A Multi-Objective Optimal Power Flow Algorithm for
AC/DC System with VSC-HVDC
Kai Gao, Chunsheng Yan, Zhengwen Li, Zijiao Han,
Bowen Tian
Guoqing Li,Yang Li
State Grid Liaoning Electric Power Company Limited
School of Electrical Engineering
Northeast Dianli University, NEDU
Jilin, China
Shenyang, China
Abstract—A new multi-objective optimal power flow (MOPF)
algorithm is proposed for AC/DC system with VSC-HVDC. First,
a MOPF model considering environmental benefits and voltage
stability is built. Then, a new approach including multi-objective
optimization and decision support is proposed to solve the model.
Specifically speaking, NSGA-II with a hybrid coding scheme is
adopted to get the Pareto-optimal solutions; the obtained
solutions are divided into three classes using fuzzy c-means
algorithm, and next the relative efficiency of the solutions
belonging to a class are accessed using super-efficiency data
envelopment analysis. In this way, the effective compromise
solutions can be obtained for decision-makers. And finally, the
application results on the IEEE 14-bus system demonstrate the
effectiveness of the proposed approach.
Index Terms-- AC/DC system, voltage source converter, optimal
power flow, NSGA-II, multi-attribute decision making
I. INTRODUCTION
Optimal power flow (OPF) has always been regarded as one
of the most important ways to ensure security and economic
operation of power systems [1]. With the interconnection of
AC/DC girds, electricity market reforms and large-scale
integration of renewable energy, the system is increasingly
running close to its stability limits. As a result, the conventional
mono-objective OPF may not be adequate to meet the needs of
an increasingly complex and stressed power grid. For this
purpose, multi-objective OPF (MOPF) has become a hot topic
in the field of OPF [2], [3].
More recently, very few attempts have been made to solve
the OPF problem for AC/DC system in the literature [8], [9].
Unfortunately, all these studies are only applicable for solving
the mono-objective OPF problem of AC/DC system. Therefore,
In this paper, a novel two-stage MOPF algorithm is
proposed for AC/DC system with HVDC. The contribution of
this paper is three-fold. First, a MOPF model is built to
uniformly coordinate the economy, safety, and environmental
requirements for AC/DC system with VSC-HVDC. Second, a
novel two-stage approach is described how to be applied to
solve the built MOPF model in detail. Third, significant results
of applying the proposed scheme to the IEEE 14-bus system is
demonstrated.
The remainder of this paper is organized as follows. First
the VSC-HVDC steady-state model is introduced in brief.
Then, the built MOPF model for AC/DC system is described in
detail. Details of the proposed two-stage approach are presented
next. Application of the proposed scheme is demonstrated using
the IEEE test systems, and finally the conclusions are made.
II. VSC-HVDC STEADY-STATE MODEL
A. Steady-state Power Characteristic
For AC/DC system with VSC-HVDC, its simplified
equivalent model is shown in Fig. 1 [11].
U c1
U s1
AC
Uc2
U s2
U dc1
U dc 2 DC
...
As a new and powerful high voltage direct current (HVDC)
transmission technology, voltage source converter based
HVDC (VSC-HVDC) has been playing an increasingly
important role in shaping the future of electric power industry
[4], [5]. Compared with other HVDC alternatives, VSC-HVDC
has many significant advantages [6], [7].
there is an urgent need for solving such a MOPF problem for
AC/DC system. The use of NSGA-II for MOPF has been
previously studied in [10]. However, the method only aims to
cope with the pure AC grids, and it cannot provide decision
support for decision-makers due to the fact that only multiobjective optimization is carried out using parallel NSGA-II.
U sn
U cn
U dcn
Figure 1. Simplified equivalent model of AC/DC system.
In this equivalent model, Zc  Gc  jBc represents the
impedance between the AC and DC network, U s  U s s is the
AC bus voltage and U c  U c c indicates the converter
voltage, the current is injected in converters by the AC grid
result in
I
Us  Uc
Gc  jBc
(1)
According to the apparent power injection given by
S s  Ps  jQs  U s I *
(2)
The active and reactive power injected into AC network are
Ps  U s2Gc  U sU c Gc cos  s   c   Bc sin  s   c  
Qs  U s2 Bc  U sU c Gc sin  s   c   Bc cos  s   c  
Pc  U Gc  U sU c Gc cos  s   c   Bc sin  s   c  
Qc  U Bc  U sU c Gc sin  s   c   Bc cos  s   c  
2
c
(4)
B. Steady-state Control Mode
When the active power absorbed by one terminal is more
than the power injected into another terminal, the voltage of DC
grid is lifted, otherwise the active power will drop. Therefore,
to keep the voltage balance, the DC voltage should be fixed in
one of the terminal in DC network (constant V-control). When
the DC voltage is steady, the DC current are proportional to the
unbalanced active power between both ends of the DC network.
Thus it is equivalent to the control for DC current and active
power. Through the above analysis, four modes are general
applied to control in AC/DC system with VSC-HVDC [11]: 1)
constant DC voltage and reactive power control; 2) constant DC
voltage and AC voltage control; 3) constant active and reactive
power control; 4) constant active power and AC voltage
control.
III. PROBLEM FORMULATION
A. Objective Functions
1) System Active Power Losses
The system active power losses of AC/DC grids is given by
n
n
i 1
j 1
O  Vi V j (Gij cosij  Bij sin ij )  PDC -loss
(5)
where O denotes the total active power losses of AC/DC grids;
Vi and V j are respectively the voltage amplitudes of bus i and
bus j ; Gij , Bij and  ij are respectively the conductance,
susceptance and phase-angle difference between bus i and bus
j ; PDC -loss is the power losses in the DC network comprising the
power losses of converter stations, reactors and DC lines.
2) Emissions of Polluting Gases
The polluting gases, such as carbon dioxide, sulfur dioxide
and nitrogen oxides, are mainly produced in the process of
power generation. Thus, the relationship between the polluting
gases emissions and the active power output of each generator
can be established [12]
NG
E   ( i PG2i  i PGi   i )
i 1
i
3) Static Voltage Stability Margin
As is well-known, there is a close relationship between the
singularity of the power flow Jacobian matrix J and static
voltage stability of power systems [13]. Consequently, the
minimum singular value of the Jacobian matrix J is suitable to
be used as a voltage stability index [14].
VSM  min eig  J 
(3)
The equations of the active and reactive power from AC
grid to DC grid are as follows:
2
c
where E is the amount of polluting gas emissions, NG is the
number of generators, PG is the power output from the i th
generator,  i ,  i and  i are respectively factors of polluting
gases emissions of the i th generator.
(6)
(7)
where eig  J  are the eigenvalues of the Jacobian matrix J.
B. Constraints
1) Equality Constraints
The related equality constraints of AC grid are
 Pgi  Pdi  Vi V j  Gij sin ij  Bij cosij   0

ji
(8)

Q

Q

V
 gi
di
i V j  Gij sin  ij  Bij cos  ij   0
ji

where Pgi , Qgi , Pdi and Qdi are respectively the active power
input, reactive power input, active load and reactive load.
The related equality constraints of DC grid are
 Pc  U c2Gc  U sU c Gc cos  s   c   Bc sin  s   c  

Qc  U c2 Bc  U sU c Gc sin  s   c   Bc cos  s   c 
 n
 P P 0
s ,i
loss

i 1

 Pdci  2U dci  Ydcij U dci  U dc j

j i

(9)

where Pc and Qc are respectively the active and reactive
powers of converters, Ps , i denotes the active power of bus i in
DC grid, Ydc indicates the admittance between bus i and bus
ij
j , Pdci is the bus active power in DC network, U dci and U dc j
are respectively the voltage between bus i and bus j .
2) Inequality Constraints
The inequality constraints in AC grids can be written as
 PGi min  PGi  PGi max

QGi min  QGi  QGi max

Vi min  Vi  Vi max
T  T  T
i
i max
 i min
QCi min  QCi  QCi max
i  1,
, NG
i  1,
, NG
i  1,
,N
i  1,
, NT
i  1,
, NC
(10)
where N , NT , and NC are respectively the total number of all
buses, adjustable taps of transformers and the reactive power
compensation buses, PGi and QGi are respectively the active
and reactive powers of i th generator, Ti denotes the
transformer tap positions, QCi are the switched-set number of
compensating capacitances, the subscript max and min are
respectively the upper and lower limits of corresponding
variables.

The related inequality constraints in DC grid are
Q  Q  r 2   P  P 2
0
s
0
 s


Z f
1

2
 S0  U s  Z   Z   , r  U s I cmax Z   Z 

tf 
f
tf
 f



 S  U 2 1  1 , r  U U 1
s 
s cm

 
 0
Z2
 Z1 Z 2 

(11)
Equation (11) is the constraint range of active and reactive
powers determined by the current and voltage in DC grid, the
center S0  P0 , Q0  of the circle that formed by each converter
active and reactive power constraint, and r represents the
radius of the circle. The I c is the upper limit of current, and
the lower limit approximately equals to zero. The Ucm indicates
the upper or the lower limit.
max
As the equivalent model of converter station shown in Fig.
2, the Z f and Z tf respectively denote the impedances to the
ground and to the AC grid, Z1 and Z 2 are respectively the
impedances to the ground after applying Y   transformation.
Judgment of termination conditions: if the conditions
are met, output the Pareto-optimal solutions; else,
update AC/DC grid data and repeat the process.
B. Decision-Makings
Here, the used decisions-making scheme includes a twostep process. First, the obtained Pareto-optimal solutions are
divided into three classes through FCM clustering; then, the
relative efficiency of decision schemes belonging to a class are
assessed by using SE-DEA. In this way, effective compromise
solutions are chosen from the Pareto-optimal solutions.
Based on the principle of minimizing input indicators and
maximizing output indicators in SE-DEA, the system active
power losses O and polluting gas emissions E are chosen as
input indicators and the static voltage stability margin VSM .
Input AC/DC system data and
NSGA-II parameters
Hybrid coding for initialize population
Compute AC/DC systems power flow
Us
Z tf
Zc
Uf
Obtain values of the objective function
Zf
Uc
Rapid non-dominated sorting and
individuals crowding distance calculating
Figure 2. Equivalent model of a converter station.
Selection, crossover and mutation operation
C. Mathematical Model
Therefore, the MOPF problem can be modeled as
min f m  u, x 
m  1, , M

j  1, , p
 s.t. h j  u, x   0

g j  u, x   0
j  p  1, , q


ui min  ui  ui max i  1, , D

Elitist strategy
yes
(12)
IV. MULTI-OBJECTIVE OPTIMIZATION AND DECISIONS-
Output the Pareto-optimal solutions
Figure 3. Flowchart of MOPF using NSGA-II.
V. CASE STUDIES
A. System Introduction
To verify the effectiveness of the proposed algorithm, a
modified IEEE 14-bus system is used as the test system, as
shown in Fig.5. This system includes 5 generators, 18 branches
and 11 loads, and the 13-14 branch changed to DC branch in
the system.
G
MAKING
G
3
2
A. Multi-objective Optimization Based on NSGA-II
NSGA-II algorithm with the elitist strategy is utilized to
solve MOPF for meshed AC/DC networks, as shown in Fig. 4.
The process mainly includes three parts:

no
Meet termination conditions?
where f   O, E , Vsm  , M is the number of object functions, h
indicates the equality constraints, g denotes the inequality
constraints of uncontrolled variables, p and q are respectively
the number of constraint h and g , u and x are respectively
the vectors constituted by controlled variables and state
variables, D is the element number of u .

Update AC/DC
system data
According to the nature of the input optimized
continuous and discrete variables, a hybrid coding
scheme is adopted to facilitate the optimization.
Using the alternating iterative method in [15], the
AC/DC power flow is calculated and the values of the
objective functions are updated for the regenerated
population at each iteration step.
G
1
5
4
G
8
7
G
6
11
10
9
12
VSC1
13
VSC2
14
Figure 4. Modified IEEE 14-bus test system with 2-terminal DC.
TABLE I.
POLLUTING GAS EMISSION COEFFICIENTS OF GENERATORS
Generator Bus Number



1
2
3
6
8
0.016
0.031
0.013
0.012
0.020
-1.500
-1.820
-1.249
-1.355
-1.900
23.333
21.022
22.050
22.983
21.313
TABLE II.
DC BUS PARAMETERS OF 2-TERMINAL DC NETWORK
R / p.u.
Nbus
13
14
X L / p.u.
0.1121
0.1121
0.0015
0.0015
Ps / p.u.
0.0429
-0.0552
U d / p.u.
2.000
2.000
Qs / p.u.
0.0078
-0.013 7
In the Table II, Nbus is the number of the AC bus connected
to VSCs, R is the equivalent resistance of the converter’s
internal losses and converter transformer losses, X L represents
the converter transformer reactance, U d is the DC-bus voltage.
B. Results and Discussion
Taking the modified IEEE 14-bus system as an example, the
optimization effects of the proposed approach are analyzed.
The control mode of VSC1 is constant DC voltage and AC
voltage control, and the control mode of VSC2 is constant
active and constant reactive power control. Note that, though
the above-mentioned control mode of the two-terminal DC
system is analyzed in this paper, the proposed approach is also
applicable to all other control modes. The distribution of
Pareto-optimal solutions in the objective function space is
shown in Fig. 6.
EXTREME SOLUTION OF 14-BUS NETWORK WITH 2TERMINAL DC NETWORK
Extreme solution
Extreme solution I
Extreme solution II
Extreme solution III
O
(MW)
8.68
11.06
12.48
E (/ten thousand
pounds/h)
0.0209
0.00045
0.0756
Vsm
0.5455
0.5351
0.5469
From Table III it can be observed that the extreme I and II
respectively reach the smallest value of the objective function
O and E, while the extreme III reaches the smallest value of the
objective function VSM . It should be pointed out that as far as
the mono-objective optimization is concerned, the obtained
three extreme solutions are optimal; but for the problem of
MOPF, they are non-inferior solutions.
Focusing on different objective functions, FCM clustering
is employed to divide the obtained Pareto-optimal solutions into
three groups with different colors, as shown in Fig.7.
0.55
0.548
0.546
0.544
0.542
0.54
0.538
0.536
0.534
0.08
0.06
0.04
0.02
0
8
10
12
14
Figure 6. Distribution of Pareto-optimal solutions of 14-bus system with 2terminal DC in the objective function space after clustering.
0.55
Then, SE-DEA is utilized to evaluate the relative efficiency
of Pareto-optimal solutions belonging to a cluster, and the
solutions with the highest efficiency are chosen as the effective
compromise solutions with the results shown in Table IV.
0.548
Static voltage stability margin
TABLE III.
Static voltage stability margin
In this work, the polluting gas emission coefficients of
generators [12] and DC bus parameters are respectively listed
in TABLE I and TABLE II.
0.546
0.544
0.542
TABLE IV.
0.54
COMPROMISE SOLUTIONS OF IEEE 14-BUS SYSTEM WITH 2TERMINAL DC
0.538
0.536
0.534
0.08
0.06
0.04
0.02
0
8
10
12
14
Figure 5. Distribution of Pareto-optimal solutions of 14-bus system with 2terminal DC in the objective function space.
As shown in Fig. 6, the mulit-objective functions,
comprising the system power losses, the emissions of polluting
gases and the static voltage stability margin for AC/DC system,
can be uniformly coordinated by using the proposed approach,
and the complete and evenly distributed Pareto-optimal
solutions can also be obtained.
Table III shows the extreme solutions corresponding to the
different objective functions in the Pareto-optimal solutions.
Effective compromise
solutions
O
(MW)
Compromise solution I
Compromise solution II
Compromise solution III
10.33
10.29
9.17
E (/ten
thousand
pounds/h)
0.0015
0.0014
0.0104
Vsm
Relative
Efficiency
0.5398
0.5393
0.5444
1.0373
1.0077
1.0749
According to the principle of SE-DEA, the compromise
solutions in Table IV are effective since their relative
efficiencies are all bigger than 1. Consequently, the effective
compromise solutions are obtained from the Pareto-optimal
solutions, facilitating decision-makers to determine the optimal
operation points according to the actual system operation needs.
Therefore, the conclusion can be drawn on the basis of the
above analysis that the proposed approach provides an effective
way to solve the MOPF problem for AC/DC system with DC.
VI. CONCLUSION
In this paper, a two-stage MOPF approach is proposed for
AC/DC system with VSC-HVDC. From the test results of IEEE
14-bus system, the conclusions can be drawn from the work:
1) Based on the VSC-HVDC steady state characteristics, a
MOPF model is built to uniformly coordinate the economy,
safety, and environmental requirements for AC/DC system
with VSC-HVDC.
2) Through the proposed two-stage approach for solving the
MOPF problem, not only the Pareto-optimal solutions with a
complete and uniform distribution can be obtained using multiobjective optimization, but also effective compromise solutions
can also be chosen for decision-makers using decision support.
3) The proposed approach may find potential applications
in intelligent planning and dispatching operation for smart
grids. Furthermore, the algorithm can be applied to any similar
multi-objective optimization problem in the field of
engineering.
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