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In order to consider the uncertainty and correlation of wind power in multiobjective transmission network expansion planning (TNEP), this paper presents an extended point-estimation method to calculate the probabilistic power flow, based on which the correlative power outputs of wind farm are sampled and the uncertain multiobjective transmission network planning model is transformed into a solvable deterministic model. A modified epsilon multiobjective evolutionary algorithm is used to solve the above model and a well-distributed Pareto front is achieved, and then the final planning scheme can be obtained from the set of nondominated solutions by a fuzzy satisfied method. The proposed method only needs the first four statistical moments and correlation coefficients of the output power of wind farms as input information; the modeling of wind power is more precise by considering the correlation between wind farms, and it can be easily combined with the multiobjective transmission network planning model. Besides, as the self-adaptive probabilities of crossover and mutation are adopted, the global search capabilities of the proposed algorithm can be significantly improved while the probability of being stuck in the local optimum is effectively reduced. The accuracy and efficiency of the proposed method are validated by IEEE 24 as well as a real system.

With the increasing complexity of power system, transmission network expansion planning needs to consider multiple objectives, such as investment cost, security, and network losses. Since these objectives may be conflicting with each other, it is hard for the a priori approaches, such as weighted method and goal programming [

There are many uncertain factors affecting the TNEP, and one of the current research focuses is wind power [

Although much work has been done to handle the influence of wind power in TNEP, the correlation between wind farms is seldom considered [

In this paper, we establish a multiobjective TNEP model considering the investment cost and network losses cost as well as the sum of the line overload capacity both under the normal operation and single contingency condition. Based on the proposed model, an extended

The rest of the paper is organized as follows. After the wind power is modeled, an extended

In the previous work, Weibull distribution is widely used to sample the wind speed [

The formulation above is a piecewise function, so it is hard to obtain the distribution of the wind turbine output. In this paper, we adopt the yearly historical outputs of wind farms as input data, and the distribution of wind farm output is described by its first four statistical moments, including the mean value, standard deviation, skewness, and kurtosis, in which the skewness is a measure of the extent to which a probability distribution of a real-valued random variable “leans” to one side of the mean, and the kurtosis is a measure of both the “peakedness” of the distribution and the heaviness of its tail. These four statistics can be obtained from the historical wind power data easily; given the historical wind speed data, the output power of wind farm can be achieved by (

In order to improve the accuracy of the modeling of wind power generation by considering the spatial correlation among the adjacent wind farms, here we use the variance-covariance matrix, which can be easily achieved from the yearly curve of output from wind farms, to characterize such correlation between wind farms.

Basic theory of point-estimate method is introduced in [

Although most of the input random variables in power system are statistically dependent, the previous work, however, has sidestepped it. To consider dependencies among the input random variables when calculating the PPF, an orthogonal transformation is used to transform the dependent input random variables into the independent ones, and then it can be processed readily through

Input the first four statistical moments of the

Decompose

Set the iteration count

Transform the first four central moments of the input variables as follows:

Calculate the concentrations

Construct the transformed points in the form

Solve a deterministic DC power flow for each of the three points

Since

Economic and security criteria are the most important factors in the transmission expansion planning. Based on [

The constraints of the above multiobjective optimization problem are as follows.

Network under normal conditions:

Network under single contingency conditions:

In the above constraints, the constraint (

A set of nondominated solutions will be obtained after solving the TNEP model. To choose the final plan, a fuzzy satisfying method based on the distance metric method is adopted to help the planner make the decision. The fuzzy sets are defined by a linear membership functions as follows:

Since the value of

Most MOEAs developed in the past decade are either good for achieving well-distributed solutions at the expense of a large computational effort or computationally fast at the expense of obtaining a not-so-accurate distribution of solutions [

Different from the existing MOEAS, the

The

Because the value of

Another performance improvement technique of the original

The adaptive adjusting curve of

Figure

Flowchart of the proposed algorithm.

In this section, a modified IEEE 24-bus test system is used to demonstrate the performance of the proposed algorithm. Assume that the system will be expanded for future conditions with the generation and load demand increasing by 2.2 times their original values, that is, load level of 6720 MW and generation level of 7490 MW. This system has 32 generators, 38 existing lines, and 89 candidate lines (including substations). The candidate lines which can be added in 29 existing right-of-ways and ten new right-of-ways are limited to three and two, respectively; up to four transformers can be installed in the substations; all data of the candidate lines can be found in [

The test system is divided into two regions, namely, the northern region with nodes 1–10 and the southern region with nodes 11–24, and three wind farms are integrated to each region. In order to investigate the effects of the wind power on the test system, five different cases are designed.

The test system is only supplied by the conventional units.

Partial conventional units are replaced by wind energy, and the total wind energy penetration is 5% of all load capacities for the years to come, and the outputs of wind farms are independent.

The correlation among wind farms is considered; other conditions are the same as Case

The total wind energy penetration is 10%; other conditions are the same as Case

The total wind energy penetration is 15%; other conditions are the same as Case

In Case

Wind power input data of test system.

Wind farm integration node | Mean (MW) | Standard deviation (MW) | Skewness | Kurtosis |
---|---|---|---|---|

1 | 103.4 | 66 | 0.9269 | 3.6858 |

2 | 100.7 | 63.2 | 1.0265 | 3.5026 |

7 | 102.1 | 65.5 | 0.9604 | 3.2001 |

15 | 101.7 | 71.7 | 1.0385 | 3.2713 |

18 | 96.1 | 69.6 | 1.1253 | 4.2175 |

23 | 102.3 | 82.7 | 1.4435 | 4.4517 |

From Table

Figure

Final results of Cases

Final Pareto front of Cases

Trade-off between investment cost and network losses cost in Cases

Final results of Cases

Final results of Cases

Final Pareto front of Cases

Trade-off between investment cost and network losses cost in Cases

Since the final results of the proposed method are a set of nondominated solutions, the decision making analysis method needs to be applied to obtain the final optimal plan. Table

Final plan of Cases

Scheme | Investment cost (M$) | Amount of line overload (MW) | Network losses cost (M$) |
---|---|---|---|

Case |
940.7418 | 2.55659 | 1235.44 |

Case |
1189.26 | 20.0876 | 1184.35 |

Case |
1279.66 | 21.5414 | 1247.85 |

Trade-off graphs of Cases

Trade-off between investment cost and the amount of line overload of Cases

Trade-off between investment cost and network losses of Cases

Final plan of Case

The proposed method is applied to a 220 kV practical power system in the south of China as well. This system has 233 buses, 452 existing lines, and 164 candidate lines. It is assumed that five years later the total installed capacity will be 4729 MW and the total load is 4158 MW. There are four wind farms integrated into the system, including three wind farms with correlation coefficient 0.6 in the west and one in the northeastern, and the correlation between wind farms within two regions is 0.1; the total penetration of wind farm is 15% of the installed capacity. Table

Wind power input data of practical system.

Wind farm | Mean (MW) | Standard deviation (MW) | Skewness | Kurtosis |
---|---|---|---|---|

1 | 200.01 | 165.33 | 1.243 | 3.674 |

2 | 165.37 | 60.73 | 1.308 | 3.970 |

3 | 300.75 | 123.8 | 0.868 | 2.494 |

4 | 63.47 | 52.78 | 1.155 | 3.198 |

Two models, one considering the correlation between wind farms while the other not, are solved by the proposed method, and the final Pareto-optimal front in two-dimensional space is shown in Figure

Trade-off graphs of practical power system.

Trade-off between investment cost and the amount of line overload

Trade-off between investment cost and network losses

Assuming the planner has high requirements on the security of transmission system and wants to minimize the network losses cost, Table

Final planning scheme of the real test system.

Desirable level | Investment cost (M$) | Amount of line overload (MW) | Network losses cost (M$) | ||
---|---|---|---|---|---|

0.2 | 1 | 1 | 18390.9 | 30.7138 | 1663.62 |

0.4 | 1 | 1 | 17740.4 | 68.3243 | 1719.06 |

0.6 | 1 | 1 | 12395 | 33.4318 | 1956.2 |

Figure

Final planning scheme of real power system.

This paper established a multiobjective TNEP model, in which the investment cost and network losses cost as well as the line overload capacity both under the normal operation and single contingency condition are considered. To handle the uncertainty and correlation of wind power in TNEP model, an extended

The input data is very simple, because only the first four statistical moments and correlation coefficients of output power of wind farms are required.

The model is more precise by considering the correlation between wind farms, and it has been demonstrated by the final results of IEEE 24 and real power system.

Since the final results are a set of nondominated solutions, planners can flexibly choose the final planning scheme according to the practical situation.

Because of considering the N-1 check in the objective function, the evaluation of objective function is time-consuming. In our next work, the proposed method will be further improved by the application of parallel computation method.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National High Technology Research and Development Program of China (863 Program) under Grant 2012AA050201.