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

Authors: Lihui Guo; Hao Bai
Journal: Mathematical Problems in Engineering
 (2015)

---

## 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.


---

## Body

## 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 (N
-
1) 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. Section2 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(1)
m
a
x
P
w
(2)
s.t.
∑
i
=
1
N
g
P
g
i
+
∑
j
=
1
N
w
P
w
j
-
P
s
=
P
l
(3)
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
(4)
I
i
j
≤
I
¯
(5)
U
i
-
U
n
≤
ε
U
n
,where P
w is allowable capacity of single wind farm; N
g and N
w are numbers of conventional generators and wind farms; P
s is network loss; P
l is total load demand; P
g
i is active power output of conventional generator i; P
g
i
¯, P
g
i
_ are upper and lower limit for active power output of conventional generator i; I
i
j is active power transferred in line i
j; I
¯ is maximum active power transferred in line i
j; U
i is voltage of node i; U
n is nominal voltage; ε 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:(6)
Δ
P
=
H
Δ
θ
,
Δ
Q
=
L
Δ
V
V
,where Δ
P and Δ
Q are increment of active power and reactive power at node; Δ
θ, Δ
V are increment of phase angle and voltage amplitude at node. H and L are sensitivity matrix.Considering some small electrical parameter, modified equation of fast decoupling PQ method can be expressed as [19](7)
Δ
P
=
V
B
′
V
Δ
θ
,
Δ
Q
=
V
B
′
′
V
Δ
V
V
.Suppose the voltageV is per unit value; the simplified equation can be written as (8)
Δ
P
=
B
′
Δ
θ
,
Δ
Q
=
B
′
′
Δ
V
,where B
′, B
′
′ are node admittance matrix; the nondiagonal and diagonal elements in B
′, B
′
′ can be formulated as(9)
B
i
j
′
=
-
x
i
j
x
i
j
2
+
r
i
j
2
,
B
i
i
′
=
∑
j
∈
i
B
i
j
′
,
B
i
j
′
′
=
-
1
x
i
j
,
B
i
i
′
′
=
-
∑
j
∈
i
B
i
j
′
′
-
b
i
,where r
i
j and x
i
j are resistance and inductance of branch i
j; b
i is grounding branch susceptance of node i.The incremental expression of phase angle and voltage amplitude can be derived from (8) and expressed as(10)
Δ
θ
=
B
g
′
-
1
Δ
P
g
+
B
l
′
-
1
Δ
P
l
,
Δ
V
=
B
g
′
′
-
1
Δ
P
g
tan
⁡
φ
g
+
B
l
′
′
-
1
Δ
P
l
tan
⁡
φ
l
,where Δ
P
g, Δ
P
l are increment of active power output for generator and active power demand for load; tan
⁡
φ
g
,
tan
⁡
φ
l are power factor of generator and load; B
g
′, B
l
′ are corresponding submatrix for generator and load in B
′; B
g
′
′, B
l
′
′ are corresponding submatrix for generator and load in B
′
′.The power flow in branch can be expressed as [20](11)
P
i
j
=
V
i
2
-
V
i
V
j
cos
⁡
θ
i
j
G
i
j
-
V
i
V
j
B
i
j
sin
⁡
θ
i
j
,
Q
i
j
=
-
V
i
V
j
G
i
j
sin
⁡
θ
i
j
-
V
i
2
-
V
i
V
j
cos
⁡
θ
i
j
B
i
j
,where P
i
j and Q
i
j are active power and reactive power transferred in branch i
j, V
i and V
j are node voltage on both ends of branch i
j; θ
i
j is phase-angle difference between nodes of branch i
j; B
i
j, G
i
j are susceptance and conductance of branch i
j.Based on theoretical basis and simplified method of decoupling method, formulation (11) can be expressed with the same form of formulation (8):(12)
Δ
P
b
=
E
Δ
θ
,
Δ
Q
b
=
F
Δ
V
,where Δ
P
b
,
Δ
Q
b are increment of active power and reactive power at branch; E
,
F 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 (13)
Δ
P
b
=
E
Δ
θ
=
E
B
g
′
-
1
Δ
P
g
+
B
l
′
-
1
Δ
P
l
=
J
g
Δ
P
g
+
J
l
Δ
P
l
,
Δ
Q
b
=
F
Δ
V
=
F
B
g
′
′
-
1
Δ
P
g
tan
⁡
φ
g
+
B
l
′
′
-
1
Δ
P
l
tan
⁡
φ
l
=
K
g
Δ
P
g
+
K
l
Δ
P
l
.Then active power and reactive power at branch can be written as(14)
P
b
=
J
g
P
g
+
J
l
P
l
,
Q
b
=
K
g
P
g
+
K
l
P
l
,where J
g, K
g are sensitivity matrix of active power output for generator; J
l, K
l are sensitivity matrix of active power demand for load.In a similar fashion, the node voltage can be expressed as(15)
V
=
T
g
P
g
+
T
l
P
l
,where T
g, T
l are sensitivity matrix between generator, load, and node voltage.


## 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(1)
m
a
x
P
w
(2)
s.t.
∑
i
=
1
N
g
P
g
i
+
∑
j
=
1
N
w
P
w
j
-
P
s
=
P
l
(3)
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
(4)
I
i
j
≤
I
¯
(5)
U
i
-
U
n
≤
ε
U
n
,where P
w is allowable capacity of single wind farm; N
g and N
w are numbers of conventional generators and wind farms; P
s is network loss; P
l is total load demand; P
g
i is active power output of conventional generator i; P
g
i
¯, P
g
i
_ are upper and lower limit for active power output of conventional generator i; I
i
j is active power transferred in line i
j; I
¯ is maximum active power transferred in line i
j; U
i is voltage of node i; U
n is nominal voltage; ε 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:(6)
Δ
P
=
H
Δ
θ
,
Δ
Q
=
L
Δ
V
V
,where Δ
P and Δ
Q are increment of active power and reactive power at node; Δ
θ, Δ
V are increment of phase angle and voltage amplitude at node. H and L are sensitivity matrix.Considering some small electrical parameter, modified equation of fast decoupling PQ method can be expressed as [19](7)
Δ
P
=
V
B
′
V
Δ
θ
,
Δ
Q
=
V
B
′
′
V
Δ
V
V
.Suppose the voltageV is per unit value; the simplified equation can be written as (8)
Δ
P
=
B
′
Δ
θ
,
Δ
Q
=
B
′
′
Δ
V
,where B
′, B
′
′ are node admittance matrix; the nondiagonal and diagonal elements in B
′, B
′
′ can be formulated as(9)
B
i
j
′
=
-
x
i
j
x
i
j
2
+
r
i
j
2
,
B
i
i
′
=
∑
j
∈
i
B
i
j
′
,
B
i
j
′
′
=
-
1
x
i
j
,
B
i
i
′
′
=
-
∑
j
∈
i
B
i
j
′
′
-
b
i
,where r
i
j and x
i
j are resistance and inductance of branch i
j; b
i is grounding branch susceptance of node i.The incremental expression of phase angle and voltage amplitude can be derived from (8) and expressed as(10)
Δ
θ
=
B
g
′
-
1
Δ
P
g
+
B
l
′
-
1
Δ
P
l
,
Δ
V
=
B
g
′
′
-
1
Δ
P
g
tan
⁡
φ
g
+
B
l
′
′
-
1
Δ
P
l
tan
⁡
φ
l
,where Δ
P
g, Δ
P
l are increment of active power output for generator and active power demand for load; tan
⁡
φ
g
,
tan
⁡
φ
l are power factor of generator and load; B
g
′, B
l
′ are corresponding submatrix for generator and load in B
′; B
g
′
′, B
l
′
′ are corresponding submatrix for generator and load in B
′
′.The power flow in branch can be expressed as [20](11)
P
i
j
=
V
i
2
-
V
i
V
j
cos
⁡
θ
i
j
G
i
j
-
V
i
V
j
B
i
j
sin
⁡
θ
i
j
,
Q
i
j
=
-
V
i
V
j
G
i
j
sin
⁡
θ
i
j
-
V
i
2
-
V
i
V
j
cos
⁡
θ
i
j
B
i
j
,where P
i
j and Q
i
j are active power and reactive power transferred in branch i
j, V
i and V
j are node voltage on both ends of branch i
j; θ
i
j is phase-angle difference between nodes of branch i
j; B
i
j, G
i
j are susceptance and conductance of branch i
j.Based on theoretical basis and simplified method of decoupling method, formulation (11) can be expressed with the same form of formulation (8):(12)
Δ
P
b
=
E
Δ
θ
,
Δ
Q
b
=
F
Δ
V
,where Δ
P
b
,
Δ
Q
b are increment of active power and reactive power at branch; E
,
F 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 (13)
Δ
P
b
=
E
Δ
θ
=
E
B
g
′
-
1
Δ
P
g
+
B
l
′
-
1
Δ
P
l
=
J
g
Δ
P
g
+
J
l
Δ
P
l
,
Δ
Q
b
=
F
Δ
V
=
F
B
g
′
′
-
1
Δ
P
g
tan
⁡
φ
g
+
B
l
′
′
-
1
Δ
P
l
tan
⁡
φ
l
=
K
g
Δ
P
g
+
K
l
Δ
P
l
.Then active power and reactive power at branch can be written as(14)
P
b
=
J
g
P
g
+
J
l
P
l
,
Q
b
=
K
g
P
g
+
K
l
P
l
,where J
g, K
g are sensitivity matrix of active power output for generator; J
l, K
l are sensitivity matrix of active power demand for load.In a similar fashion, the node voltage can be expressed as(15)
V
=
T
g
P
g
+
T
l
P
l
,where T
g, T
l 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:(16)
m
i
n
x
∈
R
sup
ζ
∈
u
⁡
f
0
x
,
ζ
s.t.
sup
ζ
∈
u
⁡
f
i
x
,
ζ
≤
0
,
i
=
1
,
…
,
m
,where x is decision variable; ζ is uncertainty data; u is uncertainty set.A general linear programming model is as follows:(17)
min
⁡
C
T
x
:
A
x
≤
b
∣
C
,
A
,
b
∈
u
∈
R
n
×
R
m
×
n
×
R
m
,where x is decision variable; C
T
x is objective function; A
x
≤
b is constraint condition.MatricesA and b can be defined as(18)
A
=
a
1
,
…
,
a
m
T
,
b
=
b
1
,
…
,
b
m
T
.Then (17) can be expressed as(19)
min
x
⁡
C
T
x
:
a
i
x
≤
b
i
,
i
=
1
,
…
,
m
,
x
∈
R
n
. The model supposes a
1
∈
u
1, a
2
∈
u
2
,
…, a
m
∈
u
m; the u
i
i
=
1
,
…
,
m is uncertainty set.The uncertainty of the coefficient matrixC can be summarized as the uncertainty of constraint matrix A, so the corresponding robust optimization model for (19) can be expressed as (20)
m
i
n
sup
⁡
C
T
x
s.t.
sup
⁡
a
i
x
≤
b
i
,
∀
a
i
∈
u
i
,
i
=
1
,
…
,
m
.The constraint condition,s
.
t
.
sup
⁡
a
i
x
≤
b
i, ∀
a
i
∈
u
i, i
=
1
,
…
,
m, is equivalent to solve the following problems [22]:(21)
max
a
i
∈
u
i
⁡
a
i
x
≤
b
i
,
∀
i
=
1
,
…
,
m
.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 setu and simplification of complicated model with specific u 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]:(22)
u
=
ς
,
κ
e
T
ς
=
0
,
ς
_
≤
ς
≤
ς
-
e
T
κ
=
0
,
κ
_
≤
κ
≤
κ
-
,where e is unit vector; ς, κ are disturbance range of uncertainty parameters in robust optimization (the uncertainty parameters cover wind speed and load demand in the paper); ς
-, κ
- and ς
_, κ
_ are upper and lower limit of ς
,
κ.


## 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:(16)
m
i
n
x
∈
R
sup
ζ
∈
u
⁡
f
0
x
,
ζ
s.t.
sup
ζ
∈
u
⁡
f
i
x
,
ζ
≤
0
,
i
=
1
,
…
,
m
,where x is decision variable; ζ is uncertainty data; u is uncertainty set.A general linear programming model is as follows:(17)
min
⁡
C
T
x
:
A
x
≤
b
∣
C
,
A
,
b
∈
u
∈
R
n
×
R
m
×
n
×
R
m
,where x is decision variable; C
T
x is objective function; A
x
≤
b is constraint condition.MatricesA and b can be defined as(18)
A
=
a
1
,
…
,
a
m
T
,
b
=
b
1
,
…
,
b
m
T
.Then (17) can be expressed as(19)
min
x
⁡
C
T
x
:
a
i
x
≤
b
i
,
i
=
1
,
…
,
m
,
x
∈
R
n
. The model supposes a
1
∈
u
1, a
2
∈
u
2
,
…, a
m
∈
u
m; the u
i
i
=
1
,
…
,
m is uncertainty set.The uncertainty of the coefficient matrixC can be summarized as the uncertainty of constraint matrix A, so the corresponding robust optimization model for (19) can be expressed as (20)
m
i
n
sup
⁡
C
T
x
s.t.
sup
⁡
a
i
x
≤
b
i
,
∀
a
i
∈
u
i
,
i
=
1
,
…
,
m
.The constraint condition,s
.
t
.
sup
⁡
a
i
x
≤
b
i, ∀
a
i
∈
u
i, i
=
1
,
…
,
m, is equivalent to solve the following problems [22]:(21)
max
a
i
∈
u
i
⁡
a
i
x
≤
b
i
,
∀
i
=
1
,
…
,
m
.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 setu and simplification of complicated model with specific u 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]:(22)
u
=
ς
,
κ
e
T
ς
=
0
,
ς
_
≤
ς
≤
ς
-
e
T
κ
=
0
,
κ
_
≤
κ
≤
κ
-
,where e is unit vector; ς, κ are disturbance range of uncertainty parameters in robust optimization (the uncertainty parameters cover wind speed and load demand in the paper); ς
-, κ
- and ς
_, κ
_ are upper and lower limit of ς
,
κ.


## 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(23)
J
g
P
g
+
J
l
P
l
-
I
¯
≤
0
,
(24)
T
g
P
g
+
T
l
P
l
-
U
n
-
ε
U
n
≤
0
.If the maximum active power at branchmax
⁡
P
b meets constraint (23), then all branches can meet the line capacity constraint, so inequality constraint (23) can be expressed as(25)
max
⁡
J
g
P
g
+
J
l
P
l
≤
I
¯
.Similarly, inequality constraint (24) can be expressed as(26)
max
⁡
T
g
P
g
+
T
l
P
l
-
U
n
≤
ε
U
n
.So the allowable capacity model based on linear programming can evolve to the box set robust optimization model:(27)
m
a
x
P
w
s.t.
∑
i
=
1
N
g
P
g
i
+
∑
j
=
1
N
w
P
w
j
-
P
s
=
P
l
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
max
⁡
J
g
P
g
+
J
l
P
l
≤
I
¯
max
⁡
T
g
P
g
+
T
l
P
l
-
U
n
≤
ε
U
n
.


### 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 Figure1:(28)
P
w
=
0
,
v
≤
v
ci
,
v
≥
v
co
⁡
P
r
v
-
v
ci
v
r
-
v
ci
,
v
ci
≤
v
≤
v
r
P
r
,
v
r
≤
v
≤
v
co
⁡
,where P
r is rated output power; v is wind speed; v
ci is cut-in wind speed; v
r is rated wind speed; v
c
o is cut-out wind speed.Figure 1
Wind turbine output power-speed curve.Wind speed can be expressed as the estimated value coupled with a distributed rangev
0
+
ς; the paper adopts box set to express the disturbance range of wind speed. The output of wind turbine can be expressed as(29)
P
w
=
P
r
v
0
-
v
c
i
v
r
-
v
c
i
+
P
r
ς
v
r
-
v
c
i
,
u
=
ς
e
T
ς
=
0
,
ς
_
≤
ς
≤
ς
-
.Thev
0 is estimated wind speed (it can be found from meteorological data); ς is an artificial decision according to actual wind condition.(1) Eliminate Equality Constraint. The constraints of mathematical formulation (27) cover absolute value, maximum value, and uncertainty parameter. It is tough to solve the problem, so the troublesome work should be converted to analytical form with convenience solution. To simplify the model, the paper substitutes the equality constraints for power balance into inequality constraints for node voltage and line capacity. The inequality constraint for line capacity firstly is converted; detailed information is listed below.The active power at branch can be expressed as(30)
P
b
=
J
g
P
g
+
J
l
P
l
=
J
g
P
g
1
⋯
P
w
⋯
P
g
N
g
+
J
l
P
l
1
⋯
P
l
n
⋯
P
l
N
l
,where N
l is number of loads.If the wind farm locates at nodem, based on equality constraint for power balance in (2), the load demand at node n can be expressed as (31)
P
l
n
=
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
P
w
-
P
s
-
∑
i
=
1
,
i
≠
n
N
l
P
l
i
.Based on (30) and (31), the equality constraint in (27) can be eliminated. The active power at branch can be expressed as (32)
P
b
=
P
w
J
g
+
J
l
n
+
∑
i
=
1
,
i
≠
m
N
g
P
g
i
J
g
+
J
l
n
+
J
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
-
J
l
n
∑
i
=
1
,
i
≠
n
N
l
P
l
i
-
J
l
n
P
s
=
P
w
J
g
+
J
l
n
+
J
l
n
Y
+
Z
,
Y
=
∑
i
=
1
,
i
≠
m
N
g
P
g
i
-
∑
i
=
1
,
i
≠
n
N
l
P
l
i
-
P
s
,
Z
=
J
g
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
J
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
.Considering the inequality constraint in (27) and branch flow in (28), the line capacity constraint can be expressed as(33)
max
⁡
P
w
J
g
+
J
l
n
+
J
l
n
Y
+
Z
≤
I
¯
.Similarly, inequality constraint for node voltage in (28) can be expressed as(34)
max
⁡
P
w
T
g
+
T
l
n
+
T
l
n
Y
-
U
n
+
M
≤
ε
U
n
,
M
=
T
g
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
T
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
.The equality constraint is eliminated and the model only has inequality constraints.(2) Handle Uncertain Parameters. Considering the uncertain parameters in inequality constraint, optimized duality theory is used to simplify the model and convert uncertainty quantities in constraints to certainty quantities, changing the model to linear programming model.Combined with (29), (33) can be expressed as(35)
max
⁡
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
Z
≤
I
¯
=
max
⁡
ς
P
r
v
r
-
v
c
i
J
g
+
J
l
n
≤
I
¯
-
J
l
n
Y
-
Z
-
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
.According to duality theory, (34) can be formulated as(36)
-
min
⁡
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
≤
I
¯
-
J
l
n
Y
-
Z
-
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
.To construct a Lagrange function,(37)
L
i
ς
,
η
,
ν
,
λ
=
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
-
η
i
ς
+
ν
i
ς
_
-
ς
+
λ
i
ς
-
ς
-
,
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
-
P
r
v
r
-
v
ci
J
g
+
J
l
n
-
η
i
+
ν
i
+
λ
i
=
0
,where η, ν, λ are Lagrange coefficients.According to duality theory, we can find(38)
min
⁡
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
=
max
η
,
γ
,
λ
⁡
min
ς
⁡
L
i
ς
,
η
,
ν
,
λ
=
m
a
x
η
,
γ
,
λ
-
ν
ς
_
-
λ
ς
-
:
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
.Then(39)
-
min
⁡
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
=
-
max
η
,
γ
,
λ
⁡
-
ν
ς
_
-
λ
ς
-
:
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
=
min
η
,
γ
,
λ
⁡
ν
ς
_
-
λ
ς
-
:
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
.Based on (39), (36) can be expressed as(40)
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
ν
ς
_
-
λ
ς
-
+
Z
≤
I
¯
.The same method is applied to node voltage constraints in (27), and thereafter the robust optimization model for allowable capacity of single wind farm only considering the uncertainty of wind speed can be expressed as(41)
m
a
x
P
r
s.t.
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
ν
ς
_
-
λ
ς
-
+
Z
≤
I
¯
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
P
r
v
0
-
v
ci
v
r
-
v
ci
T
g
+
T
l
n
+
T
l
n
Y
+
b
ς
_
-
c
ς
-
+
M
-
U
n
≤
ε
U
n
M
=
T
g
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
T
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
∇
ς
L
i
ς
,
a
,
b
,
c
=
-
P
r
v
r
-
v
ci
T
g
+
T
l
n
-
a
i
+
b
i
+
c
i
=
0
b
≥
0
,
c
≥
0
,where a, b, c are Lagrange coefficients.


### 4.2. Single Wind Farm with Uncertainty Wind Speed and Load Demand

The fluctuation of load produces difficulty in planning stage for wind farm. If the project determines the allowable capacity of wind farm based on peak load, wind power curtailment will increase when power grid has less load. According to valley load, the decision maker will determine a small allowable capacity of wind farm; it cannot make full use of clean energy.The fluctuation of load will change the sensitive matricesJ
l and T
l; if the load at node n has fluctuation, the output power of wind turbine can be expressed as(42)
P
w
=
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
Δ
J
l
n
+
J
l
n
+
Δ
J
l
n
Y
+
Z
,where Δ
J
l
n is variation of corresponding sensitive martix of load at node n. The paper defines Δ
J
l
n
=
κ
J
l
n, where κ is expressed as box set in (22).The paper uses mathematical derivation method mentioned in Section3.2 to achieve the robust optimization model for allowable capacity of single wind farm with uncertainty wind speed and load demand:(43)
m
a
x
P
r
s.t.
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
ν
ς
_
-
λ
ς
-
+
β
κ
_
-
γ
κ
-
+
Z
≤
I
¯
∇
κ
L
i
κ
,
α
,
β
,
γ
=
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
-
α
i
+
β
i
+
γ
i
β
≥
0
,
γ
≥
0
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
P
r
V
0
-
V
c
V
r
-
V
c
T
g
+
T
l
n
+
T
l
n
Y
+
b
ς
_
-
c
ς
-
+
e
κ
_
-
f
κ
-
≤
ε
U
n
+
M
∇
κ
L
i
κ
,
d
,
e
,
f
=
P
r
V
0
-
V
c
V
r
-
V
c
T
g
+
T
l
n
-
d
i
+
e
i
+
f
i
=
0
e
≥
0
,
f
≥
0
∇
ς
L
i
ς
,
a
,
b
,
c
=
0
b
≥
0
,
c
≥
0
.The model has been converted to a linear programming problem, which could be solved directly by commercial software such as ILOG CPLEX.


### 4.3. Multi Wind Farms with Uncertainty Wind Speed and Load Demand

When multi wind farms connect to power grid, if the sum of allowable capacity of single wind farm is defined as the total allowable capacity, the result will be very optimistic. We suppose the wind farms connect to power grid at nodem
1
⋯
m
n, one mathematical model is built to solve optimization problem with an objective function ∑
i
=
1
n
P
r
m
i, the total allowable capacity is P
m
a
x, and the capacity combination is P
~
r
m
1
⋯
P
~
r
m
n. There will probably be a capacity combination P
^
r
m
1
⋯
P
^
r
m
n; the capacity combination meets ∑
i
=
1
n
P
^
r
m
i
≤
∑
i
=
1
n
P
~
r
m
i. But the capacity combination will cause node voltage and line capacity to suffer out of limit. Based on the concept of min-max transfer capability in transmission interface [24], the paper proposes min-max allowable capacity P
m
i
n
-
m
a
x to represent total allowable capacity of multi wind farms. The P
m
i
n
-
m
a
x is the maximum allowable capacity of wind farm in the worst scenario; it contains the following two meanings: (1) for any capacity combination P
^
r
m
1
⋯
P
^
r
m
n, if it meets ∑
i
=
1
n
P
^
r
m
i
≤
P
m
i
n
-
m
a
x, the power system will meet constraint of node voltage and line capacity, which is required for safe and steady operation of power system; (2) under these conditions, P
m
i
n
-
m
a
x selects the larger number as much as possible, which is for reasons of maximum wind power. The bilevel optimization model can be used to solve the min-max allowable capacity. In the proposed mathematical model, the penetration level of wind farms is maximized in the upper level optimization, while the lower level optimization ensures that voltage constraints and capacity constraints are satisfied even under the worst case.Upper level optimization is(44)
m
a
x
∑
i
=
1
n
P
^
r
m
i
s.t.
∑
i
=
1
N
g
P
g
i
+
∑
j
=
1
N
w
P
w
j
-
P
s
=
P
l
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
I
P
^
r
m
1
⋯
P
^
r
m
n
≤
I
¯
U
P
^
r
m
1
⋯
P
^
r
m
n
-
U
n
≤
ε
U
n
P
g
i
-
P
l
i
-
U
i
∑
j
=
1
N
G
i
j
cos
⁡
δ
i
j
+
B
sin
⁡
δ
i
j
=
0
Q
g
i
-
Q
l
i
-
U
i
∑
j
=
1
N
G
i
j
sin
⁡
δ
i
j
-
B
cos
⁡
δ
i
j
=
0
Q
g
i
=
f
g
i
P
g
i
Q
l
i
=
f
l
i
P
l
i
.Lower level optimization is(45)
m
a
x
P
^
r
m
1
⋯
P
^
r
m
n
U
P
^
r
m
1
⋯
P
^
r
m
n
I
P
^
r
m
1
⋯
P
^
r
m
n
s.t.
∑
i
=
1
n
P
^
r
m
i
≤
∑
i
=
1
n
P
~
r
m
i
.Lower level optimization achieves the total maximum allowable capacity of multi wind farms and then searches the other capacity combinations; the sum of capacity combination should be less thanP
m
a
x. Upper level optimization checks whether the capacity combinations meet the constraints of node voltage and line capacity. The maximum capacity combination meeting the constraints is the final output result.


## 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 Figure1:(28)
P
w
=
0
,
v
≤
v
ci
,
v
≥
v
co
⁡
P
r
v
-
v
ci
v
r
-
v
ci
,
v
ci
≤
v
≤
v
r
P
r
,
v
r
≤
v
≤
v
co
⁡
,where P
r is rated output power; v is wind speed; v
ci is cut-in wind speed; v
r is rated wind speed; v
c
o is cut-out wind speed.Figure 1
Wind turbine output power-speed curve.Wind speed can be expressed as the estimated value coupled with a distributed rangev
0
+
ς; the paper adopts box set to express the disturbance range of wind speed. The output of wind turbine can be expressed as(29)
P
w
=
P
r
v
0
-
v
c
i
v
r
-
v
c
i
+
P
r
ς
v
r
-
v
c
i
,
u
=
ς
e
T
ς
=
0
,
ς
_
≤
ς
≤
ς
-
.Thev
0 is estimated wind speed (it can be found from meteorological data); ς is an artificial decision according to actual wind condition.(1) Eliminate Equality Constraint. The constraints of mathematical formulation (27) cover absolute value, maximum value, and uncertainty parameter. It is tough to solve the problem, so the troublesome work should be converted to analytical form with convenience solution. To simplify the model, the paper substitutes the equality constraints for power balance into inequality constraints for node voltage and line capacity. The inequality constraint for line capacity firstly is converted; detailed information is listed below.The active power at branch can be expressed as(30)
P
b
=
J
g
P
g
+
J
l
P
l
=
J
g
P
g
1
⋯
P
w
⋯
P
g
N
g
+
J
l
P
l
1
⋯
P
l
n
⋯
P
l
N
l
,where N
l is number of loads.If the wind farm locates at nodem, based on equality constraint for power balance in (2), the load demand at node n can be expressed as (31)
P
l
n
=
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
P
w
-
P
s
-
∑
i
=
1
,
i
≠
n
N
l
P
l
i
.Based on (30) and (31), the equality constraint in (27) can be eliminated. The active power at branch can be expressed as (32)
P
b
=
P
w
J
g
+
J
l
n
+
∑
i
=
1
,
i
≠
m
N
g
P
g
i
J
g
+
J
l
n
+
J
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
-
J
l
n
∑
i
=
1
,
i
≠
n
N
l
P
l
i
-
J
l
n
P
s
=
P
w
J
g
+
J
l
n
+
J
l
n
Y
+
Z
,
Y
=
∑
i
=
1
,
i
≠
m
N
g
P
g
i
-
∑
i
=
1
,
i
≠
n
N
l
P
l
i
-
P
s
,
Z
=
J
g
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
J
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
.Considering the inequality constraint in (27) and branch flow in (28), the line capacity constraint can be expressed as(33)
max
⁡
P
w
J
g
+
J
l
n
+
J
l
n
Y
+
Z
≤
I
¯
.Similarly, inequality constraint for node voltage in (28) can be expressed as(34)
max
⁡
P
w
T
g
+
T
l
n
+
T
l
n
Y
-
U
n
+
M
≤
ε
U
n
,
M
=
T
g
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
T
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
.The equality constraint is eliminated and the model only has inequality constraints.(2) Handle Uncertain Parameters. Considering the uncertain parameters in inequality constraint, optimized duality theory is used to simplify the model and convert uncertainty quantities in constraints to certainty quantities, changing the model to linear programming model.Combined with (29), (33) can be expressed as(35)
max
⁡
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
Z
≤
I
¯
=
max
⁡
ς
P
r
v
r
-
v
c
i
J
g
+
J
l
n
≤
I
¯
-
J
l
n
Y
-
Z
-
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
.According to duality theory, (34) can be formulated as(36)
-
min
⁡
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
≤
I
¯
-
J
l
n
Y
-
Z
-
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
.To construct a Lagrange function,(37)
L
i
ς
,
η
,
ν
,
λ
=
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
-
η
i
ς
+
ν
i
ς
_
-
ς
+
λ
i
ς
-
ς
-
,
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
-
P
r
v
r
-
v
ci
J
g
+
J
l
n
-
η
i
+
ν
i
+
λ
i
=
0
,where η, ν, λ are Lagrange coefficients.According to duality theory, we can find(38)
min
⁡
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
=
max
η
,
γ
,
λ
⁡
min
ς
⁡
L
i
ς
,
η
,
ν
,
λ
=
m
a
x
η
,
γ
,
λ
-
ν
ς
_
-
λ
ς
-
:
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
.Then(39)
-
min
⁡
-
ς
P
r
v
r
-
v
ci
J
g
+
J
l
n
=
-
max
η
,
γ
,
λ
⁡
-
ν
ς
_
-
λ
ς
-
:
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
=
min
η
,
γ
,
λ
⁡
ν
ς
_
-
λ
ς
-
:
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
.Based on (39), (36) can be expressed as(40)
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
ν
ς
_
-
λ
ς
-
+
Z
≤
I
¯
.The same method is applied to node voltage constraints in (27), and thereafter the robust optimization model for allowable capacity of single wind farm only considering the uncertainty of wind speed can be expressed as(41)
m
a
x
P
r
s.t.
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
ν
ς
_
-
λ
ς
-
+
Z
≤
I
¯
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
P
r
v
0
-
v
ci
v
r
-
v
ci
T
g
+
T
l
n
+
T
l
n
Y
+
b
ς
_
-
c
ς
-
+
M
-
U
n
≤
ε
U
n
M
=
T
g
∑
i
=
1
,
i
≠
m
N
g
P
g
i
+
T
l
∑
i
=
1
,
i
≠
n
N
l
P
l
i
∇
ς
L
i
ς
,
a
,
b
,
c
=
-
P
r
v
r
-
v
ci
T
g
+
T
l
n
-
a
i
+
b
i
+
c
i
=
0
b
≥
0
,
c
≥
0
,where a, b, c are Lagrange coefficients.


## 4.2. Single Wind Farm with Uncertainty Wind Speed and Load Demand

The fluctuation of load produces difficulty in planning stage for wind farm. If the project determines the allowable capacity of wind farm based on peak load, wind power curtailment will increase when power grid has less load. According to valley load, the decision maker will determine a small allowable capacity of wind farm; it cannot make full use of clean energy.The fluctuation of load will change the sensitive matricesJ
l and T
l; if the load at node n has fluctuation, the output power of wind turbine can be expressed as(42)
P
w
=
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
Δ
J
l
n
+
J
l
n
+
Δ
J
l
n
Y
+
Z
,where Δ
J
l
n is variation of corresponding sensitive martix of load at node n. The paper defines Δ
J
l
n
=
κ
J
l
n, where κ is expressed as box set in (22).The paper uses mathematical derivation method mentioned in Section3.2 to achieve the robust optimization model for allowable capacity of single wind farm with uncertainty wind speed and load demand:(43)
m
a
x
P
r
s.t.
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
+
J
l
n
Y
+
ν
ς
_
-
λ
ς
-
+
β
κ
_
-
γ
κ
-
+
Z
≤
I
¯
∇
κ
L
i
κ
,
α
,
β
,
γ
=
P
r
v
0
-
v
ci
v
r
-
v
ci
J
g
+
J
l
n
-
α
i
+
β
i
+
γ
i
β
≥
0
,
γ
≥
0
∇
ς
L
i
ς
,
η
,
ν
,
λ
=
0
ν
≥
0
,
λ
≥
0
P
r
V
0
-
V
c
V
r
-
V
c
T
g
+
T
l
n
+
T
l
n
Y
+
b
ς
_
-
c
ς
-
+
e
κ
_
-
f
κ
-
≤
ε
U
n
+
M
∇
κ
L
i
κ
,
d
,
e
,
f
=
P
r
V
0
-
V
c
V
r
-
V
c
T
g
+
T
l
n
-
d
i
+
e
i
+
f
i
=
0
e
≥
0
,
f
≥
0
∇
ς
L
i
ς
,
a
,
b
,
c
=
0
b
≥
0
,
c
≥
0
.The model has been converted to a linear programming problem, which could be solved directly by commercial software such as ILOG CPLEX.


## 4.3. Multi Wind Farms with Uncertainty Wind Speed and Load Demand

When multi wind farms connect to power grid, if the sum of allowable capacity of single wind farm is defined as the total allowable capacity, the result will be very optimistic. We suppose the wind farms connect to power grid at nodem
1
⋯
m
n, one mathematical model is built to solve optimization problem with an objective function ∑
i
=
1
n
P
r
m
i, the total allowable capacity is P
m
a
x, and the capacity combination is P
~
r
m
1
⋯
P
~
r
m
n. There will probably be a capacity combination P
^
r
m
1
⋯
P
^
r
m
n; the capacity combination meets ∑
i
=
1
n
P
^
r
m
i
≤
∑
i
=
1
n
P
~
r
m
i. But the capacity combination will cause node voltage and line capacity to suffer out of limit. Based on the concept of min-max transfer capability in transmission interface [24], the paper proposes min-max allowable capacity P
m
i
n
-
m
a
x to represent total allowable capacity of multi wind farms. The P
m
i
n
-
m
a
x is the maximum allowable capacity of wind farm in the worst scenario; it contains the following two meanings: (1) for any capacity combination P
^
r
m
1
⋯
P
^
r
m
n, if it meets ∑
i
=
1
n
P
^
r
m
i
≤
P
m
i
n
-
m
a
x, the power system will meet constraint of node voltage and line capacity, which is required for safe and steady operation of power system; (2) under these conditions, P
m
i
n
-
m
a
x selects the larger number as much as possible, which is for reasons of maximum wind power. The bilevel optimization model can be used to solve the min-max allowable capacity. In the proposed mathematical model, the penetration level of wind farms is maximized in the upper level optimization, while the lower level optimization ensures that voltage constraints and capacity constraints are satisfied even under the worst case.Upper level optimization is(44)
m
a
x
∑
i
=
1
n
P
^
r
m
i
s.t.
∑
i
=
1
N
g
P
g
i
+
∑
j
=
1
N
w
P
w
j
-
P
s
=
P
l
P
g
i
_
≤
P
g
i
≤
P
g
i
¯
I
P
^
r
m
1
⋯
P
^
r
m
n
≤
I
¯
U
P
^
r
m
1
⋯
P
^
r
m
n
-
U
n
≤
ε
U
n
P
g
i
-
P
l
i
-
U
i
∑
j
=
1
N
G
i
j
cos
⁡
δ
i
j
+
B
sin
⁡
δ
i
j
=
0
Q
g
i
-
Q
l
i
-
U
i
∑
j
=
1
N
G
i
j
sin
⁡
δ
i
j
-
B
cos
⁡
δ
i
j
=
0
Q
g
i
=
f
g
i
P
g
i
Q
l
i
=
f
l
i
P
l
i
.Lower level optimization is(45)
m
a
x
P
^
r
m
1
⋯
P
^
r
m
n
U
P
^
r
m
1
⋯
P
^
r
m
n
I
P
^
r
m
1
⋯
P
^
r
m
n
s.t.
∑
i
=
1
n
P
^
r
m
i
≤
∑
i
=
1
n
P
~
r
m
i
.Lower level optimization achieves the total maximum allowable capacity of multi wind farms and then searches the other capacity combinations; the sum of capacity combination should be less thanP
m
a
x. Upper level optimization checks whether the capacity combinations meet the constraints of node voltage and line capacity. The maximum capacity combination meeting the constraints is the final output result.


## 5. Study Case

In this section, the paper presents a computational study to evaluate the performance of robust optimization model for allowable capacity of single wind farm or multi wind farms. We test the improved IEEE 30-bus system, as shown in Figure2; this power system consists of 6 synchronous machines and 19 loads and its network parameters and load data can be found in [25]. Table 1 lists the output upper and lower limit on conventional generator defined by the paper. All the experiments are implemented using CPLEX 12.1, at a laptop with Intel Quad Core 2.40 GHz and 4 GB memory.Table 1
The output upper and lower limit on conventional generator.
Generation number
Output upper limit (MW)
Output lower limit (MW)
1
139
130
2
58
32
5
35
30
8
11
13
11
18
17
13
10
10Figure 2
Single-line chart of IEEE 30-bus system.The location of wind farm has influence on its allowable capability; however, the existing works do not make a quantitative analysis for the relationship between location and allowable capability. They only have a rough and recapitulative conclusion for radicalized distribution network; for example, the wind farm has larger allowable capability at the head of feeder and the allowable capability is smaller at the end of feeder [4, 5]. To make a visually quantitative analysis, the paper selects node importance degree to measure the location of different mode. Based on node importance classified by comprehensive assessment method [26], the paper calculates the importance degree of potential wind farm node and load node in IEEE 30-bus system and further selects five nodes with obvious differences of importance degree as wind farm node and load node. Tables 2 and 3 list the five wind farm nodes and load nodes.Table 2
Importance degree of grid connection point of wind farm.
Wind farm node
16
10
12
6
20
Normalized importance degree
1.000
0.935
0.806
0.694
0.511Table 3
The output upper and lower limit on conventional generator.
Load node
5
7
21
18
29
Normalized importance degree
1.000
0.874
0.601
0.384
0.296


### 5.1. Allowable Capacity of Single Wind Farm

(1) Only Considering the Uncertainty of Wind Speed. We define v
ci as 6.5 m/s, v
r 12.4 m/s, v
c
o 19.5 m/s, v
0 10.3 m/s, and ς ±5 m/s. The wind farm connects to gird at nodes 16, 10, 12, 6, and 20, respectively; the corresponding allowable capacities are listed in Table 4.Table 4
Allowable capacity of wind farm with only uncertainty of wind speed (kW).
Disturbance rangeof wind speed (m/s)
Wind farm node
16
10
12
6
20
(−1, 1)
20.7
25.21
31.0
36.9
41.7
(−2, 2)
19.1
23.4
28.8
34.8
39.3
(−3, 3)
17.8
21.1
26.4
32.4
37.0
(−4, 4)
16.0
17.6
25.2
30.1
34.5
(−5, 5)
14.6
15.3
22.5
23.9
32.4From the row data in the table, we can find that, with the increasing disturbance range of wind speed, the allowable capacity of wind farm at the same node drops continuously. The downtrend shows that if wind farm output has a big fluctuation range, power grid should improve stable robustness, so the allowable capacity of wind farm inevitably is limited. The panel data shows that, with the increasing importance degree of wind node, the allowable capacity of wind farm with the same wind speed disturbance drops continuously. The wind farm node with larger importance degree plays an important role in active power transmission of the whole system and has a stronger reactive power support ability. Compared to the node with smaller importance degree, its unit capacity will obviously have influence on voltage support and power transmission; the node voltage and line capacity are easier to suffer out of limit, so the allowable capacity of wind farm at node with larger importance degree will decrease.(2) Only Considering the Uncertainty of Load. The part selects node 5 as analysis object, whose κ is ±5%; Table 5 lists the allowable capacity of wind farm.Table 5
Allowable capacity of wind farm with only uncertainty of load (kW).
Disturbance parameterof load demand
Wind farm node
16
10
12
6
20
(−1%, 1%)
21.7
25.4
32.2
37.4
43.6
(−2%, 2%)
20.4
23.8
30.4
35.4
41.8
(−3%, 3%)
19.0
22.0
28.9
33.6
39.9
(−4%, 4%)
17.9
20.5
27.1
31.9
37.8
(−5%, 5%)
17.1
19.4
25.6
30.5
35.6Table5 shows that, with the increasing disturbance parameter of load demand, the allowable capacity of wind farm at the same node drops continuously. The bigger fluctuation of load causes a more volatile situation for power flow. The power grid should reduce the allowable capacity of wind farm to improve stable robustness.(3) Considering the Uncertainty of Different Load. In this case, κ is (−5%, 5%) and ς is (−5 m/s, 5 m/s), with different load nodes selected as analysis object, respectively; the corresponding allowable capacities are listed in Table 6 and shown in Figure 3.Table 6
Allowable capacity of wind farm with uncertainty of different loads (kW).
Load node
Wind farm node
16
10
12
6
20
5
15.5
18.9
23.7
28.1
33.7
7
16.6
20.3
25.4
30.2
36.2
21
18.2
22.2
27.9
33.1
39.7
18
19.5
23.8
29.5
35.5
42.6
29
20.9
25.4
32.0
38.0
45.5Figure 3
Allowable capacity of wind farm with uncertainty of different loads.Table6 shows that the disturbance at load node with larger importance degree will decrease the allowable capacity. The load node with larger importance degree will produce important action to global active power transmission and has a greater contribution to voltage drop, which has a larger influence on both active power transmission of the whole system and the voltage level of the region. Its fluctuation easily leads power grid to violate static security constraints; hence the allowable capacity decreases.Figure3 shows that, with the joint increasing disturbance range of wind speed and load, the operation scenario of power grid is constantly deteriorated; then the allowable capacity decreases significantly.(4) Considering the Uncertainty of Load and Wind Speed. Wind farm node 16 has wind speed fluctuation and ς is ±5 m/s; the load node 5 has fluctuation and κ is ±5%. The allowable capacity is shown in Table 7 and Figure 4.Table 7
Allowable capacity of wind farm with uncertainty of wind speed and load  (kW).
Disturbance parameterof load demand
Disturbance range of wind speed (m/s)
(−1, 1)
(−2, 2)
(−3, 3)
(−4, 4)
(−5, 5)
(−%1, %1)
25.4
24.3
23.8
22.7
21.8
(−%2, %2)
24.8
23.7
23.3
21.8
20.0
(−%3, %3)
23.5
22.6
21.9
20.3
18.8
(−%4, %4)
22.6
21.2
20.1
18.6
17.6
(−%5, %5)
21.5
19.8
18.5
17.1
15.5Figure 4
Allowable capacity of wind farm with uncertainty of wind speed and load.Table7 and Figure 4 show that, with the joint increasing disturbance range of wind farm output and load demand, the allowable capacity decreases significantly. The combination of different wind farm nodes and load nodes can represent different scenarios. In this way, the robust optimization can ensure that wind farm immunizes against all possible variables and the power grid can safely and steadily operate in the worst conditions.


### 5.2. The Comparison with Other Methods

Wind farm node 16 has wind speed fluctuation andς is (−5 m/s, 5 m/s), and load node 5 has fluctuation and κ is (−5%, 5%). The case adopts the following three methods to calculate allowance capacity of wind farm:(1)
The method is based on the determination of voltage sensitivities from a linearized power system model and ignores the uncertainty of wind speed and load [4].
(2)
The method is based on chance constrained programming, which uses Weibull distribution and normal distribution to express the uncertainty of wind speed and load, respectively [13]; the parameters for Weibull distribution are k
=
2.0 and c
=
8.5. The confidence level is 0.95.
(3)
The method proposed in the paper is based on robust liner optimization and uses box set to express the uncertainty of wind speed and load.To verify the effectiveness of allowable capacity calculated by three methods, the paper randomly produces the disturbance on wind speed and load within the disturbance range and checks the constraints of node voltage and line capacity.Method (1) ignores the fluctuation and randomness of load and wind speed and regards the rated capacity as actual capacity, which makes an optimistic and largest allowable capacity. When load 5 has disturbance, the transfer capability of branch 16-17 is beyond the maximum capacity. If the wind speed has the disturbance of −5 m/s, the output power of wind farm at node 33 will decrease and the voltage of node 19 and node 6 will be less than 0.9 pu.In method (2), as the parameters of distribution function method have influence on allowable capacity, the accurate distribution function is hard-to-get data. Considering the defective Weibull distribution and normal distribution, the result achieved by method (2) is not 100% believable and acceptable. When the wind power ramp event occurs, the wind power output increases or decreases significantly during a short period of time; the Weibull distribution cannot describe the fluctuation range of wind speed. In addition, the method does not cover constraint of node voltage; when wind speed has disturbance, the voltage of node 2 will be larger than 1.0 pu.Method (3) proposed in the paper considers the uncertainty of wind speed and load, where the allowable capacity can ensure that power gird meets the static security constraint. The result shows that the allowable capacity based on robust linear optimization model immunizes against all realizations of the uncertain data within uncertainty set containing worst-case scenario.


### 5.3. Allowable Capacity of Multi Wind Farms

Three wind farms connect to power grid at the same time;ς is (−5 m/s, 5 m/s). Load node 16 has fluctuation and κ is (−5%, 5%). The case defines four groups of wind farm nodes and Table 9 lists the maximum total allowable capacity and generation proportion of every wind farm.Tables8 and 9 show that the total allowable capacity is larger than allowable capacity of single wind farm. The allowable capacity of wind farm at nodes 16, 20, and 12 is 15.5 kW, 20.8 kW, and 25.4 kW, respectively, but the total allowable capacity is 31.6 kW. The larger the allowable capacity of single wind farm is, the larger the total allowable capacity of multi wind farms will be. For wind farm node groups (
16,20,12
) and (
16,12,20
), the allowable capacity of wind farm at nodes 12 and 20 is 25.4 kW and 39.3 kW, respectively, so total allowable capacity of wind farm node group (
16,20,12
) is less than (
16,12,20
). The wind farm node with lager allowable capacity will hold the greater generation proportion. Node 16 has a smallest allowable capacity, so it has smallest generation proportion, such as 0.21, 0.14, and 0.11 at (16, 20, 12), (16, 6, 20), and (16, 12, 20), respectively.Table 8
Allowable capacity of wind farm with different calculation method (kW).
Analysis method
Wind farm node
16
10
12
6
20
(1)
22.4
29.1
34.1
40.1
47.7
(2)
20.3
26.8
31.5
37.3
44.8
(3)
15.5
20.8
25.4
31.7
39.3Table 9
Allowable capacity of three wind farms (kW).
Wind farm node
Total allowable capacity
Generation proportion
16, 20, 12
31.6
0.21, 0.33, 0.46
16, 6, 20
52.4
0.14, 0.31, 0.55
16, 12, 20
55.9
0.11, 0.27, 0.62
12, 6, 20
57.8
0.29, 0.33, 0.38


## 5.1. Allowable Capacity of Single Wind Farm

(1) Only Considering the Uncertainty of Wind Speed. We define v
ci as 6.5 m/s, v
r 12.4 m/s, v
c
o 19.5 m/s, v
0 10.3 m/s, and ς ±5 m/s. The wind farm connects to gird at nodes 16, 10, 12, 6, and 20, respectively; the corresponding allowable capacities are listed in Table 4.Table 4
Allowable capacity of wind farm with only uncertainty of wind speed (kW).
Disturbance rangeof wind speed (m/s)
Wind farm node
16
10
12
6
20
(−1, 1)
20.7
25.21
31.0
36.9
41.7
(−2, 2)
19.1
23.4
28.8
34.8
39.3
(−3, 3)
17.8
21.1
26.4
32.4
37.0
(−4, 4)
16.0
17.6
25.2
30.1
34.5
(−5, 5)
14.6
15.3
22.5
23.9
32.4From the row data in the table, we can find that, with the increasing disturbance range of wind speed, the allowable capacity of wind farm at the same node drops continuously. The downtrend shows that if wind farm output has a big fluctuation range, power grid should improve stable robustness, so the allowable capacity of wind farm inevitably is limited. The panel data shows that, with the increasing importance degree of wind node, the allowable capacity of wind farm with the same wind speed disturbance drops continuously. The wind farm node with larger importance degree plays an important role in active power transmission of the whole system and has a stronger reactive power support ability. Compared to the node with smaller importance degree, its unit capacity will obviously have influence on voltage support and power transmission; the node voltage and line capacity are easier to suffer out of limit, so the allowable capacity of wind farm at node with larger importance degree will decrease.(2) Only Considering the Uncertainty of Load. The part selects node 5 as analysis object, whose κ is ±5%; Table 5 lists the allowable capacity of wind farm.Table 5
Allowable capacity of wind farm with only uncertainty of load (kW).
Disturbance parameterof load demand
Wind farm node
16
10
12
6
20
(−1%, 1%)
21.7
25.4
32.2
37.4
43.6
(−2%, 2%)
20.4
23.8
30.4
35.4
41.8
(−3%, 3%)
19.0
22.0
28.9
33.6
39.9
(−4%, 4%)
17.9
20.5
27.1
31.9
37.8
(−5%, 5%)
17.1
19.4
25.6
30.5
35.6Table5 shows that, with the increasing disturbance parameter of load demand, the allowable capacity of wind farm at the same node drops continuously. The bigger fluctuation of load causes a more volatile situation for power flow. The power grid should reduce the allowable capacity of wind farm to improve stable robustness.(3) Considering the Uncertainty of Different Load. In this case, κ is (−5%, 5%) and ς is (−5 m/s, 5 m/s), with different load nodes selected as analysis object, respectively; the corresponding allowable capacities are listed in Table 6 and shown in Figure 3.Table 6
Allowable capacity of wind farm with uncertainty of different loads (kW).
Load node
Wind farm node
16
10
12
6
20
5
15.5
18.9
23.7
28.1
33.7
7
16.6
20.3
25.4
30.2
36.2
21
18.2
22.2
27.9
33.1
39.7
18
19.5
23.8
29.5
35.5
42.6
29
20.9
25.4
32.0
38.0
45.5Figure 3
Allowable capacity of wind farm with uncertainty of different loads.Table6 shows that the disturbance at load node with larger importance degree will decrease the allowable capacity. The load node with larger importance degree will produce important action to global active power transmission and has a greater contribution to voltage drop, which has a larger influence on both active power transmission of the whole system and the voltage level of the region. Its fluctuation easily leads power grid to violate static security constraints; hence the allowable capacity decreases.Figure3 shows that, with the joint increasing disturbance range of wind speed and load, the operation scenario of power grid is constantly deteriorated; then the allowable capacity decreases significantly.(4) Considering the Uncertainty of Load and Wind Speed. Wind farm node 16 has wind speed fluctuation and ς is ±5 m/s; the load node 5 has fluctuation and κ is ±5%. The allowable capacity is shown in Table 7 and Figure 4.Table 7
Allowable capacity of wind farm with uncertainty of wind speed and load  (kW).
Disturbance parameterof load demand
Disturbance range of wind speed (m/s)
(−1, 1)
(−2, 2)
(−3, 3)
(−4, 4)
(−5, 5)
(−%1, %1)
25.4
24.3
23.8
22.7
21.8
(−%2, %2)
24.8
23.7
23.3
21.8
20.0
(−%3, %3)
23.5
22.6
21.9
20.3
18.8
(−%4, %4)
22.6
21.2
20.1
18.6
17.6
(−%5, %5)
21.5
19.8
18.5
17.1
15.5Figure 4
Allowable capacity of wind farm with uncertainty of wind speed and load.Table7 and Figure 4 show that, with the joint increasing disturbance range of wind farm output and load demand, the allowable capacity decreases significantly. The combination of different wind farm nodes and load nodes can represent different scenarios. In this way, the robust optimization can ensure that wind farm immunizes against all possible variables and the power grid can safely and steadily operate in the worst conditions.


## 5.2. The Comparison with Other Methods

Wind farm node 16 has wind speed fluctuation andς is (−5 m/s, 5 m/s), and load node 5 has fluctuation and κ is (−5%, 5%). The case adopts the following three methods to calculate allowance capacity of wind farm:(1)
The method is based on the determination of voltage sensitivities from a linearized power system model and ignores the uncertainty of wind speed and load [4].
(2)
The method is based on chance constrained programming, which uses Weibull distribution and normal distribution to express the uncertainty of wind speed and load, respectively [13]; the parameters for Weibull distribution are k
=
2.0 and c
=
8.5. The confidence level is 0.95.
(3)
The method proposed in the paper is based on robust liner optimization and uses box set to express the uncertainty of wind speed and load.To verify the effectiveness of allowable capacity calculated by three methods, the paper randomly produces the disturbance on wind speed and load within the disturbance range and checks the constraints of node voltage and line capacity.Method (1) ignores the fluctuation and randomness of load and wind speed and regards the rated capacity as actual capacity, which makes an optimistic and largest allowable capacity. When load 5 has disturbance, the transfer capability of branch 16-17 is beyond the maximum capacity. If the wind speed has the disturbance of −5 m/s, the output power of wind farm at node 33 will decrease and the voltage of node 19 and node 6 will be less than 0.9 pu.In method (2), as the parameters of distribution function method have influence on allowable capacity, the accurate distribution function is hard-to-get data. Considering the defective Weibull distribution and normal distribution, the result achieved by method (2) is not 100% believable and acceptable. When the wind power ramp event occurs, the wind power output increases or decreases significantly during a short period of time; the Weibull distribution cannot describe the fluctuation range of wind speed. In addition, the method does not cover constraint of node voltage; when wind speed has disturbance, the voltage of node 2 will be larger than 1.0 pu.Method (3) proposed in the paper considers the uncertainty of wind speed and load, where the allowable capacity can ensure that power gird meets the static security constraint. The result shows that the allowable capacity based on robust linear optimization model immunizes against all realizations of the uncertain data within uncertainty set containing worst-case scenario.


## 5.3. Allowable Capacity of Multi Wind Farms

Three wind farms connect to power grid at the same time;ς is (−5 m/s, 5 m/s). Load node 16 has fluctuation and κ is (−5%, 5%). The case defines four groups of wind farm nodes and Table 9 lists the maximum total allowable capacity and generation proportion of every wind farm.Tables8 and 9 show that the total allowable capacity is larger than allowable capacity of single wind farm. The allowable capacity of wind farm at nodes 16, 20, and 12 is 15.5 kW, 20.8 kW, and 25.4 kW, respectively, but the total allowable capacity is 31.6 kW. The larger the allowable capacity of single wind farm is, the larger the total allowable capacity of multi wind farms will be. For wind farm node groups (
16,20,12
) and (
16,12,20
), the allowable capacity of wind farm at nodes 12 and 20 is 25.4 kW and 39.3 kW, respectively, so total allowable capacity of wind farm node group (
16,20,12
) is less than (
16,12,20
). The wind farm node with lager allowable capacity will hold the greater generation proportion. Node 16 has a smallest allowable capacity, so it has smallest generation proportion, such as 0.21, 0.14, and 0.11 at (16, 20, 12), (16, 6, 20), and (16, 12, 20), respectively.Table 8
Allowable capacity of wind farm with different calculation method (kW).
Analysis method
Wind farm node
16
10
12
6
20
(1)
22.4
29.1
34.1
40.1
47.7
(2)
20.3
26.8
31.5
37.3
44.8
(3)
15.5
20.8
25.4
31.7
39.3Table 9
Allowable capacity of three wind farms (kW).
Wind farm node
Total allowable capacity
Generation proportion
16, 20, 12
31.6
0.21, 0.33, 0.46
16, 6, 20
52.4
0.14, 0.31, 0.55
16, 12, 20
55.9
0.11, 0.27, 0.62
12, 6, 20
57.8
0.29, 0.33, 0.38


## 6. Conclusion

Based on fast decoupled method and robust linear optimization, the paper has proposed a novel method to determine the allowable capacity of wind farm taking into account uncertainties of both loads and wind generation. The method requires no knowledge of the probability distribution for uncertainty parameters and adopts the box set to describe the fluctuation range of wind speed and load. The allowable capability determined by the method can remain feasible for the worst-case parameter variation in uncertainty set; the node voltage and line capability can meet the constraint when power grid operates in worst scenario. Furthermore min-max capability and bilevel programming methods are used to analyze the total allowable capability of multi wind farms.The fluctuation range of wind speed and load and position of wind farm node and load node with disturbances all have influence on the allowable capability of wind farm. The larger fluctuation range of wind speed and load increases variability of system parameters and variety of operation conditions. The allowable capability of wind farm should be limited to improve the stable robustness of power grid. If the wind farm node and load node with disturbances have larger importance degree, they have a larger influence on both active power transmission of the whole system and the voltage level of the region. The disturbance at these nodes will reduce the stability of the whole system, and therefore the allowable capability of wind farm is reduced.



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*Source: 790974-2015-11-03.xml*