A Mathematical Approach to Simultaneously Plan Generation and Transmission Expansion Based on Fault Current Limiters and Reliability Constraints
Abstract
:1. Introduction
- The formulation and the presentation of a G&TEP stochastic model integrating allocation and the size of the FCLs are introduced to increase the penetration of RESs, meet reliability constraints and avoid rolling blackouts.
- To avoid linearization errors, a hybridization framework is suggested to solve the G&TEP problem and handle non-linear constraints.
- Three meta-heuristic techniques, namely lévy flight distribution (LFD), the sine cosine algorithm (SCA), and the success-history-based differential evolution with semi-parameter adaptation hybrid-covariance matrix adaptation evolution strategy algorithm (LSHADE-SPACMA), are employed to solve the problem, and the results obtained are analyzed and compared.
- The suggested methodology was conducted on the realistic transmission of the Egyptian West Delta Network (WDN) to validate the capability of the proposed algorithm.
2. Problem Formulation
2.1. The Mathematical Model
- (1)
- Single-line outage:
- (2)
- Three-phase faults:
2.2. Uncertainty Modeling
3. Proposed Model
3.1. Proposed Hybridization Scheme
3.1.1. Upper-Level Problem
3.1.2. Middle-Level Problem
3.1.3. Lower-Level Problem
3.2. The Procedure of the Suggested Approach
- Step 1: Generate 1000 scenarios using the Monte-Carlo method to represent load, wind power source, and photovoltaic (PV) power source variation.
- Step 2: Select 25 scenarios using the backward reduction technique.
- Step 3: For each scenario, solve the upper level, medium level, and lower level of the G&TEP problem as follows:
- (a)
- Meta-heuristic optimization technique randomizes and to solve the upper-level problem.
- (b)
- After that, the medium-level problem is solved using the B & B algorithm to obtain DGs’ optimum location with minimum operating costs.
- (c)
- The IP method is applied to solve the lower problem to realize N-1 security.
- (d)
- If the candidate solution does not meet short-circuit and power flow constraints, it will be penalized by imposing high additional costs.
- (e)
- Do the iteration as shown in (a–d) until the iteration reaches the iteration limit.
- (f)
- Do the steps from (a–e) until the current run reaches the maximum number of runs.
- (g)
- Compare solutions of each run to select the optimal solution.
- (h)
- Go to the following scenario.
- Step 4: If the current scenario number is less than Smax, the lower bound of generation units, candidate routes, and FCL sizes are updated considering the system configuration under the previous scenario.
- Step 5: Repeat Steps 3 and 4 until Smax is reached.
4. Optimization Techniques
4.1. Sine Cosine Algorithm
4.2. LSHADE with Semi-Parameter Adaption Hybrid with CMA-ES
4.3. Lévy Flight Distribution
- Generate a population of candidate solutions.
- Define an upper bound (UB) and lower bound (LB) and a target solution.
- For each iteration, do the following:
- 1.
- For every two adjacent agents (main, neighbor), ED is calculated using the following (36):
- 2.
- If the ED is less than the threshold, the neighbor agent position is updated using (37) and (38). Based on a random value (R), if R is less than CSV, execute (37). If not, then run (38) to provide the algorithm with more chances to discover the search space.CSV is the scalar value that is compared with R in each update for the position. is the agent’s location with the fewest number of neighbors and is used as the LF direction. Lévy_ flight is a function that implements Lévy flight work in terms of step length and direction. It is implemented as follows:
- -
- Calculate the step length (leng) according to Mantegna’s algorithm.
- -
- Calculate the difference factor (DF) as follows:
- -
- Execute the actual random walk or flight with (40) such that is not outside the search space.
- 3.
- Update the main agent position using the following equation:So that:TP is the position of the best solution, and is the total target fitness of neighbors around . , and are random positive variables.
- 4.
- Bring the current agent back if it goes outside the boundaries, whether primary or neighbor.
- 5.
- Update the target solution.
5. Numerical Results and Discussion
5.1. Selection of Suitable Optimization Technique
5.2. WDN Planning
5.2.1. Robustness Measurement
5.2.2. Impact of Including FCL Allocation and Sizing in the G&TEP Problem
5.2.3. Configuration of WDN Considering N-1 Security
6. Discussion
7. Conclusions
- The results showed that the SCA is superior to LFD and LSHADE-SPACMA in terms of convergence speed and reaching the global optimum. The calculation time of SCA was 327.089 s, while it was 469.32 s and 353.231 s for LSHADE-SPACMA and LFD, respectively. The SCA gave the best solutions in 24 tested scenarios, whereas the three techniques gave the same results in one scenario. Moreover, the SCA had the lowest standard deviation of 0.4872%.
- The results demonstrated that G&TEP correlated with FCL allocation is necessary for some power systems to meet SCC constraints, and the reliance on the new circuit locations is not useful in some power systems such as the WDN. It was found that the short-circuit current if FCLs were not used, was more than 11.5 p.u.
- The robustness of the adopted approach in handling uncertainties was demonstrated. It reached 100%.
- Integrating N-1 constraint with the G&TEP model improves power system reliability and decreases the amount of LS. The results indicated that the amount of LS for a single circuit outage increased from 154.35 MW to 426.28 MW when the N-1 security was ignored.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
B&B | Branch-and-bound method |
CMA-ES | Covariance matrix adaptation evolution strategy |
DCPF | DC power flow |
DG | Distributed generation |
LS | load shedding (MW) |
FCL | Fault current limiter |
GEP | Generation expansion planning |
G&TEP | Generation and transmission expansion planning |
IP | Interior-point |
LFD | Lévy flight distribution |
LP | Linear programming |
LPSR | Linear population size reduction |
LSHADE-SPACMA | Linear population size reduction success history based differential evolution with semi-parameter adaptation hybrid with CMA-ES |
MILP | Mixed-integer linear programming |
PV | Photovoltaic |
RESs | Renewable energy sources |
SCA | Sine cosine algorithm |
SPA | Semi-parameter adaptation |
TEP | Transmission expansion planning |
WDN | Egyptian West Delta network |
N, S | Sets of buses and scenarios, respectively |
Maximum number of scenarios | |
Cost of the circuit between buses i and j | |
The cost of installing an FCL between bus i and bus j | |
Three-phase short-circuit current at bus i of scenario s and the maximum short-circuit current limit, respectively | |
The load demand (MW) at bus i for scenario s | |
Actual power generation for scenario s, the minimum capacity, and the maximum capacity of renewable energy source at bus i, respectively | |
Actual power generation of conventional power plant for scenario s, the minimum capacity, and the maximum capacity of traditional power source at bus i, respectively | |
Susceptance of route between buses i and j | |
The maximum number of circuits, and the number of new circuits for scenario s between buses i and j, respectively | |
Impedance of FCL between buses i and j | |
The minimum and the maximum impedance of FCL between buses i and j | |
Active power flow and the active power flow limit in the i-j right of way (MW), respectively | |
Three-phase short-circuit current at bus i and the maximum short-circuit current limit, respectively | |
Voltage angles at bus i | |
The cost and location of installation of the new generation station at bus i | |
The operation cost of generator i (US $/MWh). | |
The cost and amount of LS at bus i, respectively | |
Set of candidate solutions | |
Upper bound and lower bound of the decision vector, respectively | |
Mutant vector corresponding to each population member, | |
F | Scale factor |
CR | Crossover rate |
Best individual vector with the best fitness value at t generation in the population |
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Features | SCA [27] | LSHADE-SPACMA [28] | LFD [29] |
---|---|---|---|
Main idea |
|
|
|
Advantages |
|
|
|
Disadvantages |
|
Technique | Objective Function (M$) | T*comp (s) | Wilcoxon Test p-Value | Standard Deviation (%) | |||
---|---|---|---|---|---|---|---|
Min. | Max. | vs. SCA | vs. LSHADE-SPACMA | vs. LFD | |||
SCA | 285.086 | 286.181 | 327.089 | NA | 4.01 × 10−5 | 1.82 × 10−5 | 0.4872 |
LSHADE-SPACMA | 289.359 | 294.945 | 469.32 | 4.01 × 10−5 | NA | 2.61 × 10−5 | 2.0162 |
LFD | 332.209 | 341.752 | 353.231 | 1.82 × 10−5 | 2.67 × 10−5 | NA | 3.6964 |
Added Circuits | FCLs | Generation Units | Risk Scenarios | |||
---|---|---|---|---|---|---|
Location | No. Circuits | Lc | Sz (p.u) | Unit No. | Bus No. | Total |
33–53 | 1 | 8–41 | 2.544 | G.10 | 13 | 2 |
8–53 | 2 | 6–36 | 0.672 | G.14 | 29 | NA |
NA | 5–36 | 1.3352 | G.15 | 32 | ||
5–34 | 0.0001 | G.16 | 36 | |||
2–13 | 0.9487 | G.18 | 41 | |||
NA | G.23 | 53 |
Added Circuits | FCLs | Generation Units | Risk Scenarios | |||
---|---|---|---|---|---|---|
Location | No. Circuits | Lc | Sz (p.u) | Unit No. | Bus No. | Total |
33–53 | 1 | 48–49 | 0.663 | G.16 | 36 | 40 |
5–53 | 1 | 6–36 | 1.135 | G.18 | 41 | NA |
36–53 | 1 | 5–36 | 0.497 | G.19 | 42 | |
NA | 13–30 | 0.209 | G.23 | 53 |
Added Circuits | FCLs | Generation Units | Risk Scenarios | |||
---|---|---|---|---|---|---|
Location | No. Circuits | Lc | Sz (p.u) | Unit No. | Bus No. | Total. |
5–22 | 1 | 8–46 | 1.134 | G.14 | 29 | No scenarios |
7–32 | 1 | 6–36 | 1.827 | G.15 | 32 | |
23–53 | 2 | 5–36 | 0.6775 | G.16 | 36 | |
5–53 | 2 | 13–30 | 0.576 | G.18 | 41 | |
33–53 | 1 | 4–25 | 0.0602 | G.19 | 42 | |
36–53 | 2 | 7–32 | 0.941 | G.23 | 53 |
Added Circuits | FCLs | Generation Units | ||||
---|---|---|---|---|---|---|
Location | No. Circuits | Lc | Sz (p.u) | Unit No. | Bus No. | |
5–22 | 1 | 41–39, 8–41 | 3,3 | G.9 | 29 | |
6–32 | 1 | 47–48, 11–28 | 3,3 | G.15 | 32 | |
7–32 | 1 | 6–40, 5–36 | 3,3 | G.16 | 36 | |
7–36 | 1 | 5–34, 13–29 | 3,3 | G.18 | 41 | |
25–53 | 1 | 13–30, 3–14 | 3,3 | G.19 | 42 | |
23–53 | 2 | 6–42, 49–50 | 3,3 | G.23 | 53 | |
33–53 | 1 | 5–22, 6–32 | 3,3 | NA | ||
31–53 | 1 | 7–32, 36–53 | 3,3 | |||
34–53 | 1 | 25–53, 31–53 | 3,3 | |||
36–53 | 1 | 8–53, 3–17 | 3, 1.327 | |||
8–53 | 2 | 3–16, 5–31 | 1.363, 1.47 | |||
NA | 46–45, 20–21 | 0.485, 2.654 |
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Refaat, M.M.; Aleem, S.H.E.A.; Atia, Y.; Ali, Z.M.; El-Shahat, A.; Sayed, M.M. A Mathematical Approach to Simultaneously Plan Generation and Transmission Expansion Based on Fault Current Limiters and Reliability Constraints. Mathematics 2021, 9, 2771. https://doi.org/10.3390/math9212771
Refaat MM, Aleem SHEA, Atia Y, Ali ZM, El-Shahat A, Sayed MM. A Mathematical Approach to Simultaneously Plan Generation and Transmission Expansion Based on Fault Current Limiters and Reliability Constraints. Mathematics. 2021; 9(21):2771. https://doi.org/10.3390/math9212771
Chicago/Turabian StyleRefaat, Mohamed M., Shady H. E. Abdel Aleem, Yousry Atia, Ziad M. Ali, Adel El-Shahat, and Mahmoud M. Sayed. 2021. "A Mathematical Approach to Simultaneously Plan Generation and Transmission Expansion Based on Fault Current Limiters and Reliability Constraints" Mathematics 9, no. 21: 2771. https://doi.org/10.3390/math9212771
APA StyleRefaat, M. M., Aleem, S. H. E. A., Atia, Y., Ali, Z. M., El-Shahat, A., & Sayed, M. M. (2021). A Mathematical Approach to Simultaneously Plan Generation and Transmission Expansion Based on Fault Current Limiters and Reliability Constraints. Mathematics, 9(21), 2771. https://doi.org/10.3390/math9212771