Research of small oscillations of electrical
power systems using the technology of
embedding systems
Kahraman Allaev & Tokhir Makhmudov
Electrical Engineering
Archiv für Elektrotechnik
ISSN 0948-7921
Electr Eng
DOI 10.1007/s00202-019-00876-9
1 23
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https://doi.org/10.1007/s00202-019-00876-9
ORIGINAL PAPER
Research of small oscillations of electrical power systems using
the technology of embedding systems
Kahraman Allaev1 · Tokhir Makhmudov2
Received: 18 January 2019 / Accepted: 6 November 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract
The mathematical model of the regulated electrical power system in matrix form is developed in the article, the basis of which
is equations in the state space and the technology of embedding systems. The resulting mathematical model allows us to
study the static stability of an adjustable complex electrical system by determining the eigenvalues of the dynamics matrix.
Based on the method of decomposition of the initial model of a complex electrical system with the help of semiorthogonal
matrix zero divisors, the poles of the system are shifted to the desired position. The controller obtained on the basis of the
decomposition method makes it possible to increase the stability of a complex electrical system to small oscillations that
occur. The proposed technique can be used to configure automatic control systems.
Keywords Electrical system · Arrangement of poles · Decomposition of the system · Zeros of matrices
1 Introduction
2 Materials and methods
The regulated power system is a complex dynamic oscillatory system with many state variables and various connections both in the main power elements—power plant aggregates being the objects of regulation, and in their automatic
control systems. Qualitative characteristics of the vibrational
properties of power systems are manifested in temporary
transient processes.
For complex dynamical systems, known methods for
placing poles are often not applicable due to their inherent flaws, which include: poor matrix conditioning; possible
insolubility of the problem under complete controllability;
rapid growth of the dimension of the solved equations, etc.
[1–3].
Develop the mathematical model of a multi-machine electrical system for small oscillations on the basis of equations in
the state space having the form [1, 4]:
* Tokhir Makhmudov
tox‑05@yandex.com
ẋ = Ax + Bu,
(1)
y = Cx + D𝜀,
(2)
where A is the matrix of the system’s own dynamics; B is
the matrix of control actions; C is the system observation
matrix; D is the matrix of direct influence on the output of
the system; x ∈ Rn is the state vector; u ∈ Rr is the input vector; R is the set of real numbers; n ≫ 1, n > r. In what follows
we shall consider the inertial system, i.e., we assume that
D = 0 [5, 6].
The model describes the transient process in the electrical system, taking into account the balance of the moments
(powers) on the shaft of the ith unit of EPS, and has the
form [1, 4, 7]:
Kahraman Allaev
allaev099@gmail.com
𝜔
d2 𝛿 i
= 0 [PTi − PGi ],
2
Tji
dt
1
Tashkent State Technical University, ul. Universitetskaya 2,
100095 Tashkent, Uzbekistan
2
Tashkent State Technical University, 30 quarter, 28‑11,
100208 Tashkent, Uzbekistan
where ω0 = 314 rad/s. is the synchronous angular frequency;
Tji , 𝛿i , PTi , PGi are the inertia constant of the ith aggregate,
the load angle of the ith generator, the mechanical power
(3)
13
Vol.:(0123456789)
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Electrical Engineering
of the ith turbine, and the electromagnetic power of the ith
synchronous generator, respectively.
The equation of electromagnetic power of the ith synchronous generator in positional idealization has the form [4, 7, 8]:
PGi = Ei2 yii sin 𝛼ii +
n
∑
Ei Ej yij sin(𝛿ij − 𝛼ij ),
j=1,j≠i
(4)
where Ei, Ej are the emf ith and jth synchronous generators;
yii, yij are the intrinsic and mutual conductivity of the network; and αii, αij are the complementary angles.
After the corresponding transformations (3)–(4), taking
into account the damping moment, we obtain the linearized
power balance equations for the synchronous generator shaft
with respect to the absolute angles in the form [9, 10]:
⎤
⎡ n
𝜔0 ⎢�
d2 Δ𝛿i
Δ𝛿i dPGi
⎥
=
ΔEqi ⎥,
b Δ𝛿 − bii Δ𝛿i − Pdi
−
Tji ⎢⎢ j=1, ij j
dt
dEqi
dt2
⎥
⎦
⎣ j≠i
∑n
(5)
where bij = aij cos 𝛽ij ; aij = Ei Ej yij ; bii = j=1,j≠i bij ; 𝛽ij = 𝛿i0 − 𝛿j0 ,
in which δi0 and δj0 are the initial absolute angles of generators, and Pdi is the coefficient of the generalized damper
moment of the ith generator.
The importance of studies of the transient modes of the
electrical system with respect to absolute angles is noted in
[11, 12].
The peculiarity of the system of Eq. (5) is that they are
resolved with respect to the absolute angles of the system
generators. For example, for a three-generator electrical system (Fig. 1), Eq. (5) takes the form:
[
]
𝜔0
d2 Δ𝛿1
Δ𝛿1
dP1
,
−b
Δ𝛿
+
b
Δ𝛿
+
b
Δ𝛿
−
P
ΔE
−
=
11
1
12
2
13
3
d1
q1
Tj1
dt
dEq1
dt2
𝜔
d2 Δ𝛿2
= 0
Tj2
dt2
[
]
Δ𝛿2
dP2
b21 Δ𝛿1 − b22 Δ𝛿2 + b23 Δ𝛿3 − Pd2
ΔEq2 ,
−
dt
dEq2
[
(6)
]
d2 Δ𝛿3
Δ𝛿3
dP3
𝜔0
b
Δ𝛿
+
b
Δ𝛿
−
b
Δ𝛿
−
P
ΔE
−
=
32
2
33
3
d3
q3 .
Tj3 31 1
dt
dEq3
dt2
The system of Eqs. (6) of the electrical system that
reflects transient processes for small deviations is convenient, both algorithmically and computationally, in particular,
in cases of their joint solution with the equations of nodal
voltages (ENV) [13]. This is explained by the fact that the
result of the ENV solution is the voltage module of the ith
node Ui and its argument δi, determined with respect to the
balancing node [14] and used in the reduced differential
Eq. (6).
The equations of the studied complex regulated electrical
system with many inputs and outputs (MIMO—multi-input
multi-output) in the state space (1)–(2) will finally have the
form [2, 5, 7, 13, 15–18]:
ẋ = A𝛴 x + B𝛴 u,
(7)
y = Cx,
(8)
U = −Kx,
(9)
where K ∈ Rr×n is the matrix of the regulators of the excitation of synchronous generators.
It is assumed that for systems (7)–(8) there exists a feedback control of form (9) [1, 2, 10].
Define the content of AΣ and BΣ for the model of a complex EPS containing n synchronous generators with AEC.
We will solve the problem for the case when EPS generators
have automatic strong-excitation controllers (AEC-s) reacting only to deviations and the first derivatives of the regime
parameters. The time constant of the automatic controller
(Tpi = 0) is not taken into account. Then, the equation of the
output of the automatic regulator of excitation for the ith
generator has the form:
ΔUAEC =
1
�
dΔEqi
dt
]
dΔPki
k0Pi ΔPki + k1Pi
,
dt
(10)
= ΔEqi − ΔEqei ,
(11)
where Tdi is the transient time constants along the longitudinal axis with a short-circuit stator winding; ΔEqi is the
change in emf generator; and ΔEqei is the change in emf on
the rotor rings.
Equation in the excitation winding of the exciter:
′
13
[
where k is the number of parameters of the generator (electrical system) mode, which is automatically controlled by
excitation of the ith generator of the EPS, and ΔPki, k0Pi,
k1Pi are the deviation of the kth parameter of the regime, the
gain factors of the AEC along the deflection channels, and
the first derivative of the same parameters of the modes,
respectively.
Transient equation in the excitation winding:
Tdi
Fig. 1 The scheme of the three-generator electrical system
k
∑
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Te
dΔEqei
dt
(12)
= ΔUAECi − ΔEqei ,
where Te is the exciter time constants.
For small oscillations of the regime parameters on the
basis of (6)–(9), taking into account (10), it is possible to
obtain a generalized block matrix AΣ of size (4n × 4n) for the
dynamics of an electrical system with n generators having
AEC-s in the form [9, 10]:
⎡ 0nxn
⎢A
A𝛴 = ⎢ 21(nxn)
⎢ 0nxn
⎣ A41(nxn)
Inxn
A22(nxn)
0nxn
A42(nxn)
0nxn
A23(nxn)
A33(nxn)
0nxn
0nxn
⎤
⎥
0nxn
.
A34(nxn) ⎥⎥
A44(nxn) ⎦
A42(nxn)
where
⎡− 1
⎡ k1𝛿1 0 … . 0 ⎤
⎥
⎢ Te1
⎢ Te1 k
1𝛿2
⎥
⎢0
⎢0
…
.
0
Te1
=⎢
⎥, A44(nxn) = ⎢
…
.
…
.
…
.
…
.
⎥
⎢….
⎢
⎢0
⎢ 0 0 … . k1𝛿n ⎥
Ten ⎦
⎣
⎣
A21(nxn)
A22(nxn)
A23(nxn)
A33(nxn)
⎡ −Pd1
⎢0
=⎢
⎢….
⎣0
𝜔12
−𝜔22
….
𝜔n2
….
….
….
….
0
−Pd2
….
0
….
….
….
….
⎡ − 𝜕P1 𝜔0
⎢ 𝜕Eq1 Tj1
⎢0
=⎢
⎢….
⎢0
⎣
∑ Ej
𝜔
𝜔0
𝜕Pi
bii , 𝜔ij = 0 bij ,
=
sin(𝛿i0 − 𝛿j0 ).
Tji
Tji
𝜕Eqi
x
j≠i ij
j=1
(13)
At the same time, the column vector of state parameters
containing the parameters of the electrical system mode is:
The content of the input matrix BΣ entirely depends on
the excitation control law and, accordingly, the parameters
of the automatic control channels over which the excitation
system of «n» synchronous machines installed in the EPS is
controlled. For the chosen excitation control law (10), the
generalized matrix BΣ has the size 4n × n (k − m) and the
form [9]:
0
⎤
⎥
0
,
… . ⎥⎥
−Pdn ⎦
…. 0
𝜕P 𝜔
− 𝜕E 2 T 0
q2 j2
…. 0
….
0
…. ….
𝜕P
… . − 𝜕E n
qn
⎡ 1� 0 … . 0 ⎤
⎢ Td1 1
⎥
⎢ 0 T� … . 0 ⎥
d2
=⎢
⎥,
⎢…. …. …. ….⎥
⎢ 0 0 … . 1� ⎥
Tdn ⎦
⎣
A34(nxn) = −A33(nxn)
A41(nxn)
(14)
𝜔1n ⎤
𝜔2n ⎥
,
… . ⎥⎥
−𝜔nn ⎦
0
⎡ − 1�
⎢ Td1
⎢0
=⎢
⎢….
⎢0
⎣
𝜔0
Tjn
⎤
⎥
⎥
⎥,
⎥
⎥
⎦
(15)
where m is the number of mode parameters included in the
generalized matrix of the dynamics of EPS, such as the elements of the state vector of the electrical system.
In this case, the components of the BΣ are:
B41[nxn(k−m)]
⎤
⎥
…. 0
⎥
⎥,
…. …. …. ⎥
0
… . − T1� ⎥
dn ⎦
0
…. 0
− T1�
d2
⎡ k0𝛿1 0 … . 0 ⎤
⎢ Te1 k
⎥
0𝛿2
⎢0
⎥
…
.
0
Te2
=⎢
⎥,
…
.
…
.
…
.
…
.
⎢
⎥
⎢ 0 0 … . k0𝛿n ⎥
Ten ⎦
⎣
…. 0
− T1
e2
n
𝜔ii =
The components of matrix (13):
⎡ −𝜔11
⎢𝜔
= ⎢ 21
⎢….
⎣ 𝜔n1
⎤
⎥
…. 0
⎥
⎥,
…. …. …. ⎥
0
… . − T1 ⎥
en ⎦
0
B42[nxn(k−m)]
B43[nxn(k−m)]
⎡ k0P1 k1P1
⎢ Te1 Te1
= ⎢0 0
⎢… …
⎢
⎣0 0
⎡0 0
⎢ k0P2 k1P2
= ⎢ Te2 Te2
⎢… …
⎢
⎣0 0
⎤
⎥
… 0 0 ⎥,
…… … ⎥
⎥
…0 0 ⎦
…
k0Pk k1Pk
Tek Tek
…0 0 ⎤
k
k
… T0Pk T1Pk ⎥⎥
ek
ek
,
…… … ⎥
⎥
…0 0 ⎦
0
⎡0
⎢0
0
⎢… …
=⎢
⎢
k1Pn
⎢ k
⎢ 0Pn Ten
⎣ Ten
…0 0
…0 0
…… …
…
k0Pk k1Pk
Tek Tek
⎤
⎥
⎥
⎥.
⎥
⎥
⎥
⎦
13
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The generalized matrices (13) and (15) and their components with the chosen AEC-s model allow describing the
transient processes in a complex EPS with «n» generators for
small oscillations of the regime parameters.
For example, for the three-generator EPS (Fig. 1), assuming that the AEC-s reacts to the voltage and load angle deviation of the generators (Δδi, ΔUGi), as well as their first
derivatives (Δ𝛿̇ i , ΔU̇ Gi ), the equation for the output of the
automatic excitation controller for the ith generator view [3]:
ΔUAECi = k0𝛿i Δ𝛿i + k1𝛿i
dΔUGi
dΔ𝛿i
+ k0UGi ΔUGi + k1UGi
,
dt
dt
(16)
where i = 1–3.
The generalized matrices have the following dimensions:
A3 − 4n × 4n = 12 × 12 and B3 − 4n × n (k − m) = 4 * 3 × 3
(4–2) = 12 × 6 (since the number of generators is n = 3,
the number of adjustable parameters is k = 4 − (Δδi, ΔUGi,
Δ𝛿̇ i , ΔU̇ Gi ), which are used in the matrix A
­ 3 as parameters
of the state space.
[5]. In this case, the elements of the matrix A and/or their
combinations are explicitly determined, the change of which
by means of feedback allows one to provide a given arrangement of poles in a closed system [20].
Let us consider an effective method for solving the problem of the complete placement of the poles of the MIMO
system (7)–(9), which is based on the decomposition of the
model of the original system [5, 10]. Moreover, as will be
shown below, the method does not require the solution of
any special matrix equations (such as the Sylvester equation), has the same form for continuous and discrete cases
of specifying the model of the system, and has no restrictions on the algebraic and geometric multiplicity of the given
poles.
The following multi-level decomposition of the MIMO
system (1)–(2) with a pair of matrices (A, B), where A ∈ Rnxn,
B ∈ Rnxm [2, 5], is introduced.
Zero (initial) level
A0 = A, B0 = B,
(18)
First level
3 Basics of the decomposition method
A1 = B⊥ AB⊥T , B1 = B⊥ AB,
Control of systems (7)–(8) with the help of law (9) is a classical problem, and it is necessary to find a matrix K at which
the given requirements to the control process are provided
[1, 19].
In [10], the matrix of coefficients of the regulators of
excitation of synchronous generators in the general form is
obtained:
Δ𝛿
⎡ kEq11
⎢ Δ𝛿1
k
K = ⎢ Eq2
⎢…
⎢ Δ𝛿1
⎣ kEqn
Δ𝛿
Δ𝛿
Δs
Δs
… kEq1n kEq11 … kEq1n ⎤
Δ𝛿
Δs ⎥
Δs
… kEq2n kEq21 … kEq2n ⎥
,
…… … …… ⎥
⎥
Δ𝛿
Δs
Δs
… kEqnn kEqn1 … kEqnn ⎦
(17)
where kE j is the gain of the automatic regulator of excitation
(19)
f-th (intermediate) level
Af = B⊥f −1 Af −1 B⊥T
, Bf = B⊥f −1 Af −1 Bf −1 ,
f −1
(20)
L-th (final) level
AL = B⊥L−1 AL−1 B⊥T
, BL = B⊥L−1 AL−1 BL−1 ,
L−1
(21)
where symbol ⊥ denotes the so-called matrix zero divisors.
We assume that all matrices Bi in (18–21) are matrices of
full rank in columns.
In this case, the following assertion is true [2]: Suppose
that the MIMO system (7)–(8) is completely controllable,
and the matrix K ∈ Rrxm satisfies the formulas:
Zero (initial) level
qi
of the ith generator, with i, j = 1 − n. The size of the matrix
(17) is equal to 2n × 4n.
If i = j, then there is own regulation, and if i ≠ j, then there
is mutual control. Obviously, in a complex EPS, mutual
Δ𝛿
regulation is generally not used; kE j = 0, since i ≠ j. Here,
Δ𝛿
kE i
qi
(22)
First level
K1 = Φ1 B−1 − B−1 A1 , B−1 = B+1 − K2 B⊥1 ,
(23)
qi
is the gain of AEC of the ith generator by the deviation
Δs
of the absolute angle, and kE i is the slope of the generator,
qi
respectively.
The paper presents a method for stabilizing the state of a
large linear MIMO system, i.e., ensuring compliance with
the requirement for the MIMO system (7)–(8) by means of
law (9) in the sense of placing the poles (eigenvalues of the
matrix (A + BK) in the Cstab domain. The method is based
on a specific similarity transformation of the original system
13
K = K0 = Φ0 B−0 − B−0 A, B−0 = B+0 − K1 B⊥0 ,
fth (intermediate) level
Kf = Φf B−f − B−f Af , B−f = B+f − Kf +1 B⊥f ,
(24)
L-th (final) level
KL = ΦL B+L − B+L AL ,
(25)
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where B+ is the pseudo-inverse Moore–Penrose matrix, then
eig(A + BK) = ∪L+1
eig(Фi−1) the desired spectrum of a sysi=1
tem, B−i = B+i − Ki+1 B⊥i+1 [2, 9].
4 Results
Apply this method of placing the poles of the matrix of
intrinsic dynamics using the example of a three-generator
electrical system (Fig. 1).
Basic parameters: generator synchronous reactances
xdG-1 = 2; xdG-2 = 1.75; xdG-3 = 1.8; emf of the generators
EG-1 = 3.3048, EG-2 = 3.1501, EG-3 = 2.9961; the damping
coefficients ­PdG-1 = 0; ­PdG-2 = 0; ­PdG-3 = 0; initial absolute
angles of the generators δ10 = 65°; δ20 = 75°; δ30 = 60°;
transient time constants along the longitudinal axis with a
�
�
short-circuit stator winding Td G - 1 = 1 sec.; Td G - 2 = 1.5
�
sec.; Td G - 3 = 1 sec.; the inertia constant of the generators
TjG-1 = 7 s.; TjG-2 = 6 s.; TjG-3 = 5.5 s.; the time constants of
the exciters TeG-1 = 0.4 s.; TeG-2 = 0.4 s.; TeG-3 = 0.4 s.; the gain
factors of the automatic excitation controller (AEC) system
according to the deviation of the angle k0δG-1 = 10; k0δG-2 = 8;
k0δG-3 = 8; amplification factors of the AEC system by the
voltage deviation k0uG-1 = 50; k0uG-2 = 20; k0uG-3 = 50; gain
factors of the AEC system with respect to the first derivative
of the angle deviation k1δG-1 = 3; k1δG-2 = 0; k1δG-3 = 0; and
from the first derivative of the voltage deviation k1uG-1 = 0;
k1uG-2 = 0; k1uG-3 = 0.
The matrices of the own dynamics and control actions of
the electrical system A3 and B3, respectively, have the form:
51.1 ⎤
⎡ −𝜔11 𝜔12 𝜔13 ⎤ ⎡ −102.036 50.527
A21(3x3) = ⎢ 𝜔21 −𝜔22 𝜔23 ⎥ = ⎢ 59.415 −119.327 59.91 ⎥.
⎢
⎥ ⎢
⎥
65.35 −130.4 ⎦
⎣ 𝜔31 𝜔32 −𝜔33 ⎦ ⎣ 65.04
where
x31 = xd1 + xd3 +
𝜔12
xd1 xd3
,
xd2
xd1 xd2
,
xd3
x23 = xd2 + xd3 +
xd2 xd3
,
xd1
0 ⎤ ⎡0 0 0⎤
⎡ −Pd1 0
A22(3x3) = ⎢ 0 −Pd2 0 ⎥ = ⎢ 0 0 0 ⎥,
⎢
⎥
⎥ ⎢
0 −Pd3 ⎦ ⎣ 0 0 0 ⎦
⎣ 0
A23(3x3)
⎡ − 𝜕P1
⎢ 𝜕Eq1
0
=⎢
⎢
⎢
0
⎣
𝜔0
Tj1
0
0
𝜕P 𝜔
− 𝜕E 2 T 0
q2 j2
0
𝜕P3 𝜔0
− 𝜕E
0
q3 Tj3
⎤ ⎡
0
0 ⎤
⎥ ⎢ 1.676
⎥ = ⎢ 0 −10.87 0 ⎥⎥,
⎥ ⎢
0
9.32 ⎥⎦
⎥ ⎣ 0
⎦
As an example, define 𝜕E 1 using the above formula:
𝜕P
q1
E
𝜕P1
E
= 2 sin(𝛿10 − 𝛿20 ) + 3 sin(𝛿10 − 𝛿30 ),
𝜕Eq1
x12
x13
A33(3x3)
⎡ 1� 0 0 ⎤
⎥ ⎡1 0 0⎤
⎢ Td1
1
= ⎢ 0 T � 0 ⎥ = ⎢ 0 0.666 0 ⎥,
d2
⎥
⎥ ⎢
⎢
⎢ 0 0 1� ⎥ ⎣ 0 0 1 ⎦
Td3 ⎦
⎣
A34(3x3) = −A33(3x3)
k
A41(3x3)
As an example, define ω 11 and ω 12 using the above
formulas:
⎡ − 1�
0
0
⎢ Td1
1
0
= ⎢ 0 − T�
d2
⎢
⎢ 0
0 − 1�
Td3
⎣
⎤
0
0 ⎤
⎥ ⎡ −1
⎥ = ⎢ 0 −0.666 0 ⎥,
⎥
⎥ ⎢
0
−1 ⎦
⎥ ⎣ 0
⎦
⎡ T0𝛿1 0 0 ⎤ ⎡ 10 0 0 ⎤ ⎡ 25 0 0 ⎤
⎥
⎢ e1 k
⎥ ⎢ 0.4 8
0 ⎥ = ⎢ 0 20 0 ⎥,
= ⎢ 0 T0𝛿2 0 ⎥ = ⎢ 0 0.4
e2
⎢ 0 0 k0𝛿3 ⎥ ⎢ 0 0 8 ⎥ ⎢⎣ 0 0 20 ⎥⎦
⎣
⎣
0.4 ⎦
Te3 ⎦
k
𝜔
𝜔0
b11 = 0 (a12 cos 𝛽12 + a13 cos 𝛽13 )
Tj1
Tj1
[
]
𝜔0 E1 E2
EE
=
cos(𝛿10 − 𝛿20 ) + 1 3 cos(𝛿10 − 𝛿30 ) ,
Tj1 x12
x13
A42(3x3)
]
[
𝜔
𝜔 E E
𝜔
= 0 b12 = 0 a12 cos 𝛽12 = 0 1 2 cos(𝛿10 − 𝛿20 ) ,
Tj1
Tj1
Tj1 x12
A44(3x3)
𝜔11 =
x12 = xd1 + xd2 +
⎡ T1𝛿1 0 0 ⎤ ⎡ 3 0 0 ⎤ ⎡ 7.5 0 0 ⎤
⎥ ⎢ 0.4 0
⎥
⎢ e1 k
0 ⎥ = ⎢ 0 0 0 ⎥,
= ⎢ 0 T1𝛿2 0 ⎥ = ⎢ 0 0.4
e1
⎢ 0 0 k1𝛿3 ⎥ ⎢ 0 0 0 ⎥ ⎢⎣ 0 0 0 ⎥⎦
⎣
⎣
0.4 ⎦
Te3 ⎦
1
0
0
⎡ −T
⎢ e1
1
0
= ⎢ 0 −T
e2
⎢ 0
0 − T1
⎣
e3
⎤ ⎡ −2.5 0
0 ⎤
⎥ ⎢
⎥,
0
−2.5
0
=
⎥ ⎢
⎥
⎥ ⎣ 0
0 −2.5 ⎦
⎦
13
Author's personal copy
Electrical Engineering
0
0
0
1 00 0
0
0
0
0
0 ⎤
⎡
⎢
0
0
0
0 10 0
0
0
0
0
0 ⎥
⎢
0
0
0
0
0
1
0
0
0
0
0
0 ⎥⎥
⎢
51.1 0 0 0 1.676
0
0
0
0
0 ⎥
⎢ −102.036 50.527
⎢ 59.415 −119.327 59.91 0 0 0 0 −10.87 0
0
0
0 ⎥
⎢
⎥
65.04
65.35
−130.4
0
0
0
0
0
9.32
0
0
0
⎥,
A3 = ⎢
0
0
0
0 00 1
0
0 −1
0
0 ⎥
⎢
⎢
0
0
0
0 00 0
0.666 0
0 −0.666 0 ⎥
⎢
⎥
0
0
0
0 00 0
0
1
0
0
−1 ⎥
⎢
⎢
25
0
0
7.5 0 0 0
0
0 −2.5
0
0 ⎥
⎢
0
20
0
0 00 0
0
0
0
−2.5
0 ⎥
⎢
⎥
⎣
0
0
20
0 00 0
0
0
0
0
−2.5 ⎦
B41
⎡ 0
⎡ k0uG−1 ∕Te1 k1uG−1 ∕Te1 ⎤
⎥, B = ⎢⎢ k0uG−2
0
0
=⎢
⎥ 42 ⎢ Te2
⎢
0
0
⎦
⎣
⎣ 0
0
0
⎤
⎡
⎥,
0
0
B43 = ⎢
⎥
⎢
⎣ k0uG−3 ∕Te3 k1uG−3 ∕Te3 ⎦
⎡ 0
⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 0
⎢
0
B3 = ⎢
⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 125
⎢ 0
⎢
⎣ 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
50
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
125
0⎤
0⎥
0 ⎥⎥
0⎥
0⎥
⎥
0⎥
.
0⎥
0⎥
⎥
0⎥
0⎥
0⎥
⎥
0⎦
0 ⎤
k1uG−2 ⎥
,
Te2 ⎥
⎥
0 ⎦
The synthesized regulator (25) can be designed in such a
way that necessary requirements are provided in the dynamic
system: stability, damping of low-frequency oscillations, etc.
The spectrum of the matrix of the self-dynamics of the
three-generator EPS A3 with the selected parameters of
the regime and the system is equal to: 0.0016 ± 13.7172i;
0.0223 ± 12.7819i; − 2.0002 ± 1.3341i; − 3.1717; − 2.4712;
1.4194 ± 1.6989i; 0.894; 1.0297, and its 2D visualization is
shown in Fig. 2a.
As can be seen from the spectrum and the transient characteristic of the change in the deviation of the absolute load
angle of the first generator shown in Fig. 3a, the electrical
system under study for the given parameters of the regime
and the system is not stable, in view of the presence of
positive eigenvalues of the dynamics matrix, resulting in
undamped oscillations of the angle to the input of the system
of a unit pulse.
13
Define the regulator of the electrical system that provides
the pole shift to the following position: 1 ± 5i; − 4 ± 3i;
− 0.2; − 1; − 3.5; − 5.
Using expressions (18)–(25), we define the law and the
adjustment matrix.
Describe the desired eigenvalues using special matrix
constructions:
⎡ −7 + 15i
⎢
0
⎢
0
𝛷0 = ⎢
0
⎢
⎢
0
⎢
0
⎣
⎡ −2.5 − 7i
⎢
0
⎢
0
𝛷1 = ⎢
0
⎢
⎢
0
⎢
0
⎣
⎡ −1 − 5i
⎢ 0
⎢
0
𝛷2 = ⎢
⎢ 0
⎢ 0
⎢
⎣ 0
⎡ −4 + 3i
⎢ 0
⎢
0
𝛷3 = ⎢
⎢ 0
⎢ 0
⎢
⎣ 0
0
0
0
0
0 −7 + 15i
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 −2.5 + 7i
0
0
0 0 0
0
0 0 0
0
0 −0.2 0
0
0 0 0
0
0 0 0 −1 + 5i
0 0 0
0
0
0
−1
0
0
0
0 0 0⎤
0 0 0⎥
⎥
0 0 0⎥
,
0 0 0⎥
0 −3.5 0 ⎥
⎥
0 0 0⎦
0
0
0
0
0 −4 − 3i
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−5
0
Zero level of decomposition.
0⎤
0⎥
⎥
0⎥
,
0⎥
0⎥
⎥
0⎦
0⎤
0⎥
⎥
0⎥
,
0⎥
0⎥
⎥
0⎦
0⎤
0⎥
⎥
0⎥
,
0⎥
0⎥
⎥
0⎦
Author's personal copy
Electrical Engineering
(a)
(b)
15
10
10
5
5
Im
Im
15
0
0
-5
-5
-10
-10
-15
-3.5
-3
-2.5
-2
-1.5
-1
Re
-0.5
0
0.5
1
1.5
-15
-7
-6
-5
-4
Re
-3
-2
-1
0
Fig. 2 The location of the eigenvalues of the system under study on the complex axis
Fig. 3 Transient characteristics of the deviation of the absolute angle of the first generator of the three-generator system
Define the left divisor of matrix B using the null function
in the MATLAB program:
⎡ 0
⎢ 0
⎢ 0
⎢
⎢ 0
B⊥0 = ⎢ 0
⎢
⎢ 0
⎢ −1
⎢ 0
⎢
⎣ 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0⎤
0⎥
0 ⎥⎥
0⎥
0 ⎥.
⎥
0⎥
0⎥
0⎥
⎥
0⎦
⎡0
⎢0
⎢
0
B+0 = ⎢
⎢0
⎢0
⎢
⎣0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.008 0
0 ⎤
0 0
0
0 ⎥
⎥
0 0 0.02 0 ⎥
.
0 0
0
0 ⎥
⎥
0 0
0 0.008
⎥
0 0
0
0 ⎦
The first level of decomposition of the matrix of
coefficients.
Define the Moore–Penrose pseudo-inverse matrix using
the pinv function in the MATLAB program:
13
Author's personal copy
Electrical Engineering
0
⎡
⎢ 50.527
⎢ 65.35
⎢
0
⎢
0
A1 = ⎢
⎢
0
⎢
⎢
0
⎢
0
⎢
⎣ 119.327
⎡ 0
⎢ 0
⎢ 0
⎢
⎢ −125
B1 = ⎢ 0
⎢
⎢ 0
⎢ 0
⎢ 0
⎢
⎣ 0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0 −1 ⎤
0 1.676 0
0 102.036 −51.1 0 ⎥
0
0
0 9.32 −65.04 130.4 0 ⎥⎥
0
1
0
0
0
0
0 ⎥
0
0 0.666 0
0
0
0 ⎥,
⎥
0
0
0
1
0
0
0 ⎥
0
0
0
0
0
0
0 ⎥
−1 0
0
0
0
0
0 ⎥
⎥
0
0 10.87 0 59.415 59.91 0 ⎦
0 0 0 0 0⎤
0 0 0 0 0⎥
0 0 0 0 0 ⎥⎥
0 0 0 0 0⎥
0 −33.3 0 0 0 ⎥.
⎥
0 0 0 −125 0 ⎥
0 0 0 0 0⎥
0 0 0 0 0⎥
⎥
0 0 0 0 0⎦
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0⎤
0⎥
⎥
0⎥
.
0⎥
0⎥
⎥
1⎦
⎡ −209.5
⎢ 0
⎢
0
B2 = ⎢
⎢ 0
⎢ 0
⎢
⎣ 0
0 0
0 0
0 0
0 0
0 0
0 −362
0 0 0⎤
0 0 0⎥
⎥
0 −1165 0 ⎥
.
0 0 0⎥
0 0 0⎥
⎥
0 0 0⎦
Matrix the third level of decomposition:
⎡0 0 0⎤
A3 = ⎢ 0 0 0 ⎥,
⎥
⎢
⎣0 0 0⎦
The third level of decomposition of the regulator
matrix.
Define Moore–Penrose pseudo-inverse matrix
0.0048
0
⎡ 0
⎤
⎢ 0
⎥
0
0
⎢
⎥
0.0028
0
0
⎥.
B+3 = ⎢
0
0
⎢ 0
⎥
⎢ 0
0
−0.0009 ⎥
⎢
⎥
0
0
⎣ 0
⎦
Third level of decomposition of the regulator matrix:
Matrix second-level decomposition:
⎡ 0 50.527 0 102.036 −51.1
⎢ 0
0
0
0
0
⎢
0
65.35
0
−65.04
130.4
A2 = ⎢
0
0
0
0
⎢ −1
⎢ 0
0
−1
0
0
⎢
⎣ 0 119.327 0 59.415 59.91
⎡0 1 0 0 0 0⎤
B⊥3 = ⎢ 0 0 0 1 0 0 ⎥.
⎥
⎢
⎣ 0 0 0 0 −1 0 ⎦
⎡ 0 0 362 0 0 0 ⎤
B3 = ⎢ 209.5 0 0 0 0 0 ⎥.
⎢
⎥
⎣ 0 0 0 0 −1165 0 ⎦
The second level of decomposition of the matrix of
coefficients.
Determine the left divider of matrix ­B1:
⎡0
⎢1
⎢
0
B⊥2 = ⎢
⎢0
⎢0
⎢
⎣0
The third level of decomposition of the matrix of
coefficients:
Determine the left divider of matrix B2:
0 ⎤
−1 ⎥
⎥
0 ⎥
,
0 ⎥
⎥
0
⎥
0 ⎦
0
−0.0191 + 0.0143i
0 ⎤
⎡
⎢
0
0
0 ⎥
⎢
⎥
−0.0111 − 0.0083i
0
0 ⎥
.
K3 = ⎢
0
0
0 ⎥
⎢
⎢
0
0
0.0043 ⎥
⎢
⎥
0
0
0 ⎦
⎣
The second level of the matrix decomposition of the
regulator.
Define the Moore–Penrose pseudo-inverse matrix:
⎡ −0.0048
⎢
0
⎢
0
+
B2 = ⎢
0
⎢
⎢
0
⎢
0
⎣
0
0
0
0
0
0
0
0
0
0
0
0
0 −0.0009 0
0
0
0
Define the matrix B−2 :
13
0
0
⎤
⎥
0
0
⎥
0 −0.0028 ⎥
.
0
0
⎥
⎥
0
0
⎥
0
0
⎦
Author's personal copy
Electrical Engineering
0
0
0.0191 + 0.0143i
0
0
⎡ −0.0048
⎤
⎢
⎥
0
0
0
0
0
0
⎢
⎥
0
0.011 + 0.0083i
0
0
0
−0.0028 ⎥
.
B−2 = ⎢
0
0
0
0
0
0
⎢
⎥
⎢
⎥
0
0
−0.0009
0
0.0043
0
⎢
⎥
0
0
0
0
0
0
⎣
⎦
Second level of decomposition of the regulator matrix:
0.2412
0
0.3964 − 0.0811i −0.2439
0
⎡ 0.0239 + 0.0095i
⎤
⎢
⎥
0
0
0
0
0
0
⎢
⎥
0
0.3186 − 0.0083i
0
0.1641
0.1655 0.0138 + 0.0083i ⎥
.
K2 = ⎢
0
0
0
0
0
0
⎢
⎥
⎢
⎥
0
0.0561
0.0073
−0.0558
0.0969
0
⎢
⎥
0
0
0
0
0
0
⎣
⎦
The first level of decomposition of the controller matrix.
Define the Moore–Penrose pseudo-inverse matrix:
⎡0
⎢0
⎢
0
B+1 = ⎢
⎢0
⎢0
⎢
⎣0
0
0
0
0
0
0
0 −0.008 0
0
0
0
0
0
0
0
0
0
−0.03
0
0
0
0
0
0
0
0
0
0 −0.008 0
0
0
0
0
0
Define the matrix B−1 :
0
0
0
0
0
0
0⎤
0⎥
⎥
0⎥
.
0⎥
0⎥
⎥
0⎦
−0.2412
−0.0239 − 0.0095i
0
−0.008 0 .
⎡
⎢
0
0
0
0
0 .
⎢
−0.3186
+
0.0083i
0
0
0
−0.03
.
B−1 = ⎢
,
0
0
0
0
0 .
⎢
⎢
−0.0561
0
−0.0073
0
0 .
⎢
0
0
0
0
0 .
⎣
.
0
−0.3964 + 0.0811i 0.2439
0
⎤
⎥
.
0
0
0
0
⎥
.
0
−0.1641
−0.1655 −0.0138 − 0.0083i ⎥
.
.
0
0
0
0
⎥
⎥
. −0.008
0.0558
−0.0969
0
⎥
.
0
0
0
0
⎦
First level of decomposition of the regulator matrix:
0.244
0.068 + 0.072i
0
⎡ 1.8088 + 2.17i −0.4 + 0.27i
⎢
0
0
0
0
0
⎢
1.712 + 0.987i −0.164
−0.1655
0
0.176 + 0.09i
K1 = ⎢
0
0
0
0
0
⎢
⎢ 0.5329 − 0.28i
0.0558
−0.089 − 0.036i
0
0
⎢
0
0
0
0
0
⎣
.
.
.
,
.
.
.
13
Author's personal copy
Electrical Engineering
.
0
4 + 3.54i −1.83 − 2.19i
−0.24
⎤
⎥
.
0
0
0
0
⎥
.
0
0.85 + 0.492i 0.86 + 0.5i −0.31 + 0.01 ⎥
.
.
0
0
0
0
⎥
⎥
. 0.084 − 0.04i −0.53 + 0.28i 1.04 − 0.48i
−0.056
⎥
.
0
0
0
0
⎦
Zero level of decomposition of the regulator matrix:
Define the regulator matrix by formula (22):
proposed method of moving the poles of a model of an
electrical system of arbitrary complexity can be used for
operational control of EPS modes, with the choice of the
appropriate AEC law.
Thus, modern matrix methods for studying dynamical
systems and their new constructions (matrix zeros, canonization, etc.) allow us to control the transient modes of complex
⎡ −9.98 + 7.33i 12 + 1.18i 7.75 + 1.82i −2.79 + 4.41i 3.49 − 1.44i .
⎢
.
0
0
0
0
0
⎢
26.16 − 16.8i −60 + 33.77i 33.54 − 17i
−2 − 3i
4.07 + 5.65i .
K=⎢
,
.
0
0
0
0
0
⎢
⎢ −8.82 − 6.78i −10.36 − 6.8i 18.5 + 13.3i 0.67 − 0.67i 0.67 − 0.67i .
⎢
.
0
0
0
0
0
⎣
. 3.53 − 1.46i 0.95 + 0.01i
−2.62
2.27
.
0
0
0
0
. −2 − 2.97i
−0.27
−3.43 + 3.44i
−1.54
.
0
0
0
0
. −1.52 + 1i
0.09
−0.61
−0.82 − 1.07i
.
0
0
0
0
.
.
.
,
.
.
.
. −0.1 + 0.048i
0
0
⎤
⎥
.
0
0
0
⎥
.
0
−0.2 − 0.36i
0
⎥,
.
0
0
0
⎥
.
0
0
−0.08 + 0.096i ⎥
⎥
.
0
0
0
⎦
providing the pole shift of the matrix A + BK to the following
position: − 7 ± 15i; − 2.5 ± 7i; − 1 ± 5i; − 4 ± 3i; − 0.2; − 1;
− 3.5; − 5 ⊂ Cstab.
The spectrum of the desired poles of the system is shown
in Fig. 2b. Based on the eigenvalues of the dynamics matrix
obtained above and the transition characteristic shown in
Fig. 3b, we can conclude the problem of stabilizing system
(7) obtained by a pair of matrices A3 and B3, control (9), and
the controller matrix (25). The eigenvalues of the matrix
of the system have acquired the desired values, and rather
rapidly damped oscillations of the angle (Fig. 3b) indicate a
sharp improvement in the damping properties of the electrical system under investigation.
5 Conclusions
It should be noted that the model of the electrical system,
represented in the form of an A
­ Σ matrix, is effective in the
study of complex electrical systems, since it is simple and
computationally advantageous, consists, as a rule, of blocks
of zero and unit matrices, and, accordingly, is rarefied. The
13
EPS by moving the poles and to change the quality of the
dynamics of the systems under study.
References
1. Anderson PM, Fouad AA (2002) Power system control and stability, 2nd edn. Wiley, Hoboken
2. Misrikhanov MSh, Ryabchenko VN (2011) Pole placement for
controlling a large scale power system. Autom Remote Control
10:129–153
3. Bukov VN (2006) Vlojeniye system. Analiticheskiy podhod k
analizu I sintezu matrichnix system [Embedding systems. Analytical approach to the analysis and synthesis of matrix systems].F.
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