Download VOLTAGE STABILITY OF POWER SUBSYSTEM

Voltage Stability of Power Subsystem
VOLTAGE STABILITY OF POWER SUBSYSTEM
Ryszard Zajczyk / Gdańsk University of Technology
1. REACTIVE POWER OVERLOAD
1.1. Turbo-generator set characteristics
In order to describe unsteady states during overloads it is necessary to know the properties of the generating unit, including the synchronous generator, not only at near-rated voltage, but also at voltages much lower
that occur in significant overloads. An example of generating units consisting of a condensating steam turbine
with turbine governor, and asynchronous generator with generator controller is presented in Fig. 1.
�
���������������
Pgz
Z
 WP
WP
PR
K
gz
WP
Pg
 SP
Z
TB
SP
SP
PU
RT
g
NP
PI
TW
GS
Ug
Ig
U gz
RG
UW
If
z
SK
PIW
PW
Fig. 1.
Block diagram of a high-power
generating unit [10]
A high-power condensating turbine consists of high-, medium- and low-pressure parts (WP, SP, NP) and an
inter-stage steam superheater (PR). The turbine’s quantitative control is accomplished by changing the steam
flow through its individual parts by opening adjustment of the high-pressure (Zwp) and low-pressure (Zsp) control
valves. The urbine governor consists of a power controller (RP), rotational speed governor (RO), converter (P/
H), and control valves (ZR). A block diagram of turbo-generator set with control system is presented in Fig. 2.
Abstract
The paper discusses the issue of power system
reactive power overload and presents a methodology of
power subsystem voltage stability analysis. In order to
confirm the correctness of the adopted assumptions simulation calculations have been made of the subsystem’s
voltage stability. The study analyses cases of overloading
individual nodes and entire subsystems with active power,
reactive power and power with initial tg� retained. The
results have confirmed the correctness of the assumptions
adopted with regard to the manner of stability boundary
in model research.
63
Ryszard Zajczyk / Gdańsk University of Technology
64
�
�
�
� ��
�
RP
���
��
RO
�
P/H
��
SM
SL
�� ��
����������������
ZR
T
G
��
�������������������
Fig. 2. Turbo-generator set control system RP – active power controller, RO – rotational speed governor, P/H – mechanical/ hydraulic or
electro-hydraulic converter, ZR – control valves, SM – valve servo-motor, SL – valve actuator, T – turbine, G – generator [10].
High-power synchronous generators (GS) are provided with static thyristor excitation systems (TW, PW)
or machine excitation systems and multi-parameter generator controllers (RG). In either excitation system variant the multi-parameter generator controller consists of the main voltage control circuit, control system limiters, and additional elements. A block diagram of multi-parameter generator controller is presented in Fig. 3.
� ��
� ��� ���
� ��
�
��
���
���
�
�� �
–
� ��
� �������
�� �
��
� ��
–
�
�
�� ���
–
�
��
� �� ���
�
�
� � ���
�
�
��
���
��
��
�
� � � ���
� � �� �
�
�
� ���
��
� ����
��
���
�
� � � ���
�
� ���
�
��
� � �� �
� ����
����
�
� ���
� � �����
�
� ����
��
��
��
� ���� ��� ��� ��
��
�� �
� ���
�
�
� ������
Fig. 3. Structural diagram of multi-parameter generator controller [10] TRN – main voltage control circuit, UKP – current compensation
system, PSS – system stabiliser, OPS – stator current limiter, OPW – rotor current limiter, OPPW – rotor ceiling current limiter, OKM
– power angle limiter.
The generator is controlled by a controller, commonly called a voltage controller, that maintains a set
voltage. At large overload the controller fully adjusts the excitation system. As a result the excitation voltage,
and – in the steady state – the excitation current, reach their maxima. In this state U = f(Ig) characteristic is not
controlled by the controller anymore.
From the voltage stability viewpoint, characteristics of some controller components are relevant, such
as limiters of stator current, rotor current, and excitation ceiling current. Their shape (time characteristic) may
Voltage Stability of Power Subsystem
65
govern the course of voltage collapse related phenomena. An example I = f(t) time characteristic provided by
excitation ceiling current limiter and excitation maximum current limiter is presented in Fig. 4.
���
� ����
� ������� ����� ���
� ����������������
���
�
���
�����
�����
Fig. 4.
Current-time characteristic provided by limiters of excitation
ceiling and maximum current, kp – ceiling factor (1,6÷2), Ifn
– rated excitation current [10]
����
��
Based on measured voltage and current, the RG controller maintains the generator terminal voltage according to the formula:
�U g  Ugzo  I P Rk  I Q X k
where Ugz0 – set voltage, Rk , Xk – current compensation impedance.
A substitute diagram of a generator with controller is presented in Fig. 5a. The aforementioned characteristic refers to the range from idle run up to the load (in an unsteady state), at which excitation voltage Uf
reaches its maximum Ufmax. If, once the maximum excitation voltage has been achieved, the generator voltage
is still declining as an overvoltage result, the controller’s action will be ineffective, since it is not able to raise
the excitation voltage up to a level necessary to maintain the set generator voltage.
�U g  Uf max  I Q X d
A substitute diagram for this state of synchronous generator is presented in Fig. 5b.
a)
�
� ������ �
� ���
b)
� ����� ������ �
�� �
��
� ����
� �� ��� �� ���� �
��
Fig. 5. Substitute diagram of generator with controller; a) adequate state, b) at large overload for Uf = Ufmax [5,
6]
Superposition of these two characteristics produces a characteristic covering the entire range of generator voltage changes corresponding loads from the idle state up to the maximum generator load (Fig. 6).
Ryszard Zajczyk / Gdańsk University of Technology
66
�
��
� �� ��� ����
� �� ���
� ���
� �����
��
��
Fig. 6.
Extarbnel characteristic of generator with voltage controller
within the linear range (for Uf< Ufmax ) and for the maximum
excitation voltage (Uf =Ufmax) [5, 6]
1.2. Receiver characteristics
Like active power, reactive power absorbed by receivers is a function of voltage U and frequency f :
�Qo  F (U , f )
In a steady state at f = const, function Qo =F ( U , f ) has the course presented in Fig. 7
�
U
�
Un
1
2
3
Fig. 7.
Changes in received reactive power at frequency changes and at
dQ
U = const. �tg�  o
dU
Qo
Qon
1, 2, 3 – various characteristics at significant voltage reductions
[5, 6]
Relations shown in the figure may be useful also for qualitative, and – in approximation – for quantitative interpretation of unsteady states at a system, subsystem, or island overload. The quicker overload related
voltage changes, the more the actual characteristic deviates from that presented in Fig. 7. Such a discrepancy
is caused by unsteady electromagnetic states in electric motors and by the impact of drive systems’ rotating
masses.
1.3. Load level impact
To estimate voltage and reactive power changes at overload it is reasonable to explore the relationship
between these quantities:
� dU
 F (Qg , Qo )
dt
If generated and absorbed reactive powers do not balance, a stable or unstable unsteady process occurs.
In the case of unstable process – typically aperodic in general – no new steady process may be achieved and a
so-called voltage collapse occurs.
Voltage Stability of Power Subsystem
�
n
m
i 1
n
i 1
m
 Qgi   Qoi  0 then
�dU
 0 and U = const.
dt
�
If  Q gi   Qoi  0 then �dU  0 and U is increasing
i 1
i 1
dt
n
m
�
�dU
 0 and U is decreasing.
If  Qgi   Qoi  0 then
dt
i 1
i 1
If
If the overload is small, then the generator voltage control maintains the voltage near its rated value, causing, however, an excess of the admissible generator current. This excess is not – in the initial period – liquidated
by stator current limiters operating with a deliberately set time delay. A new quasi-steady state is accomplished.
�
��
��
� �� ��� ����
� �� ���
� ���
��
� �����
�����������
��� ����������
���������
� �� ��� ��
� �����
��������
Fig. 8.
Interpretation of small reactive power overloads
1 – quasi-steady state for Xk = 0; 2 – quasi-steady state
for Xk> 0; O – receiver characteristic [5, 6]
��
��
However after some time the limiters proceed to limit the current overload and cause a change in the external
generator characteristic shown in Fig. 8.
The limiter’s action may render the new steady state accomplishment impossible. A voltage collapse
develops and the subsystem can not be defended, since the existing unload automatics do not protect against
reactive power overload.
The figure presents two points of quasi-steady operation until the current is limited by the limiters. It is
also shown that as a result of the limiter’s operation the reactive power demand can not be covered and a voltage collapse develops.
At large overload the situation shown in Fig. 9 occurs.
It follows from comparison of the generator and receiver characterises that no reactive balance power
is available in the system. This results in a voltage collapse that can not be controlled without disconnection of
some receivers. Such disconnection moves “O” characteristics to the left thus enabling reactive power balancing.
�
��
��
� ���
� �� ���
��
� �� ��� ����
� �� ��� ��
� �����
�����������
��� ����������
���������
� �����
��������
��
��
Fig. 9. Interpretation of large reactive power overloads
Symbols as in the previous figures [5, 6].
67
Ryszard Zajczyk / Gdańsk University of Technology
68
2. STATISTICAL ANALYSIS OF VOLTAGE STABILITY
Voltage stability is determined for a power system’s receiving nodes. Statistical analysis of voltage stability is
based on voltage-current equations determined for any power node [10]. A system diagram is presented in Fig. 10.
�
�
�
�
�
�
�
�
���
Fig. 10.
Diagram of current distribution in k node of power
system, Ukf – k node voltage, Ulf – l node voltage,
Jkl – current between k and l nodes, Jk – current
received in k node, Jkg – generator current in k node
Zkl , Ykl – impedance and admittance of link between
k and l nodes, Yko , Ylo – admittance of Lateran branches in k and l nodes, Yk – substitute admittance of
receiver in k node [10]
��
���
���
��� �����
���
� ��
���
���
�
�
�
��
� ��
Active and reactive powers absorbed in k node are described with the following formulas:
n
n
�
Pk  U k2  Gkl   U kU l (Gkl cos φ kl  Bkl sin φ kl )
l 1
l k
l 1
l k
n
n
l 1
l k
l 1
l k
Qk  U k2  Bkl   U kU l ( Bkl cos φ kl  Gkl sin φ kl )
This formula describes active and reactive powers for all receiving nodes of the subject power system for
steady and transient states alike.
Analysing power changes in the vicinity of the steady operation point defined by (θO, UO) parameters for
all nodes, the following active and reactive power changes are determined:
Δ� P(� ,U )  P(θ ,U )  P(θ o ,U o )
Δ Q (� ,U )  Q(� ,U )  Q(� o ,U o )
System linearization and small deviations analysis of power changes ∆P and ∆Q in the function of changes
of voltage U and angle between vectors θ produces the following formulas:
� P (θ ,...,θ , U ,...U )   P1
1 1
n
1
n
θ 1

  ...
.....

  P
 Pn (θ1 ,...,θ n , U1 ,...U n )   θn1

   Q1
 Q1 (θ1 ,...,θ n , U1 ,...U n )   θ1

  ...
.....

  Qn

Q
(
θ
,...,
θ
,
U
,...
U
)
  θ1
n
1
n 
 n 1
...
P1
θ n
P1
U 1
...
...
Pn
θ n
Q1
θ n
Pn
U 1
Q1
U 1
...
Qn
θ n
Qn
U 1
i.e.:
� ΔP   P
�
 ΔQ    Q
   �
P
U
Q
U
  Δθ   J P�
 x    J
 ΔU   Q�
J PU   ΔP 
x
J QU   ΔQ
  θ1 

...   ... 


P
... Unn   θ n 
x

Q
... U 1n   U1 
...   ... 
 

Q
... U nn  U n 
...
P1
U n
Voltage Stability of Power Subsystem
Under the assumption that only reactive power changes in the system, ∆P=0, the above formula transforms to:
� 0   J Pθ
Q   J
   Qθ
J PU   θ 
x
J QU  U 
Upon transformation and elimination of angle θ, the following relationship is formulated between reactive
power changes ∆Q and voltage changes ∆U:


�Δ Q  J QU  J Qθ J P1θ J PU x U  JR U
�J R1 matrix entries on the main diagonal determine voltage sensitivity of power system’s receiving nodes
For any node k the voltage sensitivity may be determined, as well as the relationship with voltage stability
in the node:
� U
 J R1
ΔQ

�diag J
�diag J

(k ) 0
(k ) 0
�diag J R1 (k )  0 voltage-stable node
1
R
stability boundary
1
R
voltage-unstable node
3. DYNAMIC ANALYSIS OF VOLTAGE STABILITY
Dynamic analysis of voltage stability involves examination of system response to a preset input function.
System components described with differential and algebraic equations account for the basis for unsteady state
calculations. System model includes:
• Model of the transmission system described with the following equations:
x = f(x, U)
I (x, U) = YU
• Models of generating mode components such as:
• synchronous generators
• multi-parameter generator controllers
• turbines
• turbine governors
A generating node diagram with modelled components indicated is presented on Fig. 1.
For the purpose of stability analysis the system is described with a set of differential equations linearised
in the surroundings of the operation point, for which voltage stabiliity is examined. Linearised object’s generic
form may be described with:
state equation: X(t) = AX(t) + BU(t)
output equation: Y(t) = CX(t) + DU(t)
where X(t) – state variable vector, U(t) – input Signac vector, Y(t) – output Signac vector, A – state matrix,
B – input matrix, C – output matrix, D – matrix tying input signals directly influencing output.
69
Ryszard Zajczyk / Gdańsk University of Technology
70
The synchronous generator is described with the following formulas:
�I (t )  A 1B I (t )  A 1C U (t )
g
g
g g
g
g
g
state equation:
output equation: �Wg (t )  D g I g (t )
where: �I g (t )  [I d (t ), I q (t ), I f (t ), I kd (t ), I kq (t ), σ (t ), δ (t )]
T
�U Tg (t )  [U s (t ), U f (t ), M t (t )]
�WgT (t )  [U g (t ), Pg (t )]
– state variable vector
– input value vector
– output value vector
The excitation system and generator controller are described with the following formulas:
state equation:
output equation:
where:
vector, �U TRG (t )
�Y RU (t )
A

   RU
 Y SS (t )   O


1
O 
B
x  RU
A SS 
 O
B SS  RU  YRU (t )  A RU
x

B SS   YSS (t )   O
1
O 
C 
x  RU  x URG (t)
A SS 
 C SS 
T
�U f (t )  DTRU YRU (t )  EUW
U RG (t )
T
T
�YRU
(t ) – voltage control circuit state variable vector, �YSS (t ) – system stabiliser state variable
– input value vector, �U f (t ) – output value – excitation voltage change.
The condensing turbine is described with the following formulas:

1
1
state equation: �DT (t )  ATK
BTK DT (t )  ATK
CTK UTK (t )
output equation:
�M t (t )  ETT DT (t )
T
where: �DT (t ) – turbine state variable vector, �U TTK (t ) – input value vector, �M t (t ) output value – generator driving torque change
The turbine governor is described with the following formulas:
state equation:
� Y RE (t ) 
A

   RE
Y HY (t )  O


1
O 
B
x  RE
A HY 
B EH
O   YRE (t )   A RE
x

B HY  YHY (t )  O
1
O 
C 
x  RE  x URE (t )
A HY 
 O 
output equation: �Wt (t )  D RT YHY (t )
T
where �YRE
(t )
T
– state variable vector, �YHY (t ) – Valle hydraulic system state variable vector,
�U TRE (t ) – input value vector, �WtT (t )
– output value – control valve opening change.
The other matrices are described in the references cited [10].
After consideration of mutual relations between input and output values of individual objects, a generating unit’s description is obtained in the form of state equations.
Voltage Stability of Power Subsystem
4. VOLTAGE STABILITY CALCULATION METHODOLOGY
4.1. Voltage sensitivity method
Voltage sensitivity may be determined for each node of power system as the ratio of voltage change to
reactive power supplied to the node.
�
U k
wkQ 
Qk
wkP 
U k
Pk
n
n
i 1
ik
i 1
ik
n
n
n
n
i 1
ik
i 1
ik
i 1
ik
i 1
ik
U k   U ki [h  h]  U ki [h]
Qk   Qki [h  h]  Qki [h] ; Pk   Pki [h  h]  Pki [h] ; tgφ k 
Qk
Pk
Active and reactive power changes shall be determined in successful steps of determination of power distribution and voltage levels in power system in the process of dynamic simulation. Where factors are positive
(positive voltage change increases voltage), the system is voltage-stable.
Application of this method is proposed for preliminary calculation of voltage stability in a power system.
This will enable determining for a fixed system load (uniquely determined system state) the system nodes exposed to voltage stability loss.
In this method constraints may be considered of reactive power generation by synchronous generators
• constraint of absorber reactive power Qp (Qpoj)
• constraint of generated reactive Power Qg (Qind)
and constraints of transformer and autotransformer voltage ratios through consideration of the following
boundary values:
• minimum transformer voltage ratio min
• maximum transformer voltage ratio max
and related tap-changer positions.
4.2. Own values method
Where boundary characteristics (load impact on voltage stability boundary) have to be procured, analysis
of a linearised system’s own values is necessary.
Reactive power increment’s dependency on voltage changes in nodes is described by the following formula:
�U  J 1
R Q
The Jacobi matrix that appears in the formula may be determined as follows: �J R1  M� 1N , where:
M – matrtix of right-side own vectors,
N – matrtix of left-side own vectors,
Λ – diagonal matrix of own values.
1
After transformation equation 2 assumes the following form: �U  M� NQ or
n
�
mn
U   i i Q
�i
i 1
where: mi – means right-side vector i, ni – left-side vector i, λi – own value i. Own value λi and corresponding vectors: right-side mi and left-side ni make up the system’s mode i.
71
Ryszard Zajczyk / Gdańsk University of Technology
72
�m 1  n
�u  nU
and q  nQ where u and q – voltage and reactive
nU  � nQ
�
1
power mode vectors �u  Λ 1q . Formula for mode i: ui  qi . Own value λi reflects the mode’s voltage
�i
Entering to the equations formulas
1
stability (λi>0 – voltage-stable system) where nik – element k of vector ni. Ultimately for node k of power system the relation between the voltage’s reactive power derivative and own values:
�U k
Qk
n

i 1
mki nik
�i
5. SIMULATION EXAMINATION
Verification of the proposed method of voltage stability determination in receiving nodes of power system
– 110 kV grid – for the following three specific grid nodes:
• node A – typical receiving node inside 100 kV grid in rural areas,
• node B – typical receiving node inside 100 kV grid in urban agglomeration,
• node C – receiving node of 110 kV grid situated near a generating node that supplies the 110 kV grid
(generators connected to 400 kV bus bars, 110 kV grid supplied through 400/110 kV transformer with under-load voltage ratio control).
The examined sub-grid contained over 80 110 kV receiving nodes and was supplied in a few points from 220
and 400 kV grids through transformers with controlled voltage ratios. A grid diagram is presented in Fig. 11.
Voltage stability boundary in the subject sub-area was determined for the following cases:
• concurrent change of absorbed reactive and active powers with tgφ maintained in the single subject
node of 110 kV grid. Example results for nodes A, B, and C are presented in Fig. 12
• concurrent change of absorbed reactive and active powers with tgφ maintained in the subject subsystem – all nodes of 110 kV grid. Example results for nodes A, B, and C are presented in Fig. 13
• Concurrent change of absorbed reactive power in the subject subsystem - all nodes of 110 kV grid.
Example results for nodes A, B, and C are presented in Fig. 14.
The PLANS programme was used for the calculations.
�
220 kV
400 kV
�
PC ,Q C
�
110 kV
������
�������������
�
PB,QB
400 kV
������
�����
�
PA ,QA
Fig. 11.
Diagram of the subject 110 kV power
subsystem supplied from 220 and 400 kV
grids in a number of points
Voltage Stability of Power Subsystem
a)
73
b)
�
�
120
120
115
115
110
110
105
105
100
U [kV]
U [kV]
100
C
95
A
90
C
95
A
90
B
85
85
80
80
75
75
70
B
70
0
100
200
300
400
500
600
700
800
0
20
40
60
P [MW]
80
100
120
140
160
180
Q [Mvar]
Fig. 12. Voltage variability in selected nodes of 110 kV grid in the function of absorbed active power a) and reactive power b). Nodes individually loaded (each node independently).
a)
b)
�
�
120
115
110
110
105
105
C
100
B
95
90
85
85
80
80
75
75
70
5
10
15
20
25
30
A
95
90
0
C
100
A
U [kV]
U [kV]
120
115
B
70
35
0
P [MW]
2
4
6
8
10
12
Q [Mvar]
Fig. 13. Voltage variability in selected nodes of 110 kV grid in the function of absorbed active power a) and reactive power b). Concurrent
loading of receiving nodes of the subject subsystem (at the same power factor).
�
120
115
110
105
C
U [kV]
100
A
95
B
90
85
80
75
70
0
1
2
3
4
5
6
7
8
9
10
Q [Mvar]
Fig. 14. Voltage variability in selected nodes of 110 kV grid in the function of absorbed reactive power. Concurrent loading of receiving nodes
of the subject subsystem (only reactive power loading).
74
Ryszard Zajczyk / Gdańsk University of Technology
6. CONCLUSIONS
These analyses and simulation research enable formulating the following conclusions with regard to the
feasibility of various methods of voltage stability calculation in power system’s receiving modes:
1. In global calculations for an entire power system the voltage stability factors method should be applied.
Multiple repetition of simulation calculations at variable absorbed power (Q =var) will enable determination of U-Q characteristics for the nodes.
2. Subsystems should be loaded, and not individual nodes. Loading with active and reactive powers at
receiver’s set tgφ.
3. In voltage stability analyses the constraints should be considered resulting from the admissible area of
generator operation state, the impact of limiters of stator and rotor currents and of stator ceiling current, power angle in generator controller, and the constraints resulting from the impact of generating
and transmitting nodes’ group control systems [11].
4. For voltage stability examination of separated subsystems the own values method for linearised systems
may be applied. Ultimately it should be attempted to apply this method for the whole power system.
5. The only effective method of avoiding voltage collapse in cases of reactive power overloading is entering voltage elements to automatic unloading systems. The voltage element, with measurement of
time derivative, provides credible reactive power overload information [5, 6, 11].
REFERENCES
1. IEEE Guide for Synchronous Generator Modelling Practices in Stability Analyses. IEEE Std 1110-1991 (American
National Standard ANSI).
2. EEE Standard 421.5: IEEE Recommended Practice for Excitation System Models for Power System Stability Studies.
August 1992.
3. Kundur P. , Power System Stability and Control. McGraw-Hill, Inc. 1994.
4. Leon O.Chua, Pen-Min Lin, Komputerowa analiza układów elektronicznych. Algorytmy i metody obliczeniowe [Computer analysis of power systems. Algorithms and calculation methods], WNT, Warsaw 1981.
5. Lubośny Z., Szczerba Z., Zajczyk R., Analiza stanu obecnego automatyki odciążającej (SCO) w krajowym systemie
elektroenergetycznym – z punktu widzenia operatora systemu [Analysis of the present condition of self-acting frequency
unload automatics (SCO) in the national power system – from the operator point of view]. Study by EPS RESEARCH commissioned by PSE S.A., 1999
6. Lubośny Z., Szczerba Z., Zajczyk R., Automatyka realizująca obronę systemu przy awaryjnych przeciążeniach. Opracowanie nowych zasad i programu: stosowania automatyki samoczynnego odciążania w KSE – opartej na nowych algorytmach działania, skoordynowania jej z zabezpieczeniami podczęstotliwościowymi bloków, udziału sieci przesyłowych, sieci
rozdzielczych i elektrowni [Automatics that implement power system defend at emergency overloads. Development of
new methods and a programme of automatic unloading systems application in KSE national power system – based on new
functional algorithms, coordination with sub-frequency protections of generating sets, participation of transmission grids,
distribution grids, and utilities]. Stage 1 – 1999, Stage 2 – 2000. Study by EPS RESEARCH commissioned by PSE S.A.
7. Machowski J, Białek J.W., Bumby J.,R.,Power system dynamics and stability. John Wiley & Sons New York 1997.
8. Machowski J., Bernas S., Stany nieustalone i stabilność systemu elektroenergetycznego [Power system unsteady
states and stability], Warsaw, WNT, 1989.
9. Van Cutsem T., Vournas C., Voltage stability of electric power systems, Kluwer Academic Publishere, London 1998.
10. Zajczyk R., Modele matematyczne systemu elektroenergetycznego do badania elektromechanicznych stanów nieustalonych i procesów regulacyjnych [Mathematical models of power system for examination of electro-mechanical unsteady states and control processes], Gdańsk University of Technology Publications, 2003.
11. Zajczyk R., Szczerba Z., Lubośny Z., Małkowski R, Klucznik J., Kowalak R., Szczeciński P. , Dobrzyński K., Analiza stanu
obecnego i opracowanie zmian w układach regulacji napięcia i mocy biernej w elektrowniach, stacjach sieci przesyłowej
i w sieciach rozdzielczych w celu zmniejszenia ryzyka powstania awarii napięciowych w systemie elektroenergetycznym
[Analysis of the present condition and development of changes in voltage and reactive power control systems in utilities,
transmission grid stations, and distribution grids in order to reduce the risk of voltage failure development in a power system] Gdańsk University of Technology, Study commissioned by PSE Operator, Gdańsk 2007–2008.