonsingular matrix. Therefore,

(42)

where τ > 0 is a scalar. We thus obtain (35), (36) and (37) by letting

(43)

41

Now consider the simultaneous stabilization of the system

described by a set of state-space equations

(44)

Each triplet ( 𝐴𝑖 , 𝐵𝑖 , 𝐶𝑖 ), i = 1, . . . ,N represents an LTI system

linearized at a certain equilibrium point. In the same manner, we

seek a single control law 𝑢 = 𝐾. 𝑦 such that all

matrices 𝐴𝑖+𝐵𝑖 . 𝐾. 𝐶𝑖, i = 1, . . . ,N have their eigenvalues in a

prescribed region.

We can easily extend Lemma 1 to simultaneous stabilization with

closed-loop pole constraints described by the intersection of

multiple regional constraints.

42

Lemma 2:

For the finite set of plants (44), we can find a single control law

𝑢 = 𝐾. 𝑦 that simultaneously stabilizes all N plants with all the

closed-loop poles located in the intersection of

regions 𝐷(𝑝𝑗 , 𝑞𝑗 , 𝑟𝑗), j = 1, . . . ,L if and only if there exist

symmetric matrices W ∈𝑅(𝑚+𝑝)×(𝑚+𝑝) , W > 0 and 𝑃𝑖𝑗 ∈

𝑅𝑛×𝑛 , 𝑃𝑖𝑗≻ 0 that satisfy the LMIs for i = 1, . . . ,N, j = 1, . . .

,L.

(45)

and the rank condition

𝑟𝑎𝑛𝑘 𝑊 = 𝑚. (46)

If W satisfying (45) and (46) is obtained, K can be computed

by:

43

Proof of lemma 2:

Let us assume L = 1 without loss of generality. Sufficiency is trivial by

Lemma 1.

To prove necessity, suppose that a pole placement static controller K is

given for all N plants. Then, by setting R = I in (43), there exist 𝑃𝑖≻ 0

and 𝜏𝑖> 0 such that the following LMIs are feasible for i = 1, . . . ,N

(47)

Where

Since (47) holds for all τ ≥ 𝜏𝑚 ≜ max{𝜏𝑖}, a common W for (47) can be

taken as

This completes the proof.

44

Next, We want to state a numerical method for solving Rank-constrained

LMI problems like the one we encounter here.

Penalty Function Method For Rank-constrained LMI Problems:

With a slight abuse of notation, the problem to be solved has the

following generic form:

(48)

Where 𝒞 is the convex set

(49)

x is the decision vector, and W(x) ∈ 𝑅𝑛×𝑛 and R(x) ∈ 𝑅𝑚×𝑚 are matrices

that are affine functions of x. Also, r is assumed to be less than n.

The core of the PFM is to convert the problem (48) into an ordinary LMI

optimization problem by representing the difficult rank condition as a

penalty function to be minimized.

45

The rank condition is satisfied if and only if the 𝑛 − 𝑟 eigenvalues of

W are zero and that the partial sum of the eigenvalues of a

symmetric matrix is bounded above by a weighted trace of the

matrix.

Let the eigenvalues of W be 𝜆1≤ · · · ≤𝜆𝑛. The sum of the 𝑛 − 𝑟

smallest eigenvalues of W is then bounded above:

(50)

where V ∈ 𝑅𝑛×(𝑛−𝑟) is an arbitrary matrix such that 𝑉𝑇𝑉 = 𝐼𝑛−𝑟.

Thus, we define the penalty function as

(51)

further we introduce the following penalized objective function

(52)

where μ is the positive penalty parameter. Notice that the trace of W

plays an important role in minimizing the value of p(x;V) by placing

the relative weights on the eigenvalues of W since 𝑡𝑟 𝑊 = 𝜆𝑖𝑛1 .

46

We can now obtain a solution to problem (48) by sequentially

solving the following convex LMI optimization problem for k = 0,

1, . . .

(53)

where 𝜇𝑘 ≥ 𝜇𝑘−1and 𝑉𝑘 is constructed from the eigenvectors of

W(𝑥𝑘) since the eigenvectors of W can be taken to be orthonormal

to each other.

47

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48

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49

Thank You