The data of four networks that can be used in carrying out comparative studies with methods for transmission network expansion planning are given. These networks are of various types and different levels of complexity. The main mathematical formulations used in transmission expansion studies—transportation models, hybrid models, DC power flow models, and disjunctive models are also summarised and compared. The main algorithm families are reviewed—both analytical, combinatorial and heuristic approaches. Optimal solutions are not yet known for some of the four networks when more accurate models (e.g. the DC model) are used to represent the power flow equations—the state of the art with regard to this is also summarised. This should serve as a challenge to authors searching for new, more efficient methods.

This paper presents a model for use in the problem of multistage planning of energy distribution systems including distributed generation. The expansion model allows alternatives to be considered for increasing the capacity of existing substations, for installing new ones, for using distributed generation, and for the possible change to feeders in terms of addition and removing feeders sections; combining, subdividing, and load transfer between feeders; and replacement of conductors. The objective function to be minimized is the present value of total installation costs (feeders and substations), of operating and maintaining the network, and of distributed generation. The model takes operational constraints on equipment capacities and voltage limits together into account with logical constraints, aimed at reducing the search space. This paper presents: (1) an extension to the linear disjunctive formulation to represent the inclusion, exclusion, and replacement of branches and (2) a generalization of constraints related to the creation of new paths which can be applied in more complex topologies. The resulting mixed integer linear model allows the optimal solution to be found using mathematical programming methods, such as the branch-and-bound algorithm. The validity and efficiency of the model are demonstrated in Part II of this paper.

A method for optimal transmission network expansion planning is presented. The transmission network is modelled as a transportation network. The problem is solved using hierarchical Benders decomposition in which the problem is decomposed into master and slave subproblems. The master subproblem models the investment decisions and is solved using a branch-and-bound algorithm. The slave subproblem models the network operation and is solved using a specialised linear program. Several alternative implementations of the branch-and-bound algorithm have been tested. Special characteristics of the transmission expansion problem have been taken into consideration in these implementations. The methods have been tested on various test systems available in the literature.

This paper presents the computer simulation of the multistage model for distribution expansion planning with distributed generation, as described in Part I. The simulations deal with the planning of an electrical power distribution network in three stages, in five different situations: 1. each of the three stages planned independently; 2. multistage planning; 3. multistage planning with distributed generation; 4. multistage planning with distributed generation and constraints on investment; and 5. multistage planning with distributed generation considering three load levels. The influence of additional constraints is analyzed in terms of the computational effort required to find the optimum solution to the problem.

The paper assesses the steady-state performance of a five-phase induction machine fed by a modified open loop constant volt/hertz (V/f) control method, which imposes a trapezoidal induction waveform in the air gap under varying load conditions. The trapezoidal air gap induction is achieved through the imposition of an appropriate combination of the third harmonic and fundamental stator voltages. This harmonic combination is determined from the steady-state model using a mathematical optimization procedure, which allows to obtain the optimal weighting factors for each harmonic component. The optimized reference voltages lead to a trapezoidal air gap induction, which allows a better iron utilization and higher output torque for the same rms stator current when compared to sinusoidal air gap induction. The resulting air gap induction is obtained from the induced voltage of a full pitch search coil placed in the air gap. The proposed control scheme was successfully simulated and implemented on a five-phase prototype machine running under different load conditions. Experimental and simulations results show an increase in the torque/ampere relationship for loads above 50% when compared to the conventional V/f method using only the fundamental current and air gap induction. A comparison between the simulation and experimental curves presents a very good agreement that confirms and validates the parameters and model used.

The paper proposes a multiobjective optimization procedure for five-phase induction machines based on the NSGA-II (nondominated sorting genetic algorithm). In the optimization procedure, two objective functions are used (efficiency and cost of conductor material) along with 14 design variables, producing a large number of viable solutions. The design variables describe geometrical constraints, electrical and magnetic loading, and the waveshape of the airgap induction. The underlying analytical model of the machine includes the saturation of the iron parts and magnetic losses produced by higher space harmonic waves present in the airgap induction. The most relevant solutions regarding the objectives are validated through the finite element method, showing that the method proposed here is able to find practical solutions which represent a high efficiency, a reduced volume, or even a compromise between these objectives.