CC BY
8
UDC 621.3
doi: 10.20998/2074-272X.2021.1.09
B. Ayachi, T. Boukra, N. Mezhoud
MULTI-OBJECTIVE OPTIMAL POWER FLOW CONSIDERING THE MULTI-TERMINAL DIRECT CURRENT
Introduction. In recent years, transmission systems comprise more direct current structures; their effects on alternating current power system may become significant and important. Also, multi-terminal direct current is favorable to the integration of large wind and solar power plants with a very beneficial ecological effect. The novelty of the proposed work consists in the effects of the aforementioned modern devices on transient stability, thus turn out to be an interesting research issue. In our view, they constitute a new challenge and an additional complexity for studying the dynamic behavior of modern electrical systems. Purpose. We sought a resolution to the problem of the transient stability constrained optimal power flow in the alternating current / direct current meshed networks. Convergence to security optimal power flow has been globally achieved. Methods. The solution of the problem was carried out in MATLAB environment, by an iterative combinatorial approach between optimized power flow computation and dynamic simulation. Results. A new transient stability constrained optimal power flow approach considering multi-terminal direct current systems can improve the transient stability after a contingency occurrence and operate the system economically within the system physical bounds. Practical value. The effectiveness and robustness of the proposed method is tested on the modified IEEE 14-bus test system with multi-objective optimization problem that reflect active power generation cost minimization and stability of the networks. It should be mentioned that active power losses are small in meshed networks relative to the standard network. The meshed networks led to a gain up to 46,214 % from the base case. References 24, table 3, figures 11. Key words: transient stability constrained optimal power flow, multi-terminal direct current.
Вступ. В останш роки системи nepedaui електроенергп включають в себе бтьше структур посттного струму; гх вплив на енергосистему змтного струму може стати значним i важливим. Крм того, багатотермтальний постшний струм е сприятливим для ттеграцп великих втрових та сонячних електростанцш з дуже позитивним екологiчним ефектом. Новизна запропонованог роботи полягае у впливi вищезазначених сучасних пристрогв на переxiдну стабыьтсть, що виявляеться щкавим питанням для до^дження. На наш погляд, вони становлять нову проблему та додаткову складшсть для вивчення динамiчноi поведтки сучасних електричних систем. Мета. Ми шукали розв'язання задачi переxiдноi стабiльностi, обмеженог оптимальним потоком потужностi в мережах змтного/посттного струму. Збiжнiсть для забезпечення оптимального потоку енергп була глобально досягнута. Методи. Розв'язання задачi було здшснено в середовищi MATLAB за допомогою теративного комбтаторного тдходу мiж оптимiзованим обчисленням потоку потужностi та динамiчним моделюванням. Результати. Новий пiдxiд, що обмежуе переxiдну стабыьтсть, з урахуванням багатотермiнальниx систем посттного струму може покращити переxiдну стабыьтсть тсля виникнення непередбачених ситуацш та економiчно експлуатувати систему у фiзичниx межах системи. Практичне значення. Ефективтсть та надшшсть запропонованого методу перевiряеться на модифжоватй тестовт 14-шиншй системi IEEE з використанням багатоцыьовог задачi оптимiзацii, яка вiдображае мiнiмiзацiю витрат на активну генеращю електроенергп та стабыьтсть мереж. Бiбл. 24, табл. 3, рис. 11.
Ключовi слова: перехвдна стабшьшсть, обмежена оптимальним потоком потужнос™, багатотермшальний постшний струм.
Introduction. In a competitive economic environment that has placed the need for reconciliation of economic and transitional stability conditions, the cost of losing synchronism by transient instability is exceptionally high and of significant importance. For this reason, the traditional optimal power flow (OPF) approach has been extended to take into account the transient stability constraints of the system, giving a new transient stability constrained optimal power flow (TSC-OPF) approach [1]. The TSC-OPF procedures are classified in either global or sequential approaches [2, 3]. The global TSC-OPF approaches are reported in [4-7]. The sequential TSC-OPF approaches perform a standard OPF analysis to assess an optimal operating point [1]. A transient stability analysis is then carried out independently from the optimization process to check if the optimum point is transiently stable, so that the time domain simulation-related set of differential algebraic equations is not embedded in the traditional OPF problem [2, 8-10]. In recent decades, various optimization approaches have been proposed [11]. Among them, those dedicated to solve the multi-objective TSC-OPF problems
where different goals and different constraints have been taken into account [12-14].
A proposed network topology in [15-18] has been used in our work for simulations propose. The meshed topology is of great interest for our proposed methodology.
This paper is organized as follows: first part concerns modelling AC/DC power system. In the second part, the TSC-OPF problem has been formulated. In the third part, the experimental results and conclusions are presented.
AC/DC power system modelling. A generalized AC/DC system of Fig. 1 consists of three parts.
1. AC grid. We present here the expressions for active and reactive power injections. Reader may refer to [19] for more details:
P
AC
= 1
J=1
Vi
AC
v/c| • (gij cos S1J + By sin S1J ); (1)
© B. Ayachi, T. Boukra, N. Mezhoud
Q
AC
= 1
J=1
Vi
AC
vfC\\gi} sinSy - By cos Sj ), (2)
where PgjC, Q
AC gi
are active and reactive power generations at bus i respectively; ViAC, VyAC are nodal
voltage values at buses i and j respectively; 8y is angle voltage of unit i and j; Gy is the conductance; By is the susceptance; n is the total number of generators.
E1ZS1 jX,
j di
V ZO
* 2n+1 2n+
Vn+iZ0n+i ,
E 2 ZS2 jXd2..
1
V zo „
2 n+N 2n+ N
C>0 4
v^ze+ 2
1- 1-1 v zo vrz0ri—71
I 3n+1 3n+1 \S
E'ZS
JX'dn
J dn" ' v
CK=d>+
V^zo^ O +n •<
V zo
* 3n+C 3n+C
vT"" zo'n°~
V zo n n
*3n+C^ 3n+ C 1/
i_
Th
o
£
o
Q
Fig. 1. Model for combined AC and DC grid
2. HVDC converters. Assuming a lossless converter model. The active power of the AC side converter coincides with that of the DC side:
fac = fdc .
(3)
An HVDC converter station was modeled as a voltage source with variable value V00"" and angle ffon"
connected to an AC bus via a reactance XeC°"". By
varying the voltage source, it is possible to produce the desired active and reactive power flow from the DC to the AC network or vice versa
F,
T/AC T/con" DC Vi ■Vi
X
sin(o,AC -000""); (4)
eq
2, AC AC con" qDC = v, Vi ■v,
Xec
Xec
„„„In AC nconv I -cos0 -0l . (5)
nv \ '
1 eq -1*- eq
3. DC grid. The power flows over the DC lines can be calculated as follows:
VDC - VDC
pDC = Vi VJ FiJ
(6)
Problem statement and formulation. The theory of TSC-OPF is an extension of the standard OPF problem to include supplementary constraints for imaginable credible contingencies cases. In a standard form, the OPF problem is defined as in [20-22]:
min f (x, u), (7)
subject to
h(x, u ) = 0; (8)
g(x, u )< 0. (9)
where x is a vector of state variables; u is a vector of control variables; h, g are functions;fx, u) is an objective function.
In general, the intention is to minimize the objective function with the solution satisfying a number of equality and inequality constraints. But the TSC-OPF problem can be mathematically considered as a standard OPF with other inequality dynamic constraints forced by the rotor angles of generators during the transient period in study for a given set of contingencies.
1. Objective function. The objective functions f of TSC-OPF is minimization of fuel cost for all generators:
f = min J (al+bt • PgI + ci ■ Pg2) , (10)
i=i
where ng is number of generators; a, b, ci, are cost coefficients of unit i; Pgi is an active power generations by unit i.
2. Equality constraints. The equality constraints h(x, u) are the sets of the load flow equations that govern the power system:
Pf + PiDC - Pdi = 0; (ii)
qAC + QDC - Qdi = 0, (12)
where Pdi, Qdi are active and reactive power loads by unit i respectively.
3. Inequality constraints (standard OPF). The inequality constraints g(x, u) are the set of constraints that represent the system operational and security bounds like the limits on the following:
PAmCngi < Pgi < PTaXgi; where i = 1,...,ng ; (13) Q^gi < Qgi < QmCXgi; where i = 1,...,ng ; (14)
VAC?! < Vi < VACX; i = 1,....., nb; (15)
0min <0, <0max; i = 1,.....,nb ; (16)
Tmin < t < Tmax; i = 1,.....,nT , (17)
where PACg, Pa™ and Q^, QZ, are the lower and
upper limits of active and reactive power generation at bus i of AC network respectively; Qgi is reactive power generations at bus i; VC VACX are the lower and upper limits of voltage value at buses i; Vi is voltage value at buses i; nb is total number of buses; 0mm, 00" are the lower and upper limits of angle voltage; 0 is the angle voltage of unit i; Tm'°, Tm" are the lower and upper limits
of transformers tap settings; Ti is transformers tap settings of unit i; nT is total number of transformers.
4. Inequality constraints (transient stability). The transient stability problem in power system is defined by a differential algebraic equation, which can be solved by time domain simulation. The swing equation for ith generators is:
^ = ^[(( -Pel)-D( ) (18)
dt Mi
=w,-rn,,; where i = 1,....., ng (19)
dt g
dEdL =VT-l E'di + (Xqi - X'qi )iqi ^ (20)
dt Tqi0
X
dEq
q' 1~ [Efdi - Eqi - (Xdi - X'di )idi \ (21)
dt Td
di0
dE
fdi
dt
T
M
Ai
- Efdi + KAi
(Vrefi - V )J
P=P E^- Pe,)-Dp, -p );
dt m,
dSi dt
where i = 1,
(22)
(23)
(24)
where Pmi is the mechanical input generator by unit i; Pei is the electrical output generator by unit i; M is the moment of inertia of ith generator; coi is the angular speed of the rotating synchronous reference frame of unit i; a>s is the angular speed of the generator rotor of unit i; Si is rotor angle of unit i; D is the generators damping torque coefficient; E'di, Eare the internal transient
voltage of generator of unit i; Efdi is the excitation voltage of generator of unit i; idi, iqi are d-axis and q-axis courant of generator of unit i; T'd0i T'0i are the d-open circuit
and q-open circuit transient time constants of generator of unit i; X'di, X'qi are the d-transient reactance and
q-transient reactance of generator of unit i; Xdi, Xqi are the d-synchronous reactance and q-synchronous reactance of generator of unit i; Vf is voltage reference of generator of unit i.
The inequality constraints of transient stability are formulated as:
S
-S
COI
<Sm
(25)
S
COI
(26)
ng
£ Mi Si
i=l
ng '
£Mt i=l
where S is the rotor angle of unit i; SCOi is the position angle of centre of inertia (COI - centre of inertia); Smax is the maximum allowable rotor angle deviation
The selection of Smax is frequently based on experiment operation. It was generally limited to 100° to allow network having sufficient stability margin [23, 24].
5. Inequality constraints (multi-terminal direct current - MTDC). The last constraint limits the variance of the all DC nodes have their minimal and maximal voltage angle limits, in which the equipment can operate safely.
0dC, j * °DC, j * DX,j; where j = 1,....., nb,; (27)
Pdc,j < Pdc,j < PSCxj ; where j = 1,.....
, nb'
(28)
Simulation results and discussion. In this paper, the MTDC system presented for the simulation with integrated offshore wind farm is shown in Fig. 2. This system is a modified version of the IEEE 14-bus test system. A meshed DC grid including an additional generator has been added, that connects to the AC system to various buses through voltage source converters (VSC). The bus 19 is not connected to any VSC. The bus and line parameters were given in the [24]. All the simulations were performed on source codes developed in MATLAB environment running on an Intel Core i5 2.67GHz CPU and 3 GB RAM.
Fig. 2. Modified IEEE 14-bus
The fault priority list was given in Table 1. The top 4 faults in critical clearing time (CCT) are considered.
According to Table 1 and Fig. 3, it is evident that the most unfavorable case is the three-phase fault (3^). However, it should be noted that there are, for certain types of connection, cases where other types of fault are more damaging. The best known of these faults are:
• single line to ground fault (SLG), when the generator neutral is connected to ground directly or through low impedance;
• single line to ground fault, when the transformer is connected in Y-A with neutral grounded.
tP, [p.u.]
where pmi^j, PimC^xj are the lower and upper limits of
active power generation at bus j of DC buses; PDC,j is the active power generation at bus j of DC buses;
^Dmc, j, ODxClj are the lower and upper limits of angle
voltage of DC buses; dDCJ is the angle voltage of DC buses; nv is the total number of DC buses.
Fig. 3. Power injected by the generator, depending on the type of fault (LLG is double line to ground fault; LL is line to line faults; SLG is single line to ground fault; 3q> is three-phase fault)
= Wi — CO
n
g
Table 1
Priority list of fault for IEEE 14 bus system
Cases Faults Line CCT(s)
3ç SLG
1 1 1-2 0.33
2 1 1-5 0.36
3 1 1-2 0.38
4 2 1-2 0.40
The severity criteria considered in this study is the critical clearing time (CCT). This study was performed according to the 2 different faults (three-phase fault and single line to ground).
In this section, the modified IEEE 14-bus system has been used to illustrate the effectiveness of the proposed method. There are 5 cases to be discussed here, each with 2 scenarios:
• Case 1 is the standard power flow (PF);
• Case 2 is the standard OPF without transient stability constraints;
• Case 3 is OPF with transient stability constraints;
• Case 4 when IEEE 14-bus is modified, with added MTDC and without transient stability constraints;
• Case 5 using a new configuration of network with transient stability constraints.
The results are given in Table 2.
Table 2
Optimization results for IEEE 14 bus system
Case 1 Case 2 Case 3 Case 4 Case 5
3ç fault | SLG fault 3ç fault | SLG fault 3q> fault SLG fault 3ç fault | SLG fault 3q> fault SLG fault
PgU MW 232 194.41 186.63 174.97 146.41 137.63 120.06
Pg2, MW 40 36.74 37.03 37.44 25.61 34.34 39.61
Pg3, MW 0 28.61 31.53 35.41 0 0 0.07
Pg6, MW 0 0 0 1.28 0 0 8.82
Pg8, MW 0 8.52 12.44 17.58 0 0 3.23
Pg19, MW - - - - 100 100 100
Fuel cost ($/h) - 8080.77 8084.20 8102.18 5526.65 5549.29 5691.48
CCT (s) 0.33 0.38 0.51 0.45 0.57 0.47 0.28 0.37 0.31 0.39
Losses (MW) - 9.277 8.610 7.674 1.417 1.370 1.187
First, simulation results were presented with standard PF (case 1). With this generation, it was found that system transient stability was lost following the three-phase fault disturbance at bus 1 (cleared by tripping line 1-2 at 0.33 s), and was lost following single line to ground fault at bus 1 (cleared by tripping line 1-2 at 0.38 s) respectively. Visibly the system can't operate under this mode.
The second simulation (case 2), was the standard OPF without transient stability constraints; the objective function (fuel cost) reached 8080.77 $/h. But with this generation, it was found that system transient stability was lost following the fault disturbance at bus 1, as shown in Fig. 4, 5, respectively. Visibly the system can't operate under this mode because security of the network is always violated.
Case 3, scenario 1 (3^ fault), the active power of generator 1 is reduced from 194.41 MW to 186.63 MW, while those of generators 2, 3 and 8 were increased from 36.74 MW, 28.61 MW and 8.52 MW (case 2) to 37.44 MW, 31.53 MW and 12.44 MW (case 3) respectively. A fuel cost was increased from 8080.77 $/h (case 2) to 8084.20 $/h (case 3) as shown in Table 2 and Fig. 6. A consequence of satisfying the transient stability constraints is the increasing in fuel cost only by 0.042 %.
For scenario 2 (SLG fault), the active power of generator 1 is reduced from 194.41 MW to 174.97 MW. While those of generators 2, 3, 6 and 8 were increased from 36.74 MW, 28.61 MW, 0 MW and 08.52 MW to 37.44 MW, 35.41 MW, 1.28 MW and 17.58 MW (case 3) respectively. The fuel cost was increased from 8080.77 $/h (case 2) to 8102.18 $/h (case 3) as shown in Table 2 and Fig. 7. A consequence of satisfying the transient stability constraints is the increasing in fuel cost only by 0.246 %.
From Fig. 6, 7, the use of transient stability constraints in terms of OPF solution gives better results and ensures system transient stability following the fault disturbances.
-100 L
........t, [s]
"0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Fig. 4. Response of rotor angle (case 2, OPF), 3ç fault
150
5 S - Scoi
4 -<4 - Scoi
- 3 S Scoi -
2 <2 Scoi
- / 1 <1 Scoi
4
- 3
, t, [s]
0.2 0.4 0.6 0.8
Fig. 5. Response of rotor angle (case 2, OPF), SLG fault
100
50
0
100
150
0
1.2
1.4
1.6 1.8
2
o
o
5 s5 - scan
- 4 5 - <5coi -
3 <53 <5coi
2 <52 - <5coi
1/ 1 S <5co/
/ 2 5 \ /x\ "
3
4
t, [s]
100
50
-50
-100
5 5 - 5coi
4 <54 <5cO!
3 <53 <5co!
2 <52 - SCOI
, 1 5 5co-
- 2
" 4
5
" 3
, t, [s]
Fig. 6. Response of rotor angle (case 3, TSC-OPF), 39 fault Flg. 9 ResPonse of rotor angle (case 4, OPF, MTDC), SLG fault
8 60
40 -
20
-60
-80
5 55 5coI
- / \ 4 5t 5coi
3 53 - 55coi
" / \ 2 52 55coi
" 1/ \ 1- —fs 55coi
/ 5 _____ \ -
2
- 4 \ / "
3
■ \ / , t, [s]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Fig. 7. Response of rotor angle (case 3, TSC-OPF), SLG fault
Cases 4 and 5 the network is modified, and at the present the total loads is 295 MW (new load at bus 1, active power 11.6 MW) and a new generator is present on the DC bus (bus 19). The solution of cases 4 and 5, with and without transient stability constraints is given in Table 2.
The fourth simulation (case 4), the fuel cost was reduced to 5526.66 $/h. But with this generation, it was found that system transient stability was lost following the fault disturbance at bus 1, as shown in Fig. 8, 9, respectively. Visibly the system can't operating under this mode because security of the network was violated.
120
100
0
0 80
0
0
0 a. 60
w
40
S
o
20
0
0
CT
0
0
CT
£= ro -20
O
O
ol -40
-60
5 55 Scot
4 St SCOI -
3 53 ScoI
- 2 52 Scoi -
1- 5 ScoI
2 J -
5
- ^ix 3
, t, [s]
In case 5, scenario 1 (3^ fault), in order to retain the transient stability limits, the active power of generator 1 was reduced from 146.41 MW to 137.63 MW, while that of generator 2 was increased to 34.34 MW. The fuel cost was increased from 5526.66 $/h to 5549.29 $/h as shown in Table 1 and Fig. 8. A consequence of satisfying the transient stability constraints was the increase in fuel cost by 0.409 %.
For last scenario (SLG fault), the active power of generator 1 is reduced to 120.06 MW. While those of generators 2, 3, 6 and 8 were increased to 39.61 MW, 0.07 MW, 8.82 MW and 3.23 MW, respectively. A consequence of satisfying the transient stability constraints was the increase in fuel cost by 2.982 %.
From Fig. 10, 11 it was obvious that the use of TSC in OPF solution gives better results and guarantees transient stability following the fault.
120
60
Ü -20
o o
o: -40 -60 Fig
5 — 55 55coI
4 5 Scoi -
3 53 ScoI
- / 2 5 5>coi -
1 1 — 5 5coi
2 -
- 5 \ /
- 3 -
, t, [s]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10. Response of rotor angle (case 5, TSC-OPF, MTDC), 3p fault
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Fig. 8. Response of rotor angle (case 4, OPF, MTDC), 3ip fault
For the best compromise solution, we must maintain a balance between stability and economy. The solution obtained enhances the transient stability of the system at the best acceptable cost. Generally, in this situation, the cost is marginally higher as the economy is sacrificed for the improvement of transient stability. In case of lower cost solution, more emphasis is given to economy of the system and this solution is unable to improve the global transient stability of the network. For minimum cost
100
150
80
60
40
20
0
0
0.2 0.4
0.6 0.8
1.2
1.4
1.6
.8
2
80
0
80
40
20
0
5 <55 <5co/
4 <54 <5co/
3 <53 <5co/
2 <52 scot
1/ 1 S scot
_/ 2 — 4 \
5
3 -
, , t, [s]
-50
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Fig. 11. Response of rotor angle (case 5, TSC-OPF, MTDC), SLG fault
case 4, the CCT for the fault at bus 1 cleared by tripping line 1-2 becomes 0.28 s whereas it was 0.33 s for three-phase fault and 0.37 s to 0.38 s for base case for single line to ground fault, as shown in Table 3.
Table 3
CCT comparison
3q> fault SLG fault
Case 1 0.33 0.38
Case 2 0.51 0.45
Case 3 0.57 0.47
Case 4 0.28 0.37
Case 5 0.31 0.39
Finally, the solution sought for the problem of the transient stability constrained optimal power flow, in the last case, compared to all the previous solutions that maximize the transient stability index, has the highest cost as more focus is placed on optimizing the system's transient efficiency. In the event of emergency situations, this approach can be extremely useful.
It should be mentioned that active power losses are small in mixed AC/DC networks relative to the standard AC network. The AC/DC networks led to a gain up to 46,214 % from the base case. This confirms the potential need for construction of solar or wind farms, considering their advantages.
Conclusions.
It is very likely that future transmission systems will contain more multi-terminal direct current links and the effects of such system on transient stability are yet to be determined. The increase of the future system has led to a growing complexity in the study of its problem and so presents new defy to power system stability. This paper suggests a new version of the multi-objective optimal power flow, taking into account the transient stability of the power system. Two goals were concurrently considered; minimizing the cost of the fuel and optimizing the system's transient stability margin at the point of fault clearance. Furthermore, this study was performed on the basis of two different faults namely three-phase fault and single line to ground fault.
Simulations were carried out in MATLAB environment, the transient stability of the system is
compared with and without the presence of transient stability constraints. The use of transient stability constraints in terms of optimal power flow solution gives better results and also ensures system transient stability following the fault disturbance.
Conflict of interests. The authors declare no conflicts of interest.
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Received 04.11.2020 Accepted 11.12.2020 Published 25.02.2021
Bilel Ayachi1, Ph.D. of Power Engineering, Tahar Boukra1, Ph.D. of Power Engineering, Nabil Mezhoud ', Ph.D. of Power Engineering, 1 Electrotechnical Laboratory Skikda (LES), Department of Electrical Engineering, University 20 August 1955, 26 Road El Hadaiek 21000, Skikda, Algeria, e-mail: b.ayachi@univ-skikda.dz, boukratahar@hotmail.com, mezhounab@yahoo.fr
How to cite this article:
Ayachi B., Boukra T., Mezhoud N. Multi-objective optimal power flow considering the multi-terminal direct current. Electrical Engineering & Electromechanics, 2021, no. 1, pp. 60-66. doi: 10.20998/2074-272X.2021.1.09.
