Optimal Allocation of AC-DC Capacity Considering Cascading Failure Risk of AC-DC Parallel Power System Jingzhe Tu, Zehan Lu, Xiaozhe Song, Yun Liu Ahstract-There are few studies both considering the two aspects of cascading failure risk (CFR) and AC/DC capacity optimal allocation, so the problem of AC/DC capacity optimal allocation considering CFR is studied. The mathematical model of the AC/DC capacity optimal allocation problem is provided, the cascading failure Monte-Carlo simulation algorithm and the AC/DC capacity optimal allocation implicit enumeration
algorithm
are
proposed.
Simulation
based on the modified IEEE39 system is performed, in which
the AC/DC
capacity optimal
allocation
which has the smallest CFR is obtained and the impact of various factors are also analyzed. The results reflect that there exists an optimal AC/DC capacity allocation with a balanced ratio, and it can be quantitatively calculated applying the proposed method. Index
Terms-AC/DC
parallel
system;
AC/DC
capacity allocation; cascading failure; blackout I. INTRODUCTION
The characteristics of AC and DC power transmission have been widely studied by researchers all over the world, which have provided guidance to the planning and operation of power systems [1]-[2]. Compared to the AC or DC power system, the AC/DC parallel power system has both the advantages of grid construction (AC system) and fast control (DC system), so it has better application prospect in power engineering. To the ACIDC parallel system, we should not only consider the matching of DC infeed capacity and receiving-side AC system strength, but also the matching of parallel AC and DC transmission capacity. That is the problem of the AC/DC parallel capacity allocation when keeping the total capacity constant. The stability characteristics of the ACIDC parallel system is different from that of the pure AC or DC system. The stability of the ACIDC parallel system is not only related to the total capacity, but also the AC/DC capacity allocation mode [3]-[4]. Thus, different AC/DC capacity allocation modes will make the system have different stability levels. If the ACIDC capacity allocation ratio exceeds a certain balanced range, such as the strong DC and weak AC mode (with too high DC capacity) or the strong AC and weak DC mode (with too high AC capacity), will both weaken the system stability. Therefore, in order to
J. Tu is with the China Electric Power Research Institute, Beijing, China (corresponding author, e-mail:
[email protected]). Z. Lu is with the State Grid Tangshan Electric Power Supply Company, China.
X.
Song is with the State Grid Jilin Electric Power Supply Company,
Changchun, China.
Y.
Liu is with the college of Electrical Engineering, Zhejiang
University, Hangzhou, China.
978-1-4673-8040-9/15/$31.00 ©2015 IEEE
improve the stability of the AC/DC power system, it is necessary to apply the strong DC and strong AC mode (with balanced ACIDC capacity) [3]-[4]. In the practical engineering, the AC/DC capacity allocation ratio is normally decided according to experience, but there is no clear quantitative calculation method. In the literature, there is also few detailed studies on the problem of AC/DC capacity optimal allocation. There has occurred several blackouts all over the world in recent ten years [5]-[6], and comprehensive studies on this subject have been widely conducted [7]-[12]. Study results show that the probability of cascading failure and blackout cannot be neglected. Besides that the strong DC and weak AC mode or the strong AC and weak DC mode will result in normal stability problems, it may also lead to severe events such as cascading failure. Therefore, when studying the problem of AC/DC capacity optimal allocation, reducing the possibility of cascading failure should be especially considered. However, in the literature, the relative study both considering the two aspects of cascading failure risk (CFR) and AC/DC capacity optimal allocation is almost blank. So it is quite necessary and significant to study the problem of ACIDC capacity optimal allocation considering CFR. 11. AC/DC CAPACITY OPTIMAL ALLOCATION MODEL
The problem of ACIDC capacity optimal allocation can be mathematically abstracted to a standard nonlinear programing problem, and its mathematical model is quite similar to that of the stability constrained optimal power flow [13]-[14]. The only differences between them are the objective function and the constraint conditions, but the mathematical essences are the same. The mathematical model of AC/DC capacity optimal allocation problem is listed as follows. Eq. (l) is the objective function, Eq.(2) - (4) are the constraint condition 1, Eq.(5) is the constraint condition 2, and Eq.(6)-(7) are the constraint condition 3. min r(x,y,u,a)
(l)
s.t. P'L = const
(2)
P,tc = P'L a
(3)
P,u = P'L (1 - a)
(4)
u =u(x,y,a)
(S)
i=f(x,y,u,a)
(6)
O=g(x,y,u,a)
C7)
O'EI0,11
.
.
where x denotes system state variables such as the generator rotor angle and the rotation speed, y denotes system algebraic variables such as the node voltage
magnitude and the injection current, u denotes system control variables of the AC/DC emergency control & protection devices, a denotes the ratio of AC/DC capacity allocation ( 0:::; a:::; 1 ), PI. denotes the total AC/DC capacity,
�Jc denotes the DC capacity,
Pae
denotes the AC capacity, re-) denotes the function of CFR. Objective function: In order to study the ACIDC capacity optimal allocation problem from the point of preventing cascading failure and blackout, the CFR index is selected as the objective function. The CFR can be approximately expressed by the statistical average of load loss, which is related to the area included between the power law curve and the abscissa shown in Fig. I. log(P)/l
�oss
Algorithm
In practical power systems, each electric component may have fault and be switched off by corresponding ACIDC emergency control & protection devices. If the deterministic method is applied to do the N-m contingency scan, it may lead to combination explosion. Thus, the stochastic method such as the Monte-Carlo algorithm is proper to simulate the whole process of cascading failure and blackout. In each stochastic experiment, several initial contingencies are randomly selected and simultaneously imposed on the system. Applying the cascading failure simulation model considering the ACIDC emergency control & protection devices proposed in [12], the devices such as load shedding, generator tripping, AC line breaking, and DC line blocking are checked if operated or not. In the end, the total load loss in the whole process of cascading failure is counted and the CFR index can be obtained. AC/DC
Capacity
Optimal
Allocation
Implicit
Enumeration Algorithm
Fig.1 Power law curve and CFR index
denotes the system outage power,
ALGORITHM
A. Cascading Failure Monte-Carlo Simulation
B.
log(P,m)/MW
where
Ill. ACIDC CAPACITY OPTIMAL ALLOCATION
P
denotes the outage probability. Constraint condition 1: The first constraint condition is the total transmission capacity of the ACIDC parallel system is kept constant. The issues such as unit scheduling and load fluctuation are not considered here, and suppose the generation and load power of the sending and receiving side are fixed. Different ACIDC capacity allocation modes can be realized by changing DC power, then the calculation to find the optimal solution can be performed. It can be known that the DC power is the decision variable of the AC/DC capacity optimal allocation problem and the AC/DC capacity allocation mode can be characterized by DC power. Constraint condition 2: The second constraint condition is the operation characteristics of various AC/DC emergency control & protection devices. Because that the statistic average of load loss is the objective function, and the load shedding amount has direct relationship with the corresponding device operation characteristics. So the operation characteristics of various AC/DC emergency control & protection devices must be simulated in detail when studying the AC/DC capacity optimal allocation problem. Constraint condition 3: The third constraint condition is the differential equations characterizing system dynamic behaviors and the algebraic equations denoting system steady-state constraints. Power system stability problems can normally be described by a group of nonlinear differential-algebraic equations. Because that the parameter space of the decision variables is a closed set, so theoretically there must exists an AC/DC capacity optimal allocation mode with the smallest CFR. Analysis and calculation on this are totally feasible and valuable.
In practical planning and operation, the ACIDC capacity optimal allocation mode is normally selected from several candidate modes, which in fact is a discrete variable optimization problem. It will lead to failure of the derivative based algorithms, so the derivative free algorithms must be applied. The derivative free methods can adapt to complicated dynamic models of power systems, and the technical route of secondary development can be applied to reduce the difficulty of program development. It can totally meet the computation requirement when the decision variables are few. The implicit enumeration algorithm is such a representative algorithm. The implicit enumeration algorithm is a special enumeration algorithm. Theoretically, the implicit enumeration algorithm can ensure that the optimal solution can be obtained as long as the suitable discretization step is selected. 'Implicit' means that the calculation of non-optimal solution can be terminated in advance in the calculation process, so the algorithm is quite effective when the scale of the optimal problem is small [14]-[15]. When enumerating a certain parameter group of DC power, if the CFR index is found to be larger than the current minimum value, then the calculation for the current parameter group can be terminated and the next parameter group will be enumerated continually. The AC/DC capacity optimal allocation implicit enumeration algorithm is also coded by the Python language, continuously calling the inner loop illustrated in the above Section A. The CFR indices of different AC/DC capacity allocation modes are calculated, then the AC/DC capacity optimal allocation mode can be obtained. The flow chart of the AC/DC capacity optimal allocation implicit enumeration algorithm is shown in Fig.2.
ACIDC tie-line is shown in Table 1. It can be seen from the table that the total power of AC lines is close to that of DC lines, which belongs to the strong DC and strong AC mode. If the DC power is continuously increased, the strong DC and weak AC mode can be obtained. If the DC power is continuously decreased, the strong AC and weak DC mode can also be obtained. Table 1 Composition and power flow of the interface AC/DC tie lines
Fig.2 Flow chart of the ACIDC capacity allocation implicit enumeration algorithm
where t denotes the simulation time, At denotes the step size, tJllilX denotes the maximum simulation time,
�J denotes the DC power,
k
denotes the parameter
group number, kmax denotes the maximum parameter group number, Sk denotes the CFR index of the k th Smill
parameter group,
denotes the minimum CFR
index.
no.
tie-line
1 2 3 4
25-2 14-4 17-18 11-6
AC/DC DC DC AC AC
power flow
240MW 260MW 192MW 342MW
In the modified IEEE39 system, standard models of the PSS/E software are used. The generators are represented by the GENTRA model. The excitation systems are represented by the IEEET 1 model. The speed governors are represented by the IEEEGl model. The power system stabilizers (PSS) are represented by the IEEEST model. The loads are represented by the constant power model. The parameters of generators and loads are typical values. The models and parameters of the AC/DC emergency control & protection devices are shown in [12]. B. Simulation Setting and Instruction
IV. AC/DC CAPACITY OPTIMAL ALLOCATION CASE SIMULATION
A. Case System and Parameters
The case system used in this paper is the modified IEEE39 system. On the base of the standard IEEE39 system [16], two AC lines from bus 25 to bus 2 and from bus 14 to bus 4 are replaced by two DC lines transmitting the same power, constructing a ACIDC parallel system. The wiring diagram of the modified IEEE39 system is shown in Fig.3, in which the interface between the sending and receiving sides is denoted by the dotted line. The tie-lines are composed of two DC lines (25-2, 144) and two AC lines (17-18, 11-6), and the power is transmitted from the right area to the left area in the figure. \
"G;\
G
...:.3�\
\ \
..:...3
7
generators, the probability of DC line blocking, AC line breaking, and generator tripping are , and 5ndc / C5ndC + nac + ngen) , nac / (5ndc + nac + ngen) ngen / (5ndc + nac + ngen) Considering the load level and node number of the case system, two initial contingencies are randomly selected with the probability distribution above and are simultaneously imposed on the case system doing the N-2 scan. Table 2 Relationship between DC power (pu) and AC/DC capacity
�- _____ ",T ------2 7
3 _I _ __
X
� � 17
...
=1" (> /,
L-Rlf-'i- 14 ·
--' -jQ}
...
allocation mode 3 � X
�'
G
'I
�24 -36
'1' ___ 23 � ]3-G 1---12 19 '1'--'1' /
(-11
7
DC line I-pole blocking, AC line/transformer 3-phase breaking, and generator tripping are considered as the three contingency types. Considering that the contingency probability of the DC component is normally higher than that of the AC component, so the probability of DC line blocking is set five times as that of AC line breaking and generator tripping. Thus, if the system has ndc DC lines, nac AC lines, and ngen
'
-20 .10 ...
__
, _
-31" -32
o
10
'I'
22
-34- 33
do
-35
d
, I
Fig.3 Modified IEEE39 system wiring diagram
In the basic operation mode, the power flow of the
DC
per unit
power
O.20pu OAOpu O.60pu O.80pu 1.00pu 1.20pu 1.40pu 1.60pu 1.80pu
AC/DC capacity allocation mode AC power DC power strong/weak mode lOOMW 940MW 200MW 830MW strong AC & weak DC 730MW 300MW 400MW 620MW strong DC & strong AC 500MW 520MW 600MW 420MW 700MW 310MW strong DC & weak AC 800MW 210MW 900MW 100MW
It is known that the decision variable of the AC/DC capacity optimal allocation is the DC power, so it can be used to characterize different ACIDC capacity allocation modes in the case analysis. The relationship between the DC power (pu) and the AC/DC capacity allocation mode is shown in Table 2. It is also known that the CFR index
can be expressed by the statistical average of load loss. C. Case Simulation Results Analysis
Applying the cascading failure Monte-Carlo simulation algorithm and the AC/DC capacity optimal allocation implicit enumeration algorithm, the AC/DC capacity optimal allocation of the case system in the basic operation mode is calculated. The impact of different factors is also analyzed, which includes: a) initial contingency number, b) contingency probability, c) generator tripping and load shedding combination mode, d) load shedding ratio. (1) AC/DC capacity optimal allocation in the basic operation mode According to the initial contingency setting introduced, two initial contingencies are randomly selected to do the 'N-2' scan, the AC/DC capacity optimal allocation in the basic operation mode is obtained by the CFR index, the results are shown in FigA and Table 3.
weak DC mode, because the power flow transfer in large scale resulted by DC blocking is a significant cause of cascading failure and blackout. (2) impact of different factors on AC/DC capacity optimal allocation a) initial contingency number The AC/DC capacity optimal allocation mode is calculated when the initial contingency number is one (condition 1), two (condition 2), and three (condition 3). The calculation result is shown in Fig.S. 3200 .-----�--�--_____.
_
3000
_._'-
2800
1'--_. ----L ___• ___ ;---I---�---.... -_.J
3: :;; -u; 2600 Sl � 2400
i.._._-.._._._._._._._..':: -'-'''''l. _._.
j
2200
2850 .-----�--�--_____.
2oo0 '----�---�----' o 05 1 1.5 2 DC power( pu) Fig.S A CIDC capacity optimal allocation of different initial contingency numbers
2750 '----�---�----' o 0.5 1.5 2 DC power( pu) Fig.4 ACIDC capacity optimal allocation in the basic operation
b) initial contingency probability The AC/DC capacity optimal allocation mode is studied when the DC contingency probability is one time (condition 1), five times (condition 2), and ten times (condition 3) as the AC contingency probability. The calculation result is shown in Fig.6.
mode
3000
Table 3 ACIDC capacity optimal allocation in the basic operation mode DC per unit power
CFR index (load loss)
O.20pu
2801MW
O.4Opu
2761MW
O.60pu
2753MW
O.80pu
2786MW
1.0Opu
2761MW
1.20pu
2787MW
1.40pu
2765MW
1.6Opu
2830MW
1.80pu
2846MW
�
2900
:;; 2800 '"
15 ;; 2700
-·-·-
L .-.�_._. J
----l
·-·-
._._.I·-·-··
i _._. i.
� '-'-'-' 1._._.' i
,___ l---L J --
---
�
_._._ •• condition 1
2600 2500 Lo
� 05
--
� --'=======l 1 1.5 2 power( pu)
---
DC
----- condition 2 -- condition 3
-
Fig.6 ACIDC capacity optimal allocation of different contingency
It can be seen from FigA and Table 3 that DC power of O.60pu is the ACIDC capacity optimal allocation, thus the system CFR is the smallest when the DC power is 300MW and the AC power is 730MW referring to Table 2. It also verifies the effectiveness of the implicit enumeration algorithm to solve the problem of AC/DC capacity optimal allocation. It can also be known that when the DC capacity or the AC capacity is too high, the CFR index will be large. And when the ACIDC capacity is balanced, the CFR index will be small. The results also verify the engineering experience that the strong DC and strong AC mode is beneficial to improve the system stability. We can also find that the CFR index of high DC capacity is a little larger than that of high AC capacity. It shows that the strong DC and weak AC mode is more adverse to the system stability than the strong AC and
probability
c) generator tripping and load shedding combination mode The AC/DC capacity optimal allocation mode is compared when the generator tripping and load shedding mode only considers the frequency (condition 1), only considers the voltage (condition 2), and considers both the frequency and voltage (condition 3). The calculation result is shown in Fig.7.
32oo �--�---�--�--� 3000 2800
----I
;:: 2600
___...___
:;;; 2400
:
� 2200
-- condition 1
.Q
----- condition 2 •• condition 3
_._._
iil 2000 .Q
..r---,
1800 1600
14oo� o
____----..___ !---..----
I
. . ..
I . . . . � , -. · · r-·-· - L _ _ _ _. -.-. .-.-. ·-· - - --::-'c: � �=-- � ...
--
0.5
1 power( pu)
---
DC
--
1.5
-
perspective of cascading failure and blackout. Either DC capacity or AC capacity too high is adverse to system stability. The AC/DC capacity optimal allocation with the smallest CFR can be quantitatively calculated applying the method proposed in this paper. However, in the practical engineering, the AC or DC capacity is determined by numerous other requirements . If the required balanced range of the AC/DC capacity cannot be satisfied, then the AC/DC parallel interconnection mode is not reccomended for security.
2
Fig.7 AC/DC capacity optimal allocation of different generator tripping and load shedding combinations
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d) load shedding ratio The AC/DC capacity optimal allocation mode is analyzed when averagely allocates the load shedding ratios (condition 1), increases the first load shedding ratio (condition 2), and increases the first three load shedding ratios (condition 3). The calculation result is shown in Fig.8.
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. _ _jr---
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2900 ,-----�--�--_____,
. . j_ _
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-- condition 1
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----- condition 2
_._._
2700 o
05
DC
1 power( pu)
••
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[8]
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Fig.8 ACIDC capacity optimal allocation of different load shedding
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It can been summarized from Fig.5-8 that although the DC powers referring to the AC/DC capacity optimal allocation are slightly different (0.60-1.00) under different initial contingency numbers, contingency probability, generator tripping and load shedding combinations, and load shedding ratios, they all belong to the range of strong DC and strong AC, and still have the overall characteristics that the CFR is low when the AC/DC capacity is balanced. So it shows that the conclusion is general to some degree and will not be easily affected by parameter selection and control strategy. V. CONCLUSION
The mathematical model of the AC/DC capacity optimal allocation problem is provided. The cascading failure Monte-Carlo simulation algorithm and the AC/DC capacity optimal allocation implicit enumeration algorithm are proposed. Simulation based on the modified IEEE39 system is performed, in which the AC/DC capacity optimal allocation which has the smallest CFR is obtained and the impact of various factors are also analyzed. The results show that there exists an ACIDC capacity optimal allocation with balanced ratio from the
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