Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 790974, 11 pages

http://dx.doi.org/10.1155/2015/790974

## Method for Determining the Maximum Allowable Capacity of Wind Farm Based on Box Set Robust Optimization

^{1}School of Electrical Engineering, Xuchang University, Xuchang 461000, China^{2}School of Electrical & Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Received 19 July 2015; Revised 16 October 2015; Accepted 18 October 2015

Academic Editor: Peng-Yeng Yin

Copyright © 2015 Lihui Guo and Hao Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the increasing penetration of wind power, the randomness and volatility of wind power output would have a greater impact on safety and steady operation of power system. In allusion to the uncertainty of wind speed and load demand, this paper applied box set robust optimization theory in determining the maximum allowable installed capacity of wind farm, while constraints of node voltage and line capacity are considered. Optimized duality theory is used to simplify the model and convert uncertainty quantities in constraints into certainty quantities. Under the condition of multi wind farms, a bilevel optimization model to calculate penetration capacity is proposed. The result of IEEE 30-bus system shows that the robust optimization model proposed in the paper is correct and effective and indicates that the fluctuation range of wind speed and load and the importance degree of grid connection point of wind farm and load point have impact on the allowable capacity of wind farm.

#### 1. Introduction

With unceasing increase of grid-connected wind power, the fluctuation and randomness of wind power output bring aggravated negative influence to power grid such as overvoltage and overloads [1–3]. The operating state of power system will be worse when the allowable capacity of wind farm is high. The basic method to solve the problem is to redesign network structure and improve electrical equipment, but it is undoubtedly uneconomic considering the high investment and long construction time. Therefore, the feasible measure at present is to research on maximum allowable capacity of wind farm on condition of ensuring the system can operate safely and steadily with unchanged network structure and original electrical device.

The allowable capacity refers to the maximum installed capacity of wind farms connected into the distribution network on the premise of guaranteeing safety and steady operation of power grid. Many factors can determine the allowable capacity, such as voltage regulation, harmonic distortion, short-circuit current, and transient stability. It will be a very complicated problem to calculate allowable capacity if all the factors are considered, and the result may not be universal either. The existing methods usually regard wind power as distributed generation (DG) to calculate the allowable capacity from one or several aspects [4–12].

Based on the voltage sensitivities related to active and reactive power injections, [4] directly determines the maximum power of distributed generation (DG) without leading to steady-state voltage violations. The maximum voltage deviation of customers, cables current limits, and transformer nominal value can be defined as three main criteria to determine allowable capacity of photovoltaics [5]. Reference [6] analyzes sensitivity of bus voltage and lines current to determine the maximum capacity that can be injected in a specific bus and then searches all buses to find the minimum acceptable allowable capacity. Due to the nonlinear current injected by inverter-based DG units, the DG penetration level could be limited by harmonic distortion constraint [7, 8]. In [9], the maximum allowable capacity of the distributed generator in the distribution network is analyzed for meeting the requirements of relay protection and limiting the impact to short-circuit current characteristics. Reference [10] deliberates the influence of DG’s allowable capacity on traditional three-stage current protection and then builds four constraints to determine the penetration level of DG. Reference [11] uses dynamic studies to investigate the behavior of the system at different penetration level of DG, the frequency, and voltage stability of all () contingencies. With the help of an improved transient energy function, the transient energy margin can be calculated to quantify transient stability of power grid integrated with wind generation and analyze penetration level of wind generation furthermore [12].

The existing methods suppose that wind farm has a stable power output and ignores the uncertainty and fluctuation of wind speed; moreover the load demand in distribution network is in fluctuation condition. The coupled uncertainty of wind speed and load demand increases dimension of uncertain factors and complexity of the analysis. Without considering the effect of these uncertain variables, the conventional method for determining allowable capability of wind farm cannot ensure that power grid operates safely and steadily in a variety of conditions. The stochastic approach uses Weibull distribution to express the probability of wind speed and develops a chance constrained model to calculate the allowable capacity of wind farm [13], but the work has some limitations in application. First, it is difficult to identify an accurate probability distribution of the uncertainty. A large number of scenarios should be collected to get relatively accurate shape parameter and scale parameter in Weibull distribution [14, 15], which increases complexity of analysis procedure and deviation of evaluation result. Second, it is not easy to represent the wind power “ramp” events in the scenarios [16], which refer to the situations where wind power output increases or decreases significantly during a short period of time due to fast-moving weather phenomena.

Robust optimization has recently gained substantial popularity as a powerful tool to solve problems in power system with uncertainty parameter. Instead of making assumption on specific probability distribution, the robust optimization approach replaces the random parameters by predetermined uncertainty sets in the deterministic formulation [17]. The uncertainty set can be easily constructed from the historical data, such as the mean value and range of uncertainty data. The robust model immunizes against all realizations of the uncertain data within uncertainty set containing worst-case scenario, so power system can have normal operation in all uncertainty scenarios [18]. Combining robust optimization theory with fast decoupling PQ method, the paper proposes a method for determining the maximum allowable capacity of wind farm with the uncertainty of wind speed and load demand taken into consideration. The method uses box set to formulate the uncertainty data and make constraints for node voltage and line capacity. To solve the proposed model conveniently, duality theory is applied to convert uncertainty quantities in constraints to certainty quantities as well as the proposed model into linear programming model. Under the condition of multi wind farms, the paper further builds a bilevel optimization model to calculate penetration capacity. The result of IEEE 30-bus system analyzes the influence on allowable capacity of wind farm caused by fluctuation range of wind speed and load as well as the importance degree of grid connection point of wind farm and load point.

The paper is organized as follows. Section 2 introduces allowable capacity model based on linear programming. Section 3 describes robust linear optimization model and box set. Section 4 shows allowable capacity model of single wind farm based on robust linear optimization and bilevel optimization model for allowable capacity of multi wind farms. Section 5 performs a set of computational studies and reports numerical results. Section 6 concludes the paper.

#### 2. Allowable Capacity Model Based on Linear Programming

##### 2.1. Mathematical Model

The allowable capacity of wind farm connection to gird is the maximum allowable capacity of wind farm with some constraints. The mathematical model based on linear programming can be expressed as where is allowable capacity of single wind farm; and are numbers of conventional generators and wind farms; is network loss; is total load demand; is active power output of conventional generator ; , are upper and lower limit for active power output of conventional generator ; is active power transferred in line ; is maximum active power transferred in line ; is voltage of node ; is nominal voltage; is allowable voltage deviation.

##### 2.2. Load Flow Calculation

Power flow calculation has two basic calculation methods: DC model and AC model. The DC power flow model is an imprecise method and cannot calculate node voltage, so the paper adopts AC power flow model. According to physical characteristics of power grid, many decoupling flow algorithms have been introduced. The fast decoupling PQ method adjusts phase angle and voltage amplitude based on active power and reactive power, respectively:where and are increment of active power and reactive power at node; , are increment of phase angle and voltage amplitude at node. and are sensitivity matrix.

Considering some small electrical parameter, modified equation of fast decoupling PQ method can be expressed as [19]

Suppose the voltage is per unit value; the simplified equation can be written as where , are node admittance matrix; the nondiagonal and diagonal elements in , can be formulated aswhere and are resistance and inductance of branch ; is grounding branch susceptance of node .

The incremental expression of phase angle and voltage amplitude can be derived from (8) and expressed aswhere , are increment of active power output for generator and active power demand for load; are power factor of generator and load; , are corresponding submatrix for generator and load in ; , are corresponding submatrix for generator and load in .

The power flow in branch can be expressed as [20]where and are active power and reactive power transferred in branch , and are node voltage on both ends of branch ; is phase-angle difference between nodes of branch ; , are susceptance and conductance of branch .

Based on theoretical basis and simplified method of decoupling method, formulation (11) can be expressed with the same form of formulation (8):where are increment of active power and reactive power at branch; are sensitivity matrix between phase angle, voltage amplitude, and branch flow.

The incremental expression of active power and reactive power at branch can be derived from (12) and (10) and expressed as

Then active power and reactive power at branch can be written as where , are sensitivity matrix of active power output for generator; , are sensitivity matrix of active power demand for load.

In a similar fashion, the node voltage can be expressed aswhere , are sensitivity matrix between generator, load, and node voltage.

#### 3. Robust Linear Optimization

##### 3.1. Robust Linear Optimization Model

Robust linear optimization (RLP) model was proposed by Soyster in the 1970s [21], which is formulated to find an uncertainty-immunized solution within the uncertainty set. The solution remains feasible and nearly optimal for all scenarios and ensures power system is in stable operation even in the worst scenario. When decision maker needs to develop solutions under uncertainty environment, the robust optimization can guarantee the solution has feasibility and immunization against the effect of data uncertainty. The problem of determining maximum allowable capacity of wind farm in the planning stage belongs to the typical robust optimization problem.

A general robust optimization model is as follows:where is decision variable; is uncertainty data; is uncertainty set.

A general linear programming model is as follows:where is decision variable; is objective function; is constraint condition.

Matrices and can be defined as

Then (17) can be expressed as The model supposes , , ; the is uncertainty set.

The uncertainty of the coefficient matrix can be summarized as the uncertainty of constraint matrix , so the corresponding robust optimization model for (19) can be expressed as

The constraint condition, , , , is equivalent to solve the following problems [22]:

Model (21) is then transformed into a deterministic linear optimization problem, which is called the robust counterpart and is easy to solve.

##### 3.2. Uncertainty Set

The construction of uncertainty set and simplification of complicated model with specific are two issues of crucial importance in robust optimization. Considering operability and convenience in calculation, the paper adopts box uncertainty set to describe the fluctuation of uncertainty data [23]:where is unit vector; , are disturbance range of uncertainty parameters in robust optimization (the uncertainty parameters cover wind speed and load demand in the paper); , and , are upper and lower limit of .

#### 4. RLP Model for Allowable Capacity of Wind Farm

Based on (14) and (15), the line capacity constraint and node voltage constraint can be deduced to

If the maximum active power at branch meets constraint (23), then all branches can meet the line capacity constraint, so inequality constraint (23) can be expressed as

Similarly, inequality constraint (24) can be expressed as

So the allowable capacity model based on linear programming can evolve to the box set robust optimization model:

##### 4.1. Single Wind Farm with Uncertainty Wind Speed

The work first considers the allowable capacity of single wind farm with uncertainty of wind speed. The output power of wind farm mainly depends on wind speed and wind direction. With a view that the wind turbines located at same wind farm have almost the same wind speed and wind direction, we can suppose wind turbines located at the same wind farm have same output power and use an equivalent wind turbine to simulate wind farm. The output power of wind farm mainly depends on wind speed; the mathematical relation between the two variables can be approximately expressed as the piecewise function shown in Figure 1:where is rated output power; is wind speed; is cut-in wind speed; is rated wind speed; is cut-out wind speed.