Anti-disaster Transmission Expansion Planning Considering Wind Power Integration Using Ordinal Optimization Q. Xu, C. Kang and Q. Xia
D. He, and R. G. Harley1,
Jianhua Bai, Zhidong Wang,
Tsinghua University Beijing, China
[email protected]
Georgia Institute of Technology Atlanta, USA 1 Professor Emeritus, University of KwaZulu-Natal, South Africa
Hui Li, and Xin Tian State Grid Corporation of China(SGCC)
Abstract—With the rapid development of various renewable energy sources, a sustainable power system planning scheme is required throughout the world. For a power grid with largescale wind power integration, it becomes critical to achieve a better performance to supply electricity in disaster circumstances, such as ice-snow. This paper proposes an antidisaster transmission expansion planning model, and introduces the use of ordinal optimization theory to solve this combinatorial explosion problem. The ordinal optimization focuses on obtaining a good enough solution using a softened optimization object. Finally, the IEEE 39 benchmark bus system is studied and the results prove that the proposed method is valid and efficient. Index Terms--Transmission expansion planning; large-scale wind power integration; ordinal optimization; disaster scenario.
I. Subscript t (∙) ℎ(∙) , , , , , ,
Ω ,
NOMENCLATURE
Value of a variable at the tth time segment Unit operation cost function Start up and shut down cost function The forecasted load of nth bus The load shedding of nth bus Positive and negative reserve rates respectively Forecasted output of the kth wind farm Curtailment of the kth wind farm Binary decision variable: “1” if unit k is on at time t; “0” otherwise The maximum number of branches to be expanded on each existing route Cost to build the sth branch on branch l route using a normal type line
This work was supported by the Chinese National Science Fund for Distinguished Young Scholars(No.51325702).
978-1-4799-6415-4/14/$31.00 ©2014 IEEE
, ,
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# , # , # #, #"! ,
#"!
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$ , $ !'" # , #
Binary decision variable: “1” if the sth branch of m branches is expanded on branch l route; “0” otherwise Binary decision variable: “1” if the sth branch of m branches is a type of antidisaster; “0” otherwise Ratio of the normal type line cost to the anti-disaster type line cost Binary decision variable: “1” if the model is running in a disaster situation; “0” otherwise Set of disaster scenarios Binary decision variable: “1” if unit k starts up at time t; “0” otherwise Binary decision variable: “1” if unit k shuts down at time t; “0” otherwise Maximum start up times and shut down times of units The minimum on time of unit k Scheduled power of the kth conventional unit Technical maximum and minimum power output of the kth conventional unit The power flow on the branch l Positive and negative power flow limit of the existing branch l in the normal situation Ramp-up/ramp-down rate limit for unit k Positive and negative power flow limit of the existing branch l in the disaster
# , # Ω* , Ω+ , Ω- , Ω. , Ω/ 0 10 2 3 4 , , ,
56
situation Positive and negative power flow limit of the new-built branch in the normal situation Set of branches, time segments, wind farms, conventional units and system nodes, respectively Occurrence probability of the ith disaster in set Ω Disaster cost for the ith disaster in set Ω Element of incidence matrix for bus n and unit k Element of the shift matrix for branch l and bus n Up and down reserve as a response to the forecast error of , Weight coefficient for the objective function, x=1,2,3,4,5 II.
INTRODUCTION
In recent years, drastic progress has been made in the wind energy industry all over the world. However, because of the stochastic nature of wind power, large penetration of wind power has brought about tremendous challenges for the secure operation and reliability of power systems. For Denmark, it is expected that in 2020, half of the electricity will be supplied by wind. Meanwhile, there are always different kinds of disasters happening all over the world. Disasters always bring desperate consequences with a high loss. For example, China encountered an ice disaster in 2008, which put the major part of the power grid out of service. The direct economic loss was more than one billion dollars. What is worse, the everincreasing penetration levels of wind power, with its concomitant uncertainty, exacerbates the difficulty of the antidisaster operation of the power system. This in turn worsens the economic loss caused by loss of power. It is therefore essential for the power companies to start considering how to improve performance particularly during disasters. Careful transmission expansion planning can be an efficient and effective solution. Transmission expansion planning should aim to upgrade the existing power system in order to promise highly efficient electricity delivery in an optimal economic manner. Antidisaster transmission expansion planning (ATEP) aims at upgrading the power grid to operate normally in disaster circumstances. A common method to evaluate a transmission expansion planning scheme is to run system operation simulation all year round and quantify the operation cost and system reliability indexes. However, transmission expansion planning is usually a high dimension, nonlinear mixed integer optimization problem, because it is required to consider the influence of many relevant factors. LP [1], NLP [2], MIP [3], Benders decomposition method [4], etc., are usually applied to solve the problem of transmission expansion planning. All these methods aim to find a best planning scheme, yet the
complexity of the problem increases exponentially as the system size grows. Usually, it is impossible to obtain a globally convergent solution for an actual large power system. Therefore, the high computational requirement has become the bottleneck of all the above methods. Ordinal optimization (OO) [5-10] is a method to find a good enough solution with high probability instead of finding the best solution with an intolerable computational burden. OO could meet the requirements of a large practical power system. For a complex optimization problem, OO improves the computation efficiency greatly by softening the objective. This paper proposes an anti-disaster transmission expansion planning model and introduces an OO solution to it as well. The IEEE 39 bus benchmark system case is studied in this paper to prove that the proposed method is valid and efficient. This paper is organized as follows. Section III introduces the anti-disaster transmission expansion planning model. Section IV introduces the ordinal optimization theory. Section V presents a case study to show the efficacy of this planning method. Section VI concludes the paper and presents the future work plan. III.
ANTI-DISASTER TRANSMISSION EXPANSION PLANNING MODEL
A. Framework of ATEP model In this paper, two terms are added to the classic TEP model: disaster cost and wind power curtailment penalty. Thus there are totally five parts in the objective function. The paper establishes a disaster scenario set disaster , which consists of various common disaster situations. The “disaster cost” includes line damage cost and load shedding cost in disaster situations, and the load shedding cost is determined by a simplified model, which is composed of minimum load shedding objective and the constraints of unit output and power balance. Additionally, classic constraints are considered, such as unit maximum and minimum constraint, system reserve constraint, power flow constraint, power balance constraint, load following constraint and minimum on time constraint. For simplicity, two types of transmission branches are considered in this paper, i.e. regular type and anti-disaster type. The regular type will be out-of-service when the grid network encounters disaster situations, while the anti-disaster type can still be in-service. However, the anti-disaster type will cost more than the regular type. B. ATEP model 1) Objective function: m
min
1 psl I sl (1 rsl esl ) lL s 1
2
g( P
3
C
k,t
tT k U
tT k W
k,t
) h( Pk,t )
4
(1)
L
tT nD
n,t
5
idisaster
pi fi
In (1), ∑∈>? ∑: ;< (1 + ) represents the total ∑∈>? ∑ ∈>D @A# , B + ℎA# , BC expansion investment; represents the system operation cost using the optimal scheduling results; ∑∈>F ∑ ∈>E , represents the wind power curtailment penalty; ∑∈>F ∑∈>G , represents the load shedding penalty; and ∑0∈>HIJKJLMN 0 10 represents the disaster cost, each 10 is a disaster scenario cost, including the branch damage cost and load shedding cost. The weight coefficient 56 represents the importance of each part.
2) Constraints:
P
(W
+
k,t
kU
k,t
(D
- Ck,t )
n,t
Ln,t ), t T
(2)
niD
kW
I k ,t Pk Pk,t I k ,t Pk , k U , t T m
s 1
m
(4)
s 1
, l L , t T
g a
Pl ,t =
ln
nD
nk
kU
Pk,t
a
w nk
(Wk,t Ck,t )
(5)
k W
( Dn,t Ln,t ) , l L , t T
I
i k ,t
Pki +
k U
(1 u ) Dn,t
W
i k,t
k W
nD
Ln,t
I
k ,t
k U
IV.
(3)
Pl nor (1 I ) Pl dis I [1 I (1 esl )]I sl Ps Pl,t Pl nor (1 I ) Pl dis I [1 I (1 esl )]I sl Ps
Equation (2) is power balance constraint; equation (3) is the unit output constraint; equation (4) and (5) are the network constraints, which are determined by the planning scheme. In 03 0 04 0 , and ∑ ∈>I , , are the (6) and (7), ∑ ∈>I , E E spinning reserve prepared to supply up and down generation capacity in response to wind output forecast error. Equation (8) and (9) are ramping up/down constraints, which limit the unit output right after the start up and before shut down. Equation (10) and (11) are minimum on-time constraint and startup/shut-down frequency constraint respectively. Equation (12) and (13) limits the wind power curtailment and load shedding, respectively. It is worth noting that in (4) the power flow limitation of each branch is determined by the number of extra new-built branches and their types.
(6)
ORDINAL OPTIMIZATION THEORY
A. Introduction of Ordinal Optimization The Ordinal Optimization (OO) theory was proposed and developed by Y. C. Ho et. al. [5-10]. OO is utilized to solve complex optimization problems based on simulation and has been applied to various areas in power systems, such as optimal power flow (OPF) [10] and bidding strategies in markets [11]. In this paper, this theory is applied to solve the ATEP problem. The main idea of OO is to achieve good enough solutions of complex optimization problems by relaxing the optimization objective. OO divides all the problems into five types, i.e. Flat type, U type, Neutral type, Bell type and Steep type. They can be described by the ordinal performance curve (OPC), as is shown in Fig. 1, and are all non-decreasing curves.
k,tWk,t , t T
k W
Pk (1 down ) ( Dn,t Ln,t ) nD
(7)
k ,tWk,t , t T
k W
Pk ,t 1 Pk ,t (2 I k ,t I k ,t 1 ) Pk (1 I k ,t I k ,t 1 ) Rkup , k U , t T
Pk ,t Pk ,t 1 (2 I k ,t I k ,t 1 ) Pk (1 I k ,t I k ,t 1 ) Rkdown , k U , t T
I
k ,t
I k ,t 1 tk ,on
vk ,t 1 I
u
tT
k ,t
i k ,t
I
i k , t 1
t 1
I k , j 0, k U
j t tk ,on 1
Figure 1. Types of ordinal performance curves
(8)
(9)
tT
1)
Determine ΘP by uniformly and randomly sampling N solutions from search space;
2)
Use a crude and computationally fast model to estimate the performance of these N designs;
3)
Estimate the OPC type of the problem and the noise level of the crude model. The user specifies the size g of good enough set ΘQ , and the required alignment level, k;
(10)
uk ,t 1
u , vk ,t v , k U , t T
By calculating the OPC, the type can be determined, and a good enough solution with high probability can be further achieved with the following steps:
(11)
0 vk ,t , uk ,t , I k ,t 1
0 Ck,t Wk,t , k W , t T
(12)
4)
0 Ln,t Dn,t , n D , t T
Determine the size s of ΘR ;
(13)
5)
Select the observed top s designs of the N as estimated by the crude model as the selected set ΘR ;
6)
The theory of OO ensures that ΘR contains at least k truly good enough designs with probability no less than the appointed level 0.95.
B. Apply OO theory to the ATEP model Usually a crude ATEP model, which is denoted as the CATEP model in this paper, is required to be established firstly to solve an OO problem. Thus, the proposed ATEP model above can be denoted as the precise ATEP model (P-ATEP). For P-ATEP, the system simulation will be 365 days long. In order to reduce calculation, one typical day is selected in each month to represent the whole month, and the operation cost of each month equals the cost of the selected day multiplied by the day number of the month. Hence, there are totally 12 typical days, and the operation cost of the whole year can be roughly evaluated by running a small number of system operation simulations. V.
CASE STUDY
In this section, a case study is presented to verify the efficiency of the proposed model. This paper uses the IEEE 39-bus test system, shown in Fig. 2 as an example. All parameters are taken from Matpower 4.0 [12] and are shown in Table I. The total capacity of ten conventional units is 7367 MW, and five wind farms are added to the system, which are located at bus 1 (100 MW), bus 2 (200 MW), bus 3 (100 MW), bus 25 (100 MW) and bus 28 (150 MW). Therefore, the total wind power capacity accounts for 8.1% of the total system capacity.
is set to be 300 MVA in a normal situation and 100 MVA in the ice disaster situation. The newly built branches of regular type will be out-of-service in the ice disaster situation. In addition, it is assumed that a half of load under each bus could be shut down in the ice disaster situation. TABLE I.
BASIC PARAMETERS OF THE MODEL Type
Value
1.5
($)
100,000
m
2
SSS #
Unit capacity
#
50% of unit capacity
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100 20
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1
5
