Abstract
In this work, a space transformational invasive weed optimization (ST-IWO) algorithm is applied for the solution of single and multi-objective optimal power flow problem. The ST-IWO technique integrates the invasive weed optimization (IWO) and the space transformation search (STS) techniques. The IWO technique is a population-based stochastic algorithm inspired by nature, while the STS is an evolutionary technique inspired by opposition based learning. The STS compares a solution with its opposition to find a better solution which reduces the computational efforts and search direction moves toward the promising region to overcome the premature convergence problem. In order to deal with conflicting objectives of multi-objective problem, the non-interactive approach is applied. In this approach, the decision maker has prior preference information, which eases the selection of the best non-dominated solution. To authenticate the performance of ST-IWO technique, it is tested on standard benchmark functions and three standard IEEE bus systems. The achieved results are compared with recently published results and performance found satisfactory. The statistical test (t test) has been executed to confirm the robustness of the technique.
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Abbreviations
- \(V_{i} ,V_{k}\) :
-
Voltage magnitudes corresponding to bus i and bus k, respectively
- \(\delta_{i} ,\delta_{k}\) :
-
Phase angles of voltages corresponding to bus i and bus k, respectively
- \(G_{j}\) :
-
Transmission line conductance connecting bus i and bus k
- \(a_{l} ,b_{l} ,c_{l} ,d_{l}\) and \(e_{l}\) :
-
Fuel cost coefficients of the lth generator
- \(V_{L,i}\) :
-
Magnitude of voltage at load bus i
- \(P_{G,l}^{\max } ,P_{G,l}^{\min }\) :
-
Upper, lower real power bounds for the lth generator, respectively
- \(V_{L,i}^{\max } ,V_{L,i}^{\min }\) :
-
Upper, lower load bus voltage bounds for the bus i, respectively
- \(P_{G,l}\) :
-
Real power output of the lth generator
- \(Q_{j}^{\max } ,Q_{j}^{\min }\) :
-
Maximum, minimum VAR limits introduced by the jth shunts, respectively
- \(P_{L,i} ,Q_{L,i}\) :
-
Demand of real, reactive power at bus i, respectively
- \(P_{G,i} ,Q_{G,i}\) :
-
Available real, reactive power generated at bus i, respectively
- \(D\) :
-
Decision variables
- \(\sigma_{it}\) :
-
Standard deviation (present iteration)
- \(\sigma^{{{\text{final}}}}\) :
-
Standard deviation (final)
- V reference :
-
Reference voltage magnitude in p.u.
- NT :
-
Total transformers
- \(P_{L}\) :
-
Total system transmission losses
- \(A_{r} (it)\) :
-
Randomly selected solution from the archive A
- \({\text{Seeds}}_{\max }\),\({\text{Seeds}}_{\min }\) :
-
Maximum, minimum values of produced seeds for a weed, respectively
- \(\mu_{{C\;{\text{worst}}}} ,\mu_{{C\;{\text{best}}}}\) :
-
Worst, best fitness of weed
- \(n\) :
-
Modulation index
- \(A_{n} ,B_{n}\) :
-
Lower and upper values of nth dimension decision variable, respectively.
- NB:
-
Total buses
- \(Y_{ik}\) :
-
Transmission line admittance connecting bus i and bus k
- p 1 , p 2 , p 3 and p 4 :
-
Penalty coefficients of the relevant constraints
- NPQ:
-
Total load buses
- \(\alpha_{l} ,\beta_{l} ,\gamma_{l} ,\eta_{l}\) and \(\lambda_{l}\) :
-
Emission coefficients of the lth generator
- \(N\) :
-
System buses except slack bus
- \(Q_{G,l}^{\max } ,Q_{G,l}^{\min }\) :
-
Upper, lower reactive power bounds for the lth generator, respectively
- \(P_{{G,\;{\text{slack}}}}^{\lim } ,V_{L,i}^{\lim }\), \(Q_{G,l}^{\lim }\) :
-
Upper or lower value for the slack bus real power, voltages of load bus, reactive powers of thermal generator units, respectively
- \({\text{NC}}\) :
-
Total shunt compensators
- \(t_{k}^{\max } ,\;t_{k}^{\min }\) :
-
Upper, lower tap setting bounds for the kth transformer, respectively
- \(S_{j}^{\max } ,\;S_{j}\) :
-
Maximum, actual apparent power of branch j, respectively
- \(V_{G,l}^{\max } ,V_{G,l}^{\min }\) :
-
Upper, lower voltage magnitude bounds for the lth generator, respectively
- \(F_{k}^{\max } ,F_{k}^{\min }\) :
-
Maximum, minimum kth objective function values, respectively
- NL:
-
Total transmission lines
- NG:
-
Total generators
- \(M\) :
-
Number of objectives
- \(P_{{G,{\text{slack}}}}\) :
-
Real power generated by slack bus
- \(Q_{G,l}\) :
-
Reactive power generated at lth generator
- \(\mu_{{\text{C}}} (X_{i} )\) :
-
Fitness of weed i
- \(it^{\max }\) :
-
Maximum number of iterations
- \(\sigma^{{{\text{initial}}}}\) :
-
Standard deviation (initial)
- \(\omega\) :
-
Inertia weight
- F :
-
Scaling parameter
- \(\mu_{{\text{C}}}\) :
-
Scalarized cardinal priority ranking
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Kaur, M., Narang, N. Optimal Power Flow Solution Using Space Transformational Invasive Weed Optimization Algorithm. Iran J Sci Technol Trans Electr Eng 47, 939–965 (2023). https://doi.org/10.1007/s40998-023-00592-y
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DOI: https://doi.org/10.1007/s40998-023-00592-y