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This paper presents a novel mathematical model for the transmission network expansion planning problem. Main idea is to consider phase-shifter (PS) transformers as a new element of the transmission system expansion together with other traditional components such as transmission lines and conventional transformers. In this way, PS are added in order to redistribute active power flows in the system and, consequently, to diminish the total investment costs due to new transmission lines. Proposed mathematical model presents the structure of a mixed-integer nonlinear programming (MINLP) problem and is based on the standard DC model. In this paper, there is also applied a specialized genetic algorithm aimed at optimizing the allocation of candidate components in the network. Results obtained from computational simulations carried out with IEEE-24 bus system show an outstanding performance of the proposed methodology and model, indicating the technical viability of using these nonconventional devices during the planning process.

Transmission network planning begins with the establishment of power demand growth scenarios, in accordance with forecasts along the time. Given these scenarios, one can verify the eventual need to broaden and to strengthen the network. In case electric service conditions are not satisfied, there should be proposed a plan that has coherence among the power supply availability, demand, and installation of new equipments in the network. Integration of these new equipments in the network, aimed at maintaining suitable technical and operating conditions, requires planning of the allocation of such reinforcement.

Main objective of the transmission expansion planning is to obtain the

Formulation of a mathematical representation for the transmission expansion planning problem begins with some assumptions, where accuracy and complexity are considered in the model construction. Regularly, the problem is represented by a Mixed-Integer Nonlinear Programming (MINLP) problem that presents many local optima solutions for real-life systems. This high number is due to the possible expansion plans that shows the association of the specified optimal operational mode. Therefore, a basic problem consists in defining the least-cost expansion alternative that satisfies all operating constraints.

In static long-term transmission expansion planning (typically with a planning horizon of more than 5 years), all investments are carried out in a single-year planning horizon, whereas for the multistage it is divided into several stages.

Static planning is aimed at searching

In the technical literature, DC and transportation models are static mathematical models often used to solve the transmission expansion planning problem. These models consider only the addition of transmission lines (TLs) and conventional transformers.

Here a novel transmission expansion strategy is proposed. An improved model considers the inclusion of a new kind of device, in this case, a flexible alternating current transmission system (FACTS) device.

The literature concerning the use of FACTS devices is wide. However, most papers treat only the operational improvement by using FACTS devices [

Feasibility of employing the PS as a candidate component in the long-term transmission expansion planning process is analyzed regarding a static and centralized planning model. Nevertheless, the present model can be extended to the multistage planning [

Instead of trying to consider all functions concerning the planning problem in a single model, this work is focused solely on the core of the network synthesis; for which the mathematical modeling and the solution technique is addressed.

The proposed model is based on the DC model, which is the most employed one in planning problems; consequently, only the active power flow is considered. Other aspects such as performance analysis (reliability and stability analysis, reactive planning, AC power flow, and short-circuit calculation) relevant to transmission expansion are beyond the scope of this paper. Nonetheless, in general, after obtaining a basic solution, all those analyses can be carried out.

This section introduces the classical mathematical model and the proposed one.

The mathematical formulation of the DC model for transmission network expansion planning problem, when considering solely the installation of transmission lines and/or conventional transformers, assumes the following form:

subject to

The objective function (

The first set of constraints (

Set of constraints (

Constraint (

From the operational research standpoint, system (

For some types of algorithms utilized for the transmission network planning problem, it is more suitable to carry out alterations to the basic modeling for allowing the application of the solution techniques. An alteration commonly used is the insertion of new variables that represent the load shedding associated with all load buses of the system. This resource can also be seen as an artificial generation aimed at turning the problem always viable during computational implementations [

When PS is considered in the transmission network planning problem, the DC model assumes the following form:

subject to

The objective function (

In constraint (

Installation of PS in one or more lines can be represented as the combination of the buses’ angles and the angle supplied by the equipment.

Angle between terminal voltages of a transmission line can be modified by installing a PS. Therefore, power flow equations (KVL) are affected when these devices are inserted. Thus, the function of PS appears in the KVL, which redirects the active power flow.

The PS is considered a component with negligible reactance that can be placed in series with a transmission line or a conventional transformer.

In this work, the angular difference of the PS was considered an unbounded variable; however, a limit can be set without modifying the mathematical model significantly.

Presence of variable

Set of constraints (

Therefore, the proposed mathematical model is more complex than the classical one, due to the characteristics of a PS.

In traditional models, a transmission line or a conventional transformer in a path

Transmission line.

The PS is considered with zero impedance connected in series with a transmission line or a conventional transformer in path

Transmission line and phase-shifter transformer.

It is also considered that when a path

Problem (

An example consisting of a network with three buses is presented in order to illustrate the application of the PS. System data is shown in Tables

Lines data of the 3 buses system.

Line | Reactance (pu) | Maximum flow (MW) |
---|---|---|

1-2 | 0.333 | 35 |

1–3 | 0.500 | 40 |

2-3 | 0.500 | 40 |

Generation and demand data of the 3-buse system.

Bus | Generation (MW) | Demand (MW) |
---|---|---|

1 | 70 | 0 |

2 | 0 | 60 |

3 | 0 | 10 |

Results obtained by performing the linearized DC power flow are illustrated in Figure

Three-bus system.

Notice that there is a load shedding of

Load flow solution with a phase shifter.

Thus, this example shows how a PS is able to redirect the active power flow. This property will be employed in the long-term transmission planning, in which basic components are transmission lines, conventional transformers, and PS.

PSs have the ability to redirect active power flows in the network. This feature provides a dynamic operational mode since it makes increasing the utilization of existing circuits possible. Consequently, as it can be verified from relation (

Another important aspect is the use of relaxed models. In general, optimal solutions for relaxed models are not feasible for more accurate or constrained ones. Thus, it is probable that the optimal solution obtained by the transportation model, where the KVL (

The primary objective of this work is to verify the operation of PS and the technical feasibility of considering such type of equipment in long-term transmission expansion planning. In case a reduced-cost PS is employed, the optimal solution will be the same of the transportation model with addition of PS. On the other hand, higher costs will inhibit the presence of PS in the optimal solution, tending to the solution given by classical DC model. Finally, if the costs are competitive to transmission lines, an intermediary solution will be provided.

Metaheuristic algorithms are specially suited for problems that present large search space with many local optima, such as the transmission expansion planning problem. The nonlinearity of the problem concerning the KVL is higher than the conventional model, thus degenerating even more the performance of more accurate methods. For instance, simulated annealing, genetic algorithms, and tabu search represent efficient methods for solving such problems. This work employed a modified version of the genetic algorithm presented in [

This section presents the genetic algorithm developed for the planning problem considering the addition of PS.

Each individual in a population (chromosome) is a proposed solution for the problem. In this work, an individual is encoded considering only the integer and binary variables. Remaining variables (continuous variables) are obtained from the linear programming (LP) solution. Thus, transmission lines and transformers are represented by decimal encoding (variable

An example of this chromosome of length

Encoding proposal (chromosome).

In [

The objective function of any solution proposal is found by solving an LP problem. The LP determines the exact values of the operational variables, which makes verifying the operation feasibility of a determined investment proposal possible, that is, whether the system presents load shedding to the implemented expansion proposal. Considering that an investment proposal

subject to

For each solution proposal, the objective function is calculated with the following expression:

In the genetic algorithm, every solution proposal is considered, including the infeasible ones. The infeasible configurations (with load shedding) are eliminated gradually by selection process, since these configurations are penalized by parameter

The selection is based on tournament with

The single point crossover was employed in this work. The crossover point is chosen randomly and a descendant, which has a parcel of its parents from the crossover point, is created. The random point was generated from an interval of 1 to (

Single point crossover.

The mutation operator acts in the following form. Considering transmission lines, the application of mutation operation means the addition or removal of one transmission line added during the optimization process (

The mutation operation should be executed respecting the following conditions:

the maximum number of transmission lines in the path;

before adding a PS in the selected path, the existence of a transmission line must be checked; if it is an empty path, a transmission line has also to be added;

when a transmission line is removed, the corresponding PS is removed, in case it exists.

Concluding, a PS can only be inserted to an existing transmission line, whereas the number of PS is equal to the number of transmission lines in a branch. In Figure

Mutation.

The general structure of the implemented genetic algorithm is similar to that presented in [

Set the control parameters and generate the initial population. Make the initial population the current population.

Calculate the objective function of the current population by solving one LP for each element (topology) of the current population. Update the incumbent solution whenever possible.

If the stopping criterion is satisfied, stop the process. Otherwise go to step 4.

Execute selection by tournament with

Execute one point crossover.

Implement specialized mutation.

Form the current population and go to step 2.

Some details of the algorithm are presented in the next section.

We present, briefly, details of the algorithm and some improvements made to the genetic algorithm.

Generation of the initial population is made by a controlled random process. Basically, it defines the number of paths in which the transmission lines are added and the maximum number of PS. Regarding transmission lines are defined randomly number of branches where the lines are inserted, position, and number of transmission lines (subject to the limits of added lines). In the case of PS, the number of branches and the position are selected randomly. In general, experience in transmission planning indicates that the number of branches to be added should be small, whereas the number for PS should be even smaller.

The performance coefficient (

After ranked, the circuits are separated into two groups with different size (75%, 25%), whereas the largest one presents the most interesting transmission lines in terms of capacity usage. The initial population is formed by 80% up to 100% of elements belonging to the largest group. Another option to the initial population generation is to employ constructive heuristic algorithms as in [

The employed crossover rate was

Mutation is executed in the following way: the power flow of each topology is stored in four matrices considering the load level of each circuit. They are separated in intervals of 25, 50, 75, and 100% of capacity. The load level is calculated by means of the relationship (

Mutation operation is executed based on the probability of 70% of circuit removal and 30% of circuit addition. In case that circuit addition operation is selected, a random selection is carried out over the branches from the matrix with most loaded branches (up to 75%). If the matrix is empty, the next load level matrix is searched. For the circuit removal operation, the search starts from the least loaded branches (matrices with branches on 25% and 50% of its capacity). With this strategy, transmission lines with low utilization are removed from the system, whereas in regions where higher utilization is observed lines are inserted.

The algorithm stops when a defined total number of iterations is reached or when there is no improvement of the incumbent solution after a specified number of iterations.

The population diversity is controlled by changing the mutation rate. A measurement of diversity is given by (

The diversification rate is calculated after mutation. If this rate is below 50%, there is used a mutation rate of 0.6; otherwise a rate of 0.1 is considered. This mechanism was applied aimed at maintaining diversity and at exploring new search spaces.

Another strategy used was the elitism, in which the parent topologies are compared with the descendants and then the two best topologies are preserved in the current population.

In order to analyze the performance of the proposed genetic algorithm and to demonstrate viability of the mathematical model with PS, tests were carried out with the IEEE-24 bus system. This system has 41 circuit paths, 8550 MW of load, generation capacity of 10215 MW, and five different generation plans whose data is present in [

For all simulations, a fixed cost for the PS was adopted, that varies among 2 and 120 million of dollars. In all the tests was used a value of

The simulations were divided in three stages. In the first stage, equipments of low value were added. In the following stage, equipments of high value were added. Finally, in the last stages, intermediate costs were adopted; that is, in each stage different values were fixed for the PS.

With the purpose of testing the algorithm, the first simulations set the cost of PS as low as

Five transmission expansion plans considering low-cost phase-shifter transformers.

Circuits | Plan | Plan | Plan | Plan | Plan | |||||

TL | PS | TL | PS | TL | PS | TL | PS | TL | PS | |

1 | ||||||||||

1 | ||||||||||

1 | ||||||||||

1 | ||||||||||

1 | 1 | 1 | 1 | 1 | ||||||

2 | 2 | 1 | 2 | 2 | ||||||

1 | 1 | |||||||||

1 | ||||||||||

1 | 1 | |||||||||

1 | 1 | |||||||||

1 | 1 | |||||||||

1 | ||||||||||

1 | 1 | 1 | 1 | |||||||

1 | ||||||||||

2 | 2 | 1 | 1 | |||||||

1 | 1 | |||||||||

1 | 1 | |||||||||

1 | ||||||||||

2 | ||||||||||

1 | 1 | |||||||||

Partial cost | ||||||||||

Total cost |

A very interesting fact in the simulations with low-cost PS refers to transmission lines. For the five plans illustrated in Table

Results also confirmed that the optimal solution for the transportation model is not feasible for the DC model since KVL constraints are violated. Nonetheless, during the simulation, this problem was overcome by adding low-cost PS in strategic branches.

Here there is considered

Five transmission expansion plans considering high-cost phase-shifter transformers.

Circuits | Plan | Plan | Plan | Plan | Plan | |||||

TL | PS | TL | PS | TL | PS | TL | PS | TL | PS | |

1 | 1 | |||||||||

1 | 1 | 1 | ||||||||

1 | 1 | 1 | 1 | 1 | ||||||

2 | 2 | 1 | 2 | 2 | ||||||

1 | ||||||||||

1 | 1 | 1 | 1 | |||||||

1 | ||||||||||

1 | 1 | 1 | 1 | 1 | ||||||

1 | 1 | |||||||||

2 | 2 | 1 | 1 | |||||||

1 | ||||||||||

2 | 2 | |||||||||

1 | ||||||||||

Partial cost | ||||||||||

Total cost |

Again, allocation of lines deserves importance. Simulations carried out with high-cost PS indicate that there was no addition of PS in none of the five plans of expansion of the system. The justification for the absence of PS, in all the topologies, is that they are too expensive. Thus, they did not take part of the optimal solutions. Consequently, there is a tendency of solely adding transmission lines.

Another interesting fact that deserves emphasis is that the results are the same found to the conventional DC model.

PSs with arbitrary intermediary costs were employed in order to produce expansion proposals with intermediary values when compared to extreme solutions obtained in previous simulations. The PS was considered with cost of

Plan

Circuits | Plan | Plan | Plan | |||

TL | PS | TL | PS | TL | PS | |

1 | 1 | 1 | ||||

1 | 1 | 1 | ||||

2 | 2 | 2 | ||||

1 | ||||||

1 | ||||||

1 | 1 | |||||

1 | ||||||

1 | ||||||

1 | 1 | 1 | ||||

2 | 2 | 2 | ||||

1 | 1 | 1 | ||||

1 | 1 | 1 | ||||

Partial cost | ||||||

Total cost |

The proposed algorithm found alternative optimal solutions for the plan

It is important to mention that all tests have been carried out for PS with nonrealistic cost values in order to test, from the theoretical point of view, the feasibility of modeling such devices as expansion components for electrical systems. Additional network transmission planning bibliography can be found in [

Nowadays, modern elements, such as FACTS devices, are playing an important role in transmission systems. In this way, inclusion of such devices jointly with classical components is of importance for the transmission expansion planning problem. Thus, this work was aimed at presenting the technical feasibility of considering phase-shifter transformers as components for the long-term transmission expansion planning, jointly with conventional transformers and transmission lines.

A novel methodology was proposed for the inclusion of phase-shifter transformers in the mathematical model that represents the transmission planning problem. The proposed model is more complex than the model DC. However, the mathematical problem was solved adequately with genetic algorithms.

Tests have shown the model consistency as well as the high performance of the algorithm.

This contribution extends the utilization of classical components during the expansion-planning problem, to modern elements, such as the FACTS devices.

Investment costs (US$)

Cost of a circuit that can be added in path

Number of circuits added in path

Node-branch transposed incidence matrix of the system

Active power flow composed by elements

Generation composed by elements

Demand of the buses

Total active power flow through path

Susceptance of one circuit in path

Base case total number of circuits

Phase angle of bus

Active power flow limit of one circuit in path

Generation level of bus

Generation capacity limit of bus

Number of circuits that can be added in path

Set of all paths

Fixed cost of a PS in path

Represents the presence (1) or not (0) of a PS in path

Penalty factor due to load shedding

Artificial generator at load bus

Set of buses with load

Angular difference of a PS in path

Number of PS added in path

Total number of paths in the network

Number of the circuits added in path

Represents the presence (1) or not (0) of a PS in path

Load shedding costs of a configuration (US$)

Performance coefficient of one transmission line in path

Crossover rate

Mutation rate

Flow utilization coefficient of one transmission line in path

Population diversification ratio (%)

Total number of repeated configurations in the current population

Total number of configurations of the population.

This work was supported by CAPES, CNPq, and Fundação Araucaria.