Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model
Abstract
:1. Introduction
- i.
- The description of a stochastic mixed-integer convex (SMIC) model for the optimal placement and sizing of D-STATCOMs in electrical distribution networks. The proposed SMIC model considers nine scenarios involving low-, medium-, and high-load-generation levels.
- ii.
- The addition of binary-polynomial constraints to the proposed SMIC model in order to model the power transfer losses of D-STATCOMs. These constraints are convexified using second-order cone relaxation.
- iii.
- The objective function of the proposed SMIC model is composed of two terms: minimizing the annual energy loss costs and the annualized investment costs related to installing a new D-STATCOM, which implies a multi-objective problem. A weight factor is used to solve the problem, as these terms are in conflict.
- iv.
- Three simulation cases are proposed to demonstrate the effectiveness of the stochastic convex model, and it is compared to three solvers available in the General Algebraic Modeling System (GAMS) software.
2. Mathematical Model
2.1. Objective Function
2.2. Set of Constraints
2.2.1. Grid-Connected D-STATCOMs
2.2.2. Power Balance Equation
2.2.3. Power Flow Equation
2.2.4. Operating Regulation
2.3. Interpretation of the Mathematical Model
3. Convex Reformulation
3.1. Approximation to a Linear Function for D-STATCOM Costs
3.2. Conic Representation of the Power Flow Equations
3.3. Conic Representation of VSC Power Losses
3.4. Proposed Mixed-Integer Convex Model
3.5. Stochastic Mixed-Integer Convex Model
4. Test System
4.1. Modified IEEE 33-Bus Test System
4.2. Modified IEEE 69-Bus Test System
4.3. Modified IEEE 85-Bus Test System
4.4. Load-Generation Scenario
5. Numerical Implementation
- C1:
- The proposed model was compared with GAMS, only considering the installation of three D-STATCOMs.
- C2:
- The impact on the total annual operating costs of the number of D-STATCOMs available for installation was assessed by varying the number of devices from 0 to 5.
- C3:
- The Pareto front was formed by analyzing the trade-off between the costs of the D-STATCOMs and the annual energy loss costs, which are conflicting objectives.
5.1. Analysis of Case 1 (C1)
- The proposed SMIC model for the three test systems finds the best solutions (global optima) with objective function values of USD 46,212.30, USD 44,082.85, and USD 48,581.90. These solutions were obtained from the exact costs function (1). Comparing the benchmark cases to the best solutions revealed significant reductions of 29.25%, 60.89%, and 52.54% for the modified IEEE 33-, 69-, and 85-bus test systems, respectively.
- For the modified IEEE 33-bus test system, it can be noted that the BONMIN solver reached the same solution as the proposed SMIC model. In contrast, the CONOPT and GUROBI solvers achieved the worst solutions (local optima), reducing the annual operating costs by 25.81% and 29.21%, respectively.
- As for the modified IEEE 69-bus test system, the BONMIN solver did not yield any feasible solution due to convergence issues. On the other hand, the GUROBI solver found the same solution as the proposed SMIC model, and the CONOPT solver reached a local optimum, reducing the annual operating cost by 59.79%.
- For the modified IEEE 85-bus test system, it is evident that none of the solvers could attain the global optimum of the problem, and only local optima were found. In this case, the CONOPT solver reported a better solution than other solvers, which indicates that none of the solvers outperformed the other.
5.2. Analysis of Case 2 (C2)
5.3. Analysis of Case 3 (C3)
- There is a trade-off between the annual costs of energy losses and those associated with investment costs in D-STATCOMs, as improving one objective leads to the deterioration of the other and vice versa. Specifically, the two extreme solutions are characterized by: (i) annual energy losses costs of USD , USD , and USD per year for the modified IEEE 33-, 69-, and 85-bus test systems, respectively, with zero investments (i.e., ); and (ii) increased investments of USD , USD , and USD per year of operation (i.e., ), which results in the lowest cost of energy losses for the three test systems.
- The optimal solution for each test system in Table 4 is achieved via the proposed SMIC model, which corresponds to the minimum cost reported in the multi-objective case (i.e., when ). This indicates that the scaling factor of the objective function for the single-objective case does not significantly affect the final result of the convex model. However, the main advantage of having a Pareto front is that it offers a range of possibilities in order for a utility company to choose the most suitable option based on its investment capabilities.
6. Conclusions and Future Works
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Methodology | Objective Function | Year | Ref. |
---|---|---|---|
Genetic algorithm | Minimization of power losses | 2011 | [16] |
Artificial neural networks | Mitigation of voltage sags under faults | 2012 | [17] |
Immune algorithm | Minimization of power losses and reduction of investment and operating costs | 2014 | [18] |
Particle swarm optimization | Minimization of power losses and voltage profile improvement | 2014 | [19] |
Ant colony optimization | Minimization of power losses and voltage profile improvement | 2015 | [10] |
Sensitivity indices | Minimization of power losses and voltage profile improvement | 2015 | [20] |
Harmony search algorithm | Minimization of power losses | 2015 | [21] |
Heuristic search algorithm | Minimization of power losses | 2016 | [22] |
Imperialist competitive algorithm | Minimization of energy costs and voltage profile improvement | 2017 | [23] |
Discrete-continuous vortex search algorithm | Investment and operating costs reduction | 2017 | [15] |
Modified crow search algorithm | Reducing line losses, maximizing economic benefits, improving voltage profiles, and reducing pollution levels | 2018 | [24] |
Hybrid analytical-coyote | Minimization of active power losses and voltage profile improvement | 2019 | [25] |
Modified sine-cosine algorithm | Minimization of power losses and voltage profile improvement | 2020 | [26] |
GAMS software for the solution of the exact MINLP model | Reduction in investment and operating costs | 2021 | [27] |
Scenario | Load-Generation | Probability () |
---|---|---|
1 | Low/low | 0.2210 |
2 | Low/medium | 0.0443 |
3 | Low/high | 0.0676 |
4 | Medium/low | 0.2767 |
5 | Medium/medium | 0.0554 |
6 | Medium/high | 0.0845 |
7 | High/low | 0.0845 |
8 | High/medium | 0.0332 |
9 | High/high | 0.0507 |
Par. | Value | Unit | Par. | Value | Unit |
---|---|---|---|---|---|
0.1390 | USD/kWh | T | 365 | Days | |
0.50 | h | 0.30 | USD/MVAr | ||
−305.10 | USD/MVAr | 127,380 | USD/MVAr | ||
6/2190 | 1/Days | 10 | Years |
Method | Location (Bus) | Size (k) | (USD/year) | (USD/year) | (USD/year) |
---|---|---|---|---|---|
IEEE 33-bus test system | |||||
Ben. case | 65,324.25 | 65,324.25 | 0.00 | ||
CONOPT | 48,464.31 | 39,803.98 | 8660.33 | ||
BONMIN | 46,212.30 | 36,115.04 | 10,097.26 | ||
GUROBI | 46,243.04 | 36,105.07 | 10,137.98 | ||
SMIC | 46,212.30 | 36,115.04 | 10,097.26 | ||
IEEE 69-bus test system | |||||
Ben. case | 112,740.90 | 112,740.90 | 0.00 | ||
CONOPT | 45,329.51 | 39,803.98 | 9223.70 | ||
GUROBI | 44,082.85 | 33,810.49 | 10,272.36 | ||
SMIC | 44,082.85 | 33,810.49 | 10,272.36 | ||
IEEE 89-bus test system | |||||
Ben. case | 102,369.39 | 102,369.39 | 0.00 | ||
CONOPT | 49,062.63 | 33,795.46 | 15,267.17 | ||
BONMIN | 51,045.58 | 36,882.55 | 14,163.03 | ||
GUROBI | 66,816.56 | 56,144.56 | 10,672.00 | ||
SMIC | 48,581.90 | 32,853.47 | 15,728.43 |
Factor () | (USD/year) | (USD/year) | (USD/year) |
---|---|---|---|
IEEE 33-bus test system | |||
0.0 | 65,324.25 | 65,324.25 | 0 |
0.1 | 65,324.25 | 65,324.25 | 0 |
0.2 | 47,707.61 | 37,002.80 | 9005.45 |
0.3 | 46,341.15 | 36,390.38 | 9950.77 |
0.4 | 46,276.73 | 36,252.71 | 10,024.01 |
0.5 | 46,212.31 | 36,115.05 | 10,097.26 |
0.6 | 46,379.65 | 36,172.90 | 10,206.75 |
0.7 | 46,395.17 | 36,172.01 | 10,223.16 |
0.8 | 46,410.70 | 36,171.13 | 10,239.57 |
0.9 | 46,412.49 | 36,272.77 | 10,339.72 |
1.0 | 104,613.34 | 32,121.99 | 72,491.35 |
IEEE 69-bus test system | |||
0.0 | 112,740.90 | 112,740.90 | 0 |
0.1 | 112,740.90 | 112,740.90 | 0 |
0.2 | 44,757.04 | 35,202.5 | 9554.54 |
0.3 | 44,754.02 | 34,336.33 | 10,417.69 |
0.4 | 46,332.37 | 33,331.98 | 13,000.39 |
0.5 | 44,082.85 | 33,810.49 | 10,272.36 |
0.6 | 44,101.20 | 33,577.67 | 10,523.53 |
0.7 | 44,119.56 | 33,344.86 | 10,774.70 |
0.8 | 44,230.72 | 33,501.36 | 10,729.36 |
0.9 | 48,705.06 | 30,205.07 | 18,499.99 |
1.0 | 50,821.03 | 30,109.31 | 20,711.72 |
IEEE 85-bus test system | |||
0.0 | 102,369.39 | 102,369.39 | 0 |
0.1 | 102,369.39 | 102,369.39 | 0 |
0.2 | 90,641.97 | 88,022.15 | 2619.82 |
0.3 | 56,682.66 | 47,597.36 | 9085.30 |
0.4 | 51,604.91 | 39,705.79 | 11,899.12 |
0.5 | 48,581.90 | 32,853.47 | 15,728.43 |
0.6 | 49,777.82 | 32,370.34 | 17,407.48 |
0.7 | 49,701.84 | 28,554.90 | 21,146.94 |
0.8 | 52,111.75 | 28,893.70 | 23,218.05 |
0.9 | 52,287.42 | 28,302.63 | 23,984.79 |
1.0 | 66,766.11 | 27,425.10 | 39,341.01 |
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Gil-González, W. Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model. Electronics 2023, 12, 1565. https://doi.org/10.3390/electronics12071565
Gil-González W. Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model. Electronics. 2023; 12(7):1565. https://doi.org/10.3390/electronics12071565
Chicago/Turabian StyleGil-González, Walter. 2023. "Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model" Electronics 12, no. 7: 1565. https://doi.org/10.3390/electronics12071565
APA StyleGil-González, W. (2023). Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model. Electronics, 12(7), 1565. https://doi.org/10.3390/electronics12071565