An online line switching methodology with look-ahead capability to alleviate power system overloads based on a three-stage strategy

Electrical Power and Energy Systems 115 (2020) 105500

Contents lists available at ScienceDirect

Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

An online line switching methodology with look-ahead capability to alleviate power system overloads based on a three-stage strategy ⁎

T

⁎

Zhengwei Shena, , Hsiao-Dong Chiangb, , Yong Tangc, Ning Zhoud a

School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA c State Key Laboratory of Power Grid Safety and Energy Conservation, China Electric Power Research Institute, Haidian District, Beijing 100192, China d Department of Electrical and Computer Engineering, State University of New York at Binghamton, Binghamton, NY 13091, USA b

A R T I C LE I N FO

A B S T R A C T

Keywords: Line switching Look-ahead capability Overloads Three-stage strategy

An online line switching methodology to alleviate line overloads with look-ahead capability is proposed in this paper. This novel online methodology is based on a three-stage strategy, including screening, ranking, and detailed analysis and assessment stages for fast speed (online application) and accuracy. The proposed online methodology performs the tasks of rapidly identifying effective candidate lines, ranking the effective candidates, detailed analysis of the top ranked candidates, and supplying a set of effective line switching solutions for a current operating point. Both the current and the look-ahead post-switching power systems, after executing the proposed line switching action, meet the operational and engineering constraints. One distinguishing feature of the proposed methodology is that it provides a set of high-quality line switching solutions. The results provided by the exact AC power flow are used as a benchmark to compare the accuracy of the proposed three-stage methodology. The effectiveness and fast speed of the proposed line switching methodology with look-ahead capability are evaluated on the IEEE 39-bus and 2746-bus power systems with promising results.

1. Introduction Overloads and voltage violation problems can seriously affect the stable operations of power systems. Bus-bar splitting, load shedding, generation rescheduling, and line switching are effective actions to relieve overload problems. It is well known that switching transmission lines can change the state of the power system. It may change the distribution of power flows, short circuit currents, transmission losses, transient stability, voltage profiles, and the voltage stability of power systems. There is no additional cost required since line switching is just an operational action with minimal impact on generation and load demands. It was found that applying line switching to an IEEE 118-bus test system case results in a 24.9% cost savings as compared with generation dispatch [1]. In addition to economic benefits, line switching can be an effective measure for enhancing power system operations and stability compared [2,3]. Compared with other methods, line switching has been the usual practice and offers savings in both cost and time [2–5]. Line switching can be utilized as a corrective mechanism to relieve line overloads, voltage violations, reduce line losses, and in OPF

⁎

studies, see for example [7–10,14–18]. Ruiz & Foster [9,10] showed that alternative methods based on sensitivity analysis have been very successful in reducing the computational times in an OPF setting. One of the pioneer work on line switching was presented in [11]. A scheme of corrective line switching measures to provide a feasible load flows for both the base and contingency cases was proposed in [12]. A busbar splitting scheme for preventing transmission overloading and reducing load curtailment was proposed in [13]. Line switching can also be utilized to enhance power system dynamical behaviors such as enhancing small-signal stability [19], enhancing transient stability [20,21], enhancing static voltage stability margin [4,6]. The incorporation of line switching into unit commitment is a challenging task, see for example [22]. The incorporation of line switching action in the real-time contingency analysis was studied in [23]. A new multi-period formulation for static security-constrained transmission switching was proposed in [24]. The effectiveness of line switching depends on the selection of lines to be switched off. However, there is a gap in the design of effective line switching for online applications in large-scale power systems. Hence, this paper seeks to design an effective line switching measure to relieve the overloads of large power systems in an online manner to execute the

Corresponding authors. E-mail addresses: [email protected] (Z. Shen), [email protected] (H.-D. Chiang).

https://doi.org/10.1016/j.ijepes.2019.105500 Received 3 April 2019; Received in revised form 20 July 2019; Accepted 19 August 2019 Available online 27 August 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

effectiveness of the line switching solutions. Define ΔPi and ΔQi as the vectors of active and reactive loads and the generation variation between the look-ahead vector and the current vector, respectively. Mathematically, we propose the following line switching problem formulation:

designed measure in online mode. In order to further improve the speed and accuracy, a novel online line switching methodology with lookahead capability is proposed to relieve overloads in this paper. Instead of dealing with the combinatorial character of OTS, the proposed methodology combines linear and nonlinear methods to achieve the goal of online application to relieve overloads. The proposed methodology employs a three-stage strategy: (i) a screening stage, (ii) a ranking stage, and (iii) a detailed analysis and assessment stage for fast and accurate results. The screening stage of the proposed methodology is designed for the quick selection of effective lines (to be switched) from a list of credible candidate lines whose disconnection leads to relieving overloads. The ranking stage ranks these effective candidates from the list of candidate lines selected at the screening stage. The detailed analysis and assessment stage performs a detailed evaluation of the several top candidates from the ranking stage to assess multiple high-quality line switching solutions that can relieve overloads for the current power systems, while both the current and the look-ahead postswitching power systems satisfy the operational and engineering constraints. An online line switching method based on a three-stage strategy to relieve overloads was proposed in [25]. This paper extends the method proposed in [25] to improve both the speed and quality of the solutions. Distinguishing features of the proposed methodology are summarized as follows:

min Num ( )

subject to: (i) Power flow equations for λ = 0 and λ = 1: n

Pi + λ ΔPi = Vi (λ )

∑ Vj (λ)(Gij cosθij (λ ) + Bij sinθij (λ )) (3)

j=1 n

Qi + λ ΔQi = Vi (λ )

∑ Vj (λ)(Gij sinθij (λ ) − Bij cosθij (λ )) (4)

j=1

(ii) Operating constraints for λ = 0 and λ = 1:

|Sij (λ )| ≤ Sijmax

i, j ∈ Ω;

Vimin ≤ Vi (λ ) ≤ Vimax

i ∈ Ω;

(5-a) (5-b)

where Pi and Qi are the active and reactive power injections of bus i, respectively; Vi and Vj are the voltage magnitudes of buses i and j, respectively; and θij (λ ) is the difference in the voltage angles between buses i and j when λ = 0 and λ = 1, respectively. Ω = [1, 2, 3⋯,n] is the set of buses of the power system. Eqs. (3) and (4) are the parameterized power flow constraints that represent the current and 30-minute lookahead post-switching power systems when λ = 0 and λ = 1, respectively. Sijmax is the line apparent power flow limit of line i-j. Vimin and Vimax are the minimum and maximum voltage magnitudes at bus i, respectively. Sij (λ ) and Vi (λ ) denote the apparent power flow on line i-j and the voltage magnitude at bus i of the current and look-ahead postswitching power systems when λ = 0 and λ = 1, respectively. The objective function (2) seeks an effective network topology  for the current power system such that overloading is relieved with a minimum number of lines switched off (for minimum economic cost). In addition, there will be no violation of the current and look-ahead post-switching power systems. The current and look-ahead postswitching power systems must be satisfied with the constraints of power flow Eqs. (3) and (4) and operational and engineering constraints (5-a) and (5-b).

(1) The proposed methodology can find multiple high-quality line switching solutions, which may include the ‘best’ solution, to relieve overloads in an online manner and ensure the effectiveness of the line switching solution for the look-ahead power system. Actually, the ‘best’ here does not necessarily imply the global optimal solution; it may represent high-quality local optimal solutions. (2) The proposed methodology is fast and suitable for online application and can be effective for large-scale systems. The effectiveness of the proposed online line switching methodology is evaluated on the IEEE 39-bus and 2746-bus power systems with various loading conditions in MATLAB. 2. Problem formulations We consider a comprehensive power system quasi-steady-state model of the following general form:

fb (x , λ ) = f (x ) + λb = 0

(2)

 ∈N

3. Solution methodology

(1)

where fb (·) denotes the active and reactive power flow balance equations representing current and look-ahead base-case power systems when λ = 0 and λ = 1, respectively. x is the vector of state variables, b = b1 − b0 represents the power injection variation at each bus, and b0 and b1 are the vectors of power injections for the current power system and the look-ahead power system, respectively. Suppose the online state estimation and the corresponding real-time power flow calculation indicate that overloads have occurred. Under this circumstance, operators need to take immediate control to remove the overloading condition and ensure that no other static violations such as overloading and voltage problems exist. Assume that the operators choose line switching as the control action. The network topology of a power grid is altered after line switching. Given the current network topology 0 , we seek to identify a highquality network topology  ∈ N (i.e., post-switching case) to relieve overloads by switching off lines for the current power system. We let Num ( − 0 ) represent the number of line switches needed to alter the current network topology 0 to the high-quality network topology  . Taking into account the cyclical changes in short-term load levels, the quality of the solution can be further improved. Therefore, we look ahead 30 min for the post-switching power systems to assess the

The proposed methodology employs a three-stage strategy that contains screening, ranking, and detailed analysis and assessment. The solution methodology utilized in each stage is presented. Stage 1 (screening) utilizes a sensitivity-based method to achieve the goal of rapid screening. Stage 2 (ranking) utilizes the DC power flow method to achieve the goal of fast and accurate ranking, while Stage 3 (detailed analysis and assessment) utilizes AC power flow to assess line switching solutions for the current power system, and detailed analysis to assess the effectiveness of these solutions for the look-ahead post-switching power systems. The architecture of the proposed online line switching methodology with look-ahead capability is displayed in Fig. 1. 3.1. Stage 1: Screening This stage performs the task of identifying candidate lines whose disconnection may lead to transmission overload reduction. In this stage, a linear method is employed to rapidly estimate the variation in line overloading of the power system due to the switching out action of each candidate [26]. Assume that the network has n buses and b branches. We can relieve overloads by switching line k-m when line i-j is overloading. Switching 2

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

Fig. 1. Architecture of the proposed three-stage solution methodology.

Then Eq. (7) can be rewritten as

out a transmission line changes the current system operating state and the power flow of the post-switching power system. The real power flow on line i-j can be rewritten as

Pij = Vi Vj Yijcos (δij + ωj − ωi ) − Vi2 Yij cosδij

βij − k = aij mik + bij mjk

Define the sensitivity impact of switching line k-m impact on the overloading line i-j as follows

(6)

where ωi and ωj are the angles of buses i and j, respectively; and Yij and δij are the magnitude and angle of the ijth element of the Y matrix of the bus. Define the impact factor βij − k as the change in the real power flow (ΔPij ) of line i-j due to unit change in the power injection (ΔPk ) at any bus k. Mathematically, βij − k can be written as

βij − k =

βij − km = ξ1 βij − k + ξ2 βij − m

(7)

By using the Taylor series approximation, (6) can be written as (ignoring second and higher order terms)

ΔPij =

∂Pij ∂ωi

Δωi +

∂Pij ∂ωj

Δωj +

∂Pij ∂Vi

ΔVi +

∂Pij ∂Vj

ΔVj

(18)

where ξ1 and ξ2 are the weight coefficients, and ξ1 + ξ2 = 1 (0 ≤ ξ1 ≤ 1, 0 ≤ ξ2 ≤ 1). The virtual power injections at buses k and m can be used to describe the line k-m switched off, as shown in Fig. 2. The screening stage uses the sensitivity formula (17) to calculate βij − k for all buses in the system to rapidly identify effective candidate lines based on Eq. (18) (except for the overloaded line). Considering the error it may incur, all of the candidates with |βij − km | > ε (ε is a predefined value from 0 to 1) are captured, and sent to Stage 2 for further ranking their effectiveness. This may increase the workload of Stage 2 to some extent, but will help to ensure the quality of the solutions.

ΔPij ΔPk

(17)

(8)

In this paper, we have rewritten Eq. (8) as

ΔPij = aij Δωi + bij Δωj + cij ΔVi + dij ΔVj

(9)

The coefficients appearing in (9) can be obtained using the partial derivatives of real power flow (6) with respect to variables ω and V as

aij = Vi Vj Yijsin (δij + ωj − ωi )

(10a)

bij = −Vi Vj Yijsin (δij + ωj − ωi )

(10b)

cij = Vj Yijcos (δij + ωj − ωi ) − 2Vi Yij cosδij

(10c)

dij = Vi Yijcos (δij + ωj − ωi )

(10d)

To get βij − k , the following Jacobian relationship has been used:

J J ⎡ ΔP ⎤ = [J ] ⎡ Δω ⎤ = ⎡ 11 12 ⎤ ⎡ Δω ⎤ ⎣ ΔV ⎦ ⎣ J21 J22 ⎦ ⎣ ΔV ⎦ ⎣ ΔQ ⎦

(11)

Neglecting P − V coupling, Eq. (11) can be simplified as (12)

ΔP = [J11 ][Δω] Then we have

Δω = [J11]−1 [ΔP ] = [M ][ΔP ]

(13)

Eq. (13) can be rewritten as follows: n

Δωi =

∑ mik ΔPk

i ∈ Ω. (14)

k=1

And Eq. (9) can be approximated as

ΔPij = aij Δωi + bij Δωj

(15)

Combining Eqs. (14) and (15), we get n

ΔPij = aij

n

∑ mik ΔPk + bij ∑ mjk ΔPk k=1

k=1

(16)

Fig. 2. Line k-m switched off and modeled by virtual power injections. 3

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

Δθij = −Cψlk θkm

3.2. Stage 2: Ranking

Thus, the real power flow variation on line i-j caused by the switched out line k-m can be expressed as

This stage performs the task of ranking the effectiveness of each candidate line identified at Stage 1. Since the DC power flow model is adequate for evaluating real power problems, the DC power flow method is employed in Stage 2 to estimate the power flow of the overloading line with each candidate line switched out and to rank the effectiveness of each candidate line based on the calculated postswitching power flow. With line k-m switched out, the reactance matrix variation ΔX post can be obtained by the following:

ΔX post = (B post )−1 − (B pre )−1

ΔPij − km = bij Δθij =

= −CX k (M k )T (B post )−1

3.3. Stage 3: Detailed analysis and assessment To perform detailed analysis of the several top candidates ranked at Stage 2, we apply AC power flow studies in this stage to compute the exact post-switching power flow. There are no science-based rules regarding how many top candidates should be picked in Stage 3. Our rule of thumb is 5-10 top candidates for a small system and 10-20 top candidates for a large-scale system. The optimal network topology of the post-switching power system, along with the required action of line switching, are assessed based on the exact calculation of the AC power flow. Stage 3 performs detailed assessment with a focus on accuracy instead of speed. While using active power can greatly improve efficiency, the apparent power is chosen in Stage 3 for accuracy since the thermal limits are related to apparent power. Thus, in this stage, we define the performance index NAM as follows:

(20)

where

(M k )T = [0, …, 1, …, −1, …, 0] ↑ ↑ k m

(21)

(B pre )−1M k

(22) −1

1 C=⎛ + (M k )T X k ⎞ ⎝ Δbkm ⎠ ⎜

⎟

NAM =

where Δbkm is the susceptance variation of line k-m. Subject to

then

Δbkm 1 + Δbkm ψkk

Combing the above equations, lowing:

(23)

ΔX post

can be obtained by the fol-

ΔX post = −CX k (X k )T

Sij _max − Sij _post Sij _max

∗ 100

(31)

where Sij _max and Sij _post are the maximum and actual power flow of line i-j with line k-m switched out. NAM represents the normalized apparent power margin of the overloading line with line k-m switched off. By using (31), each line switching candidate is assessed. To further assess the effectiveness of the proposed switching scheme, we look-ahead 30 min for the post-switching power systems and analyze its effectiveness for the line switching solutions. If violations still exist with a candidate solution, we will eliminate the candidate solution from the line switching solution list and reorder the remaining solutions by using (31), with Sij _post as the actual power flow of line i-j with line k-m switched out of the look-ahead post-switching power system under these circumstances. Then effective line switching solutions are assessed.

(M k )T X k = ψkk

C=

(30)

(19)

ΔX post = ((B pre )−1 − CX k (M k )T (B pre )−1) − (B pre )−1

=

−bij ψlk Δbkm Pkm 1 + Δbkm ψkk bkm

where Pkm is the real power flow of line k-m. We calculate the alleviation contribution ΔPij − km of each line on the violated line using (30) and rank them in order.

where B pre is the susceptance matrix of the base case, and B post is the susceptance matrix with line k-m switched out. Hence, ΔX post can be written as [27,28]

Xk

(29)

(24)

The DC power flow equations of the base case can be written as

P = Bθ 4. The overall solution methodology

where P , B , and θ are the real power injections, the susceptance matrix, and the voltage phase angles, respectively. When line k-m has been switched out, the voltage phase angle variation can be expressed as

Δθ post = (B post )−1P − (B pre )−1P

A step-by-step description of the proposed three-stage line switching methodology for online application with look-ahead capability is summarized as follows and shown in Fig. 3.

(25)

The voltage phase angle of line i-j can be written as

Δθij = Δθi − Δθj = (M l )T Δθ post

Step 1: Input the online data, including the generation schedule, load demands, state estimation, network topology, and candidate lines, for online line switching action. Step 2: Run the AC power flow according to the given operating state. If overloads exist, go to step 3; otherwise, stop and output the base-case assessment results. Step 3: Generate candidate lines and go to step 4. Step 4: Identify effective lines by (18). Step 5: Find effective candidate lines, and then send them to step 6 for ranking. Otherwise, increase one more switching lines by one and go to step 3. Step 6: Rank all the effective lines in order by (30), then select several top lines for the next stage. Step 7: Run the AC power flow to compute the post-switching power flow corresponding to each top-ranked line switching, and provide the ordered line switching solutions by (31).

(26)

where

(M ι )T = [0, …, 1, …, −1, …, 0] ↑ ↑ i j

(27)

Hence, Δθij can be expressed as:

Δθij = (M l )T ΔX postP = −C ((M l )T X k )((M k )T θ)

(28)

Subject to

(M l )T X k

= ψlk

(M k )T θ = θk − θm ≜ θkm Eq. (28) can be rewritten as 4

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

Table 1 Results for a 39-bus test system in relieving a single-line overload by switching out a single line (Stages 1 and 2). Line switching methodology Effective candidates 1-2 2-25 4-5 4-14 5-8 6-7

6-11 7-8 8-9 10-11 10-13 13-14

14-15 2-3 16-17 17-27 26-27 3-18

Highly ranked candidates

P5 − 6 (MW)

5-8 6-11 4-5 10-11 /

235.26 262.88 377.00 348.74

methodology. Furthermore, the speed of the proposed method is compared with other methods. 5.1. Single line switching The IEEE 39-bus system has 46 branches, with the maximal power flow of line 5-6 being 450 MVA and 900 MVA for the other transmission lines. The power flow was run at current operating conditions, and an overloading was found on line 5-6. By applying the proposed methodology, several solutions are assessed to relieve the overloading of line 5-6. The screening and ranking results are shown in Table 1, the line switching solutions for the current and look-ahead post-switching power systems of Stage 3 are shown in Table 2, and the CPU time for this example is displayed in Table 3. We have the following observations from the results:

• Stage 1: By using β • • •

Fig. 3. Flow chart of the proposed methodology.

Step 8: Assess the effectiveness of the line switching solutions based on a 30-minute look-ahead for the post-switching power system. Step 9: Output the ordered high-quality solutions effective switching lines are found; otherwise, utilizes other control actions to relieve overloading.

ij − km , 21 effective candidate lines were identified from 45 candidate lines in which 18 line candidates are listed in the Table 1. Stage 2: The ΔPij − km of each effective candidate from stage 1 is calculated in order to select the top 4 candidates and rank them in order: lines 5-8, 6-11, 4-5, and 10-11. Stage 3: (i) AC power flow is used to check for any overloads at the current operating point with the top 4 lines switched out individually. Then lines 5-8, 6-11, 10-11, and 4-5 are assessed to relieve overload on line 5-6 for the current power system. Stage 3: (ii) To further analyze the effectiveness of the line switching solutions, we look-ahead 30 min for the post-switching power systems. Then lines 5-8, 6-11, 10-11, and 4-5 are obtained to relieve overload for the 30-minute look-ahead post-switching power systems (λ = 1) . With each of the four lines switched out, the power flow on line 5-6 is 235.48MVA (NAM = 47.67), 266.09MVA (NAM = 40.87), 349.74MVA (NAM = 22.28), and 380.09MVA (NAM = 15.54) for the current power system, respectively, the load margins are 1013.2 MW, 1030.5 MW, 1030.1 MW, 1064.7 MW for the post-switching power systems, respectively.

Table 2 Results for a 39-bus test system in relieving a single-line overload by switching out a single line (Stage 3). Stage 3 Detailed analysis and assessment

λ=0

5. Numerical schemes In this section, the proposed online line switching methodology with look-ahead capability is applied to the IEEE 39-bus and 2746-bus power systems to validate its effectiveness and accuracy. The proposed methodology was implemented in MATLAB 2014a on a ThinkPad PC with Intel Core 2.50-GHz i5-7200U CPU and 8 GB of memory. The results provided by the exact AC power flow are used as a benchmark to compare the accuracy of the proposed three-stage

λ=1

Top candidates

NAM

S5 − 6 (MVA)

Top candidates

NAM

S5 − 6 (MVA)

5-8 6-11 10-11 4-5

47.67 40.87 22.28 15.54

235.48 266.09 349.74 380.09

5-8 6-11 10-11 4-5

41.80 33.38 13.48 6.82

261.88 299.33 389.34 419.33

“λ = 0 ” represents the current power systems. “λ = 1” represents the look-ahead post-switching power systems. 5

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

Table 3 CPU time required for the case of single line switching (seconds). The proposed methodology

The method in reference [8]

Table 5 Results for a 39-bus test system in relieving a single-line overload by switching out a multiple-line (Stages 1 and 2).

Speed-up

Line switching methodology Stage 1

Stage 2

Stage 3

Total

0.0193

0.0007

0.7543

0.7743

Effective candidates 1.6972

To evaluate the accuracy of the proposed methodology, all 45 candidate lines are switched out individually, then lines 5-8, 6-11, 1011, and 4-5 are identified to relieve overload by using AC power flow. With the four lines switched out individually, the power flow on line 56 is 235.48MVA, 266.09MVA, 349.74MVA, and 380.09MVA, respectively. It should be emphasized that this is the same as the solutions assessed by the proposed methodology. Even though this work is focused on the thermal limit constraint, operational considerations related to this constraint should be taken into account, such as the cost of implementing a line switching, and the impact of line switching on voltage stability, transient stability, or small-signal stability. Hence, a final decision should involve other relevant operational considerations. The total CPU time of the proposed methodology in this study is 0.7743 s, whereas the method from reference [8] takes 1.6972 s. The speed-up of 54.38% is compared with reference [8] method. The above results illustrate the effectiveness and fast speed of the proposed three-stage methodology. We next evaluate the robustness of the proposed methodology to relieve overloads at various loading conditions. If all loads are increased by 10%, then we have S5 − 6= 597.01MVA, which demonstrates that approximately a 20.61% overload has occurred on line 5-6. By using the proposed methodology, we found four line switching solutions to relieve the overload for the current power system, then analyzed their effectiveness for look-ahead post-switching power systems. The effective line switching solutions are lines 5-8, 611, 10-11, 4-5, the power flow on line 5-6 is 261.88MVA (NAM = 47.09), 299.33MVA (NAM = 39.53), 389.34MVA (NAM = 21.35), 419.33MVA (NAM = 15.29) for the current postswitching power system, respectively, as summarized in Table 4. The load margins are 830.1 MW, 846.0 MW, 845.5 MW, 876.9 MW for the post-switching power system, respectively.

P5 − 6 (MW)

4-5 6-11 6-11 21-22 4-5 28-29 4-5 15-16 6-11 15-16 /

678.19

NAM

S5 − 6 (MVA)

5-8 6-11 10-11 4-5

47.09 39.53 21.35 15.29

261.88 299.33 389.34 419.33

5-8 6-11 10-11 4-5

41.02 31.81 12.28 6.50

291.96 337.52 434.22 462.82

10-11

3-18

6-11

16-24

21-22

10-13

17-18

28-29

2-3

7-8

8-9

23-24

13-14

15-16

697.19 804.89 760.51 683.31

λ=1

Top candidates

NAM

S5 − 6 (MVA)

Top candidates

NAM

S5 − 6 (MVA)

4-5

23.06

692.44

14.42

770.20

21.89

702.96

12.86

784.25

20.84

712.48

11.41

797.32

13.93

774.62

4.87

856.19

8.68

821.86

4-5 6-11 6-11 15-16 6-11 21-22 4-5 28-29 /

Table 7 CPU time required for the case of multiple-line switching (seconds). The proposed methodology Stage 1

Stage 2

Stage 3

Total

0.0203

0.0009

0.7656

0.7868

The method in reference [8]

Speed-up

2.1789

63.89%

The above results illustrate the effectiveness and fast speed of the proposed methodology for multiple-line switching solutions.

• 24 candidates are identified from 44 candidate lines at Stage 1, and

• •

λ=1 Top candidates

4-5

6-11 6-11 15-16 6-11 21-22 4-5 15-16 4-5 28-29

Stage 3 Detailed analysis and assessment

S5 − 6 (MVA)

4-14

λ=0

Table 4 Results for a 39-bus test system in relieving a single-line overload by switching out a single line (loads increased).

NAM

5-8

Stage 3 Detailed analysis and assessment

In this IEEE 39-bus system, the maximal power flow of each line has been 900 MVA. The outage of line 6-7 will cause an 8.22% overload on line 5-6. If switching off a single line cannot effectively relieve the overloading, we add one more switched line, then utilize the proposed methodology to provide a set of multiple-line switching solutions. The obtained solutions are summarized in Tables 5 and 6, the CPU time is displayed in Table 7. We have the following observations from the results:

Top candidates

2-25

Table 6 Results for a 39-bus test system in relieving a single-line overload by switching out a multiple line (Stage 3).

5.2. Multiple-line switching

λ=0

Highly ranked candidates

54.38%

6

then sent to stage 2 for ranking. Due to space limitations, Table 5 displays 18 effective line candidates in Stage 1. At Stage 2, the ΔPij − km of each line (24 effective candidates from stage 1) is calculated to select the top 5 lines, which are then combined in pairs. Ultimately, the top 5 multiple-line switching solutions are selected to relieve overload by calculating ΔPij − km again. They will be sent to Stage 3 for detailed analysis and assessment. Lines 4-5 and 6-11, lines 6-11 and 15-16, lines 6-11 and 21-22, lines 4-5 and 15-16, lines 4-5 and 28-29 are assessed to relieve overload for the current power system by using AC power flow at stage 3. To further analyze the effectiveness of the line switching solutions, we employ a 30-minute look-ahead for the post-switching power systems and find that, with lines 4-5 and 15-16 switched off, new overloads can occur. Thus, effective line switching solutions are obtained to relieve overload: lines 4-5 and 6-11, lines 6-11 and 1516, lines 6-11 and 21-22, lines 4-5 and 28-29. For each of the successful line switching solutions, the power flow on line 5-6 is 692.44MVA (NAM = 23.06), 702.96MVA (NAM = 21.89), 712.48MVA (NAM = 20.84), 821.86MVA (NAM = 8.68) of the

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

current power system, respectively. The load margins are 724.4 MW, 606.4 MW, 761.0 MW, 622.7 MW for the post-switching power systems, respectively.

Table 9 Results for a 39-bus test system in relieving a multiple-line overload by switching out a single line (Stage 3). Stage 3 Detailed analysis and assessment

To verify the effectiveness of the obtained solutions, we perform an exhaustive search with all 44 candidate lines combined in pairs, then switched out. Switching solutions 4-5 and 6-11, lines 6-11 and 21-22, lines 4-5 and 28-29, lines 6-11 and 15-16, lines 6-11 and 8-9, lines 8-9 and 4-5 are obtained to relieve overload by using AC power flow. Due to the errors in stage 1, some solutions may be lost. Although the exhaustive search method produces two solutions that the proposed method does not find, the effective line switching solutions obtained by the proposed methodology still can relieve overload in this case. The CPU time of the proposed methodology in this case is 0.7868 s, whereas the method from reference [8] takes 2.1789 s. The speed-up is 63.89%.

λ=0

λ=1

Top candidates

NAM

S5 − 6 (MVA) S5 − 8 (MVA)

Top candidates

NAM

S5 − 6 (MVA) S5 − 8 (MVA)

6-11

23.41 2.26

689.30 747.73

6-11

20.83 1.01

712.52 757.24

Table 10 CPU time required for the case of relieving multiple-line overloads (seconds). The proposed methodology Stage 1

Stage 2

Stage 3

Total

0.0187

0.0006

0.3401

0.3594

The method in reference [8]

Speed-up

1.2195

70.53%

5.3. Relieving multiple-line overloads In this IEEE 39-bus system, the maximal power flow of line 5-8 is 765 MVA with the other lines being 900 MVA. An outage of line 6-7 will cause an 8.22% overload on line 5-6 and a 1.55% overload on line 5-8. The online line switching results are displayed in Tables 8 and 9, and the CPU time requested by the proposed approach is displayed in Table 10. At stage 1, βij − km is used to identified effective candidate lines whose disconnection may relieve the overloading of lines 5-6 and 5-8, respectively. The sets of effective candidates to relieve lines 5-6 and 5-8 are defined as C5-6 and C5-8 at stage 1, respectively. Set Cm is the joint of sets C5-6 and C5-8, which are the final effective candidate lines to relieve overloading on both lines 5-6 and 5-8 at the same time at Stage 1. There are 8 candidates identified from a total of 43 candidate lines at stage 1. The top 2 candidates (6-11 and 4-14) are selected from stage 2 and sent to stage 3 for detailed analysis and assessment. Line 6-11 is identified to relieve overload for the current power system at Stage 3, and its effectiveness on a 30-minute look-ahead post-switching power system is verified. With line 6-11 switched out, the power flow on lines 5-6 and 58 is 689.30MVA (NAM = 23.41) and 747.73MVA (NAM = 2.26) of the current power system, respectively. The load margin is 3435.6 MW for the post-switching power system. In comparison, we perform an exhaustive search where all 43 candidate lines are switched out individually. Line 6-11 is then obtained to relieve the overload by using AC power flow. The solutions obtained by the proposed methodology are consistent with the exhaustive search approach. The CPU time of the proposed methodology for this design is 0.3594 s, whereas the method from reference [8] takes 1.2195 s. The speed-up is 70.53%. The results showing the effectiveness and fast speed of the proposed methodology for multiple-line overloads.

Table 11 Results for the 2746-bus test system in relieving a single-line overload by switching out a single line (Stages 1 and 2). Line switching methodology Effective candidates 8-14 14-29 190-201 185-151 207-168 235-237

461-475 780-622 268-397 186-192 410-25 45-9

533-268 578-240 431-410 25-192 53-45 533-462

Highly ranked candidates

P28 − 25 (MW)

11-4 410-397 134-20 431-368 1932-416 446-416

25-192 410-25 186-192 410-397 53-45 431-410

Table 12 Results for the 2746-bus test system in relieving a single-line overload by switching out a single line (Stage 3). Stage 3 Detailed analysis and assessment

λ=0

λ=1

Top candidates

NAM

S5 − 6 (MVA)

Top candidates

NAM

S5 − 6 (MVA)

25-192 410-25

26.59 5.05

440.44 569.71

25-192 /

18.28

490.35

Table 13 CPU time required for the case of 2746-bus system (seconds). The proposed methodology

Table 8 Results for a 39-bus test system in relieving a multiple-line overload by switching out a single line (Stages 1 and 2).

Stage 1

Stage 2

Stage 3

Total

1.3320

1.8356

5.0187

8.1863

The method in reference [8]

Speed-up

73.5131

88.86%

5.4. The 2746-bus system

Line switching methodology Effective candidates

Highly ranked candidates

P5 − 6 (MW) P5 − 8 (MW)

3-4 4-5 4-14 6-11 2-3 23-24 2-25 8-9

6-11

677.57 734.43 1148.3 761.64

4-14

The proposed online methodology for line switching was evaluated on a 2746-bus power system containing 3514 transmission lines with the thermal limit on line 28-25 being 600MVA and the other lines being 700 MVA. The line switching solutions obtained after applying the proposed methodology are summarized in Tables 11–13:

/

• Stage 1: There are 103 candidates identified from 2835 candidate lines.

7

Electrical Power and Energy Systems 115 (2020) 105500

Z. Shen, et al.

• Stage 2: The 103 candidates are ranked, then the top 6 candidates • •

Declaration of Competing Interest

are selected for detailed analysis and assessment to be performed at stage 3: lines 25-192, 410-25, 186-192, 410-397, 53-45, and 431410. Stage 3: (i) For each top candidate line, AC power flow is performed to assess the effectiveness of each candidate. Consequently, lines 25192 and 410-25 are assessed to be effective for relieving overload for the current power system. Stage 3: (ii) Analysis of the look-ahead post-switching power system (i.e., switching line 410-25) found a new overload. Thus, the line switching solution (25-192) is effective and the corresponding power flow of line 28-25 is 440.44MVA (NAM = 5.05) with the load margin being 1185.7 MW of the post-switching power system.

The authors declared that there is no conflict of interest. Acknowledgments This work was supported by the National Key Research and Development Program of China [grant number 2016YFB0900602]; and the National Science Foundation (USA) {grant number 1508986]. References [1] Fisher EB, O’Neill RP, Ferris MC. Optimal transmission switching. IEEE Trans Power Syst Aug. 2008;23(3):1346–55. [2] Hedman WK, Oren SS, O’Neil PR. A review of transmission switching and network topology optimization. Proc IEEE Power Energy Soc General 2011:1–7. [3] Rolim JG, Machado LJB. A study of the use of corrective switching in transmission systems. IEEE Trans Power Syst Feb. 1999;14(1):336–41. [4] Wang L, Chiang HD. Toward online bus-bar splitting for increasing load margins to static stability limit. IEEE Trans Power Syst Sep. 2017;32(5):3715–25. [5] Henneaux P, Kirschen DS. Probabilistic security analysis of optimal transmission switching. IEEE Trans Power Syst Jan. 2016;31(1):508–17. [6] Wang L, Chiang HD. Toward online line switching for increasing load margins to static stability limit. IEEE Trans Power Syst Jul. 2015;31(3):1744–51. [7] Shao W, Vittal V. Corrective switching algorithm for relieving overloads and voltage violations. IEEE Trans Power Syst Oct. 2005;20(4):1877–85. [8] Shao W, Vittal V. BIP-based OPF for line and bus-bar switching to relieve overloads and voltage violations. In: Proc IEEE power systems and exposition; 2006. p. 2090–5. [9] Ruiz PA, Foster JM, Rudkevich A, Caramanis MC. Tractable transmission topology control using sensitivity analysis. IEEE Trans Power Syst Aug. 2012;27(3):1550–9. [10] Ruiz PA, Foster JM, Rudkevich A, Caramanis MC. On fast transmission topology control heuristics. In: Proc IEEE power and energy society general; Jul. 2011. p. 1–9. [11] Bacher R, Glavitsch H. Network topology optimization with security constraints. IEEE Trans Power Syst Nov. 1986;1(4):103–11. [12] Li M, Luh PB, Michel LD, Zhao Q, Luo X. Corrective line switching with security constraints for the base and contingency cases. IEEE Trans Power Syst Aug. 2008;23(3):125–33. [13] Mazi AA, Wollenberg BF, Hesse MH. Corrective control of power system flows by line and bus-bar switching. IEEE Trans Power Syst Aug. 1986;1(3):258–64. [14] Khanabadi M, Ghasemi H, Doostizadeh M. Optimal transmission switching considering voltage security and N-1 contingency analysis. IEEE Trans Power Syst Feb. 2013;28(1):542–50. [15] Wrubel JN, Rapcienski PS, Lee KL, Gisin BS, Woodzell GW. Practical experience with corrective switching algorithm for on-line applications. IEEE Trans Power Syst Feb. 1996;11(1):415–21. [16] Hedman KW, O’Neil RP, Fisher EB, Oren SS. Optimal transmission switching with contingency analysis. IEEE Trans Power Syst Aug. 2009;24(3):1577–8. [17] Hedman KW, O’Neil RP, Fisher EB, Oren SS. Optimal transmission switching-sensitivity analysis and extensions. IEEE Trans Power Syst Aug. 2008;23(3):1469–79. [18] Hou LR, Chiang HD. Toward online line switching method for reducing transmission loss in power systems. Proc IEEE Power Energy Society General Nov. 2016:1–5. [19] Li C, Chiang HD, Du Z. Investigation of an effective strategy for computing smallsignal security margins. IEEE Trans Power Syst Sep. 2018;33(5):5437–45. [20] Mak TWK, Hentenryck PV, Hiskens IA. A nonlinear optimization model for transient stable line switching. Am Control Conference (ACC) May 2017:2085–92. [21] Owusu-Mireku R, Chiang H-D. A direct method for the transient stability analysis of transmission switching events. Proc IEEE Power Energy Society General Aug. 2018:1–5. [22] Khodaei A, Shahidehpour M. Transmission switching in security-constrained unit commitment. IEEE Trans Power Syst Nov. 2010;24(4):1937–45. [23] Sahraei-Ardakani M, Li X, Balasubramanian P, Hedman KW, Abdi-Khorsand M. Real-time contingency analysis with transmission switching on real power system data. IEEE Trans Power Syst May 2016;31(3):2501–2. [24] Liu C, Wang J, Ostrowski J. Static switching security in multi-period transmission switching. IEEE Trans Power Syst Nov. 2012;27(4):1850–8. [25] Liu WL, Chiang HD. Toward on-line line switching method for relieving overloads in power systems. Proc IEEE Power Energy Society General Oct. 2015:1–5. [26] El-Abiad AH, Stagg GW. Automatic evaluation of power system performance-effects of line and transformer outages. AIEE Trans Power Apparat Syst Feb. 1963;81(64):712–5. [27] Alsac O, Stott B, Tinney WF. Sparsity-oriented compensation methods for modified network solutions. IEEE Trans Power Apparat Syst May 1983;PAS-102(5):1050–60. [28] Vemuri S, Usher RE. On-line automatic contingency selection algorithms. IEEE Trans Power Apparat Syst Feb. 1983;PAS-102(2):346–54.

To evaluate the accuracy of the proposed methodology, all 2835 candidate lines are switched out individually, and lines 25-192 and 410-25 are found to relieve overload by using AC power flow. This is consistent with the solutions obtained by the proposed methodology. The total CPU time of the proposed methodology in this study is 8.1863 s, whereas the method from reference [8] takes 73.5131 s, an improvement in speed by the proposed methodology of 88.86%. The above results show the efficiency and effectiveness of the proposed methodology for online application in large-scale systems. 6. Conclusion This paper proposes a novel online methodology of line switching for relieving base-case power system overloads with look-ahead capability while satisfying the operational and engineering constraints of post-switching power systems. The proposed methodology employs a three-stage strategy that contains a screening stage, ranking stage, a detailed analysis and assessment stage. The proposed methodology balances speed (for online application) and accuracy (for the solution quality) by combining linear and nonlinear methods to achieve the goal of online application to relieve overloads. To ensure look-ahead capability, the proposed methodology obtains a set of line switching solutions for the current operating point first, then makes a 30-minute look-ahead assessment of the post-switching systems to further examine the effectiveness of these solutions. The results provided by the exact AC power flow is used as a benchmark to compare the accuracy of the proposed methodology. The speed of the proposed methodology is compared with the method from reference [8]. One distinguishing feature of the proposed methodology is that it can find multiple high-quality line switching solutions, which may include the ‘best’ solution, to relieve overloads in an online manner and ensure the effectiveness of the line switching solution for the lookahead power system. Numerical schemes and methods have been developed and implemented for each stage of the proposed methodology. It has been evaluated on the IEEE 39-bus and 2746-bus power systems with promising results. A numerical study conducted on the 2746-bus power system reveals the fast speed and effectiveness of the proposed methodology on largescale systems, showing its good potential for online application. Our future work includes the consideration of renewable uncertainties in the present problem formulations. It appears to us that expressing renewable uncertainties as multiple scenarios can be a viable approach. In addition, our future work will also include consideration of line reconnection instead of only line disconnection. This consideration, even though will significantly increase the dimension of the search space, may lead to solutions with higher quality. Furthermore, the impact of line switching on power system reliability is also important to analyse.

8