Optimal Available Transfer Capability Assessment Strategy for Wind Integrated Transmission Systems Considering Uncertainty of Wind Power Probability Distribution

Wind power prediction research shows that it is difficult to accurately and effectively estimate the probability distribution (PD) of wind power. When only partial information of the wind power probability distribution function is available, an optimal available transfer capability (ATC) assessment strategy considering the uncertainty on the wind power probability distribution is proposed in this paper. As wind power probability distribution is not accurately given, the proposed strategy can efficiently maximize ATC with the security operation constraints satisfied under any wind power PD function case in the uncertainty set. A distributional robust chance constrained (DRCC) model is developed to describe an optimal ATC assessment problem. To achieve tractability of the DRCC model, the dual optimization, S-lemma and Schur complement are adopted to eliminate the uncertain wind power vector in the DRCC model. According to the characteristics of the problem, the linear matrix inequality (LMI)-based particle swarm optimization (PSO) algorithm is used to solve the DRCC model which contains first and second-order moment information of the wind power. The modified IEEE 30-bus system simulation results show the feasibility and effectiveness of the proposed ATC assessment strategy.


Introduction
Wind power energy has become one of the fastest growing renewable energies in recent years.However, the intermittency, randomness and unpredictability of wind power increase the power volatility of the transmission system [1].With increasing uncertainty in the transmission system, the available transfer capability (ATC) of the transmission system will be influenced greatly.Currently, the wind power prediction accuracy, which to a certain extent has a prediction error, is limited, and it is difficult to obtain the accurate probability distribution function wind power.To improve the security of the transmission system, the wind power uncertainty should be considered in the ATC assessment.Therefore, the probability distribution uncertainty of wind power should be considered to carry out robust and economical ATC assessment strategies [2,3].

Traditional Chance Constrained Optimal Available Transfer Capability (ATC) Assessment Model
For optimal ATC assessment for wind integrated transmission systems considering the uncertainty on the wind power probability distribution chance constrained programming is employed to avoid the operational scheme being limited by small probability events and satisfy the security constraints of the transmission system.The chance constrained optimal ATC assessment model is as follows: the objective function is to maximize the all tie-line power output between a given interface, say, from area a to area b [29]: ATC = max ∑ P a→b (1) where P a→b denotes all tie-line power output from area a to area b.

Active and Reactive Power Balance Constraint
P W + P G − P D − S T I J P L = 0 (2) where P W and Q W denote the active and reactive power output of wind farms, respectively; P G and Q G denote the active and reactive power output of conventional generators, respectively; P D and Q D denote active and reactive loads, respectively; P L and Q L denote the active and reactive power flow, respectively; Q c indicates the actual capacity of reactive power compensators; S IJ denotes the node branch incidence matrix, which is defined as: where P G,max and P G,min denote the upper and lower active power limits of conventional generation, respectively; Q G,max and Q G,min denote the upper and lower reactive power limits of conventional generation, respectively.
where P W,max and Q W,max represent the maximum active and reactive power output of corresponding wind farms, respectively.

Chance Constraints of Node Voltage
Energies 2016, 9, 704 4 of 18 where V, V max and V min represent the actual voltage amplitude, upper limit and lower limit of voltage amplitude, respectively.Pr φ is the probability under φ, where φ is a placeholder for a probability distribution; β is the confidence level.

Chance Constraints of Branch Power
where P L,max indicates the upper limit of branch power.Random variables P W and Q W exist in the chance constrained optimal ATC assessment model, thus the node voltage and branch power constraints become complicated.The traditional approach to solve the above chance constraints is to use Boole's inequality [28] to convert these problems into a single chance-constrained problem.The Monte-Carlo sampling method is adopted under the given probability distribution function.However, this method is only efficient if the wind power probability distribution is defined and comprehensively described.

Distributional Robust Chance Constrained Model
Due to the fact that the wind speed prediction accuracy is limited, the wind speed fluctuates significantly at different seasons over a year, even during different hours over a day.Besides that, geography and climate influence the wind speed a lot.There is lots of uncertain information about wind power and the decision-makers can not get all the information about the probability distribution of wind power.Instead, partial information on wind power such as the first and second order moments can be collected from the historical data.Denote the active output expectations and covariance matrix of m wind farms are µ = [µ 1 , . . .,µ m ] T and Γ, respectively.Define as the set of all the distributions whose mean and covariance are µ and Γ.
The wind probability distribution function may be any one of set Φ Ξ (µ, Γ), and the predetermined node voltage and branch power constraints should be satisfied under all possible wind power probability distributions.
Therefore, the distributional robust chance constrained model is adopted to describe the optimal ATC assessment problem.When the probability distribution of random variables is uncertain distributional robust chance constrained programming is an effective way to solve such problems [22].
In the node voltage constraints in Equation ( 8) and branch power constraint in Equation ( 9), since the probability distribution of the random vector P W and Q W are uncertain, P W and Q W can be any possible realization under the set Φ Ξ (µ, Γ), therefore wind power output influences the node voltage and branch power.Therefore, the infimum is introduced in Equations ( 8) and ( 9) in the distributional robust chance constrained model to express that for any possible probability distribution function, branch power and node voltages should satisfy the constraints, which is shown in Equations (10) and (11), where in f φ∈Φ Ξ (µ,Γ) Pr φ {A} represents the minimum probability of A under all possible probability distributions.
If the probability distribution of wind power isn't specified, the optimal ATC assessment programs should meet the pre-set confidence level for node voltage and branch power under any possible probability distribution function of set Φ Ξ (µ,Γ).The difficulty in solving the model is how to handle constraints in Equations (10) and (11) with system node voltage and branch power constraints satisfied when only first and second moment of wind power PD are given.

Solving Method for Distributional Robust Chance Constrained Optimal Available Transfer Capability (ATC) Assessment Model
It is impossible to accurately estimate the wind power probability distribution function in the optimal ATC assessment model as analyzed above.The solving process for distributional robust chance constrained ATC assessment model is as follows: firstly, the wind power vector is separated into active and reactive power balance Equations ( 2) and (3).Secondly, employing the node voltage and reactive power model [30,31], the branch power constraints and node voltage constraints are expressed as a function of wind power.Thirdly, dual optimization, S-lemma and Schur complement are adopted to eliminate the random variable.Finally, the probabilistic based constraints are converted into deterministic constraints.

Reformulation of Distributional Robust Chance Constraints
Rewrite the node voltage constraints in Equation (10) as follows: Employing the node voltage and reactive power model [31], ∆V = J∆Q, where ∆V is the voltage vector increment, ∆Q is the node reactive power injection increment, J is the contraction of Jacobian matrix V -Q.The Newton-Raphson power flow calculation method is adopted in this paper.The core idea is to gradually linearize the nonlinear equations using the Taylor series expansion method.If the initial value is chosen appropriate, which means the first derivative of the initial value is relatively small in the equation, and the second order and higher order term can be omitted.Here in the equation ∆V = J∆Q, ∆Q is the last correction term which can meet the real value of the power balance equation.
The specific power flow calculation of ∆V = J∆Q is as follows: (1) Give the initial value of Q(λ) and V(λ), where λ = 0 at the beginning, it denotes the initial value.
(4) Use ∆V(λ) and ∆Q(λ) to correct Q(λ) and V(λ), then obtains Q(λ + 1) and V(λ + 1), let λ = λ + 1, go back to Step (2). Here, Substitute ∆V into the node voltage constraint in Equation ( 12) and eliminate the random variable.Equation ( 12) can be reformulated as Equation (13), where Prob 1 (Q G ) denotes the maximum probability value which satisfies the node voltage constraint under all possible wind power probability distributions: T , rewrite Equation (13) as: Energies 2016, 9, 704 T , where T T I J is the power transmission distribution coefficient matrix [31].The active power flow P L can be formulated as follows [31]: Thus, recall in f φ∈Φ Ξ (µ,Γ) Pr φ {A}, denote Prob 2 (P G ) as the maximum probability value which satisfies the branch active power constraint under all possible wind power probability distributions:

Determination Model for Distributional Robust Chance Constraint
According to the dual theory [20], Prob 1 (Q G ) and Prob 2 (P G ) in Equations ( 14) and ( 16) correspond to the optimal value of sub-optimization problem in Equations ( 17) and ( 18), respectively.The detailed derivation is shown in Appendix A: where Tr(•) denotes the trace operations, N = [Γ + µµ T , µ; µ T , 1], M k1 and M k2 are symmetric matrices containing all the dual variables.In a similar way, employing the dual optimization, S-lemma and Schur complement [20,21], the random wind power in the optimization model is eliminated [28], and Equations ( 17) and ( 18) are converted into linear matrix inequalities as shown in Equations (19) and (20).The detailed derivation is shown in Appendix B: Energies 2016, 9, 704 7 of 18 where in matrix W l 1 ∈ R (m+1)×(m+1) the (l 1 , l 1 )-th element is 1, and the (l 1 , m + 1)-th and (m + 1, l 1 )-th element is −Q W,max,l 1 /2, the rest of the elements is 0, and there are m wind farms.The maximum reactive power output of the l 1 -th wind farms is Q W,max,l 1 .
In matrix W l 2 ∈ R (m+1)×(m+1) , the (l 2 , l 2 )-th element is 1, and the (l 2 , m + 1)-th and (m + 1, l 2 )-th element is −P W,max,l 2 /2, the rest of the elements is 0, and there are m wind farms.The maximum active power output of the l 2 -th wind farms is P W,max,l 2 .
After the above derivations, it is obvious that the wind power P W and Q W do not exist in the optimization problem in Equation (21), and only the first and second moment information of the wind power probability distribution function is needed.

Linear Matrix Inequality (LMI)-Based Particle Swarm Optimization (PSO) Algorithm for Available Transfer Capability (ATC) Assessment Problem
The optimal ATC assessment in Equation ( 21) only needs the first and second order moments of the wind power probability distribution information.It is relatively difficult to solve the nonlinear problem as it includes LMI and nonlinear constraints.The PSO algorithm [32] is used in this paper to solve the problem in Equation (21).
Particle swarm optimization was first proposed by Eberhart and Kennedy.The design of the PSO algorithm is as follows: initialize a group of random particles, each particle represents a candidate solution.The particles update themselves through tracking the personal best position and the optimal solution for the entire population called global minimum, the position of the best particle that give the best fitness value in the entire population.The PSO algorithm has a clear searching direction and it converges quickly at the beginning, however, the PSO algorithm can easily fall into local minimum of convergence speed and precision during the course of evolution.Thus we combine the immune Energies 2016, 9, 704 8 of 18 algorithm and PSO, which is expected to address the convergence speed shortcomings of the local minimum of the PSO.
In the PSO updating process, the particles which have high fitness value will be retained.However, if such particles are too concentrated, it can easily to fall into a local optimum.Those particles which have lower fitness value but maintain good evolutionary trends will not be eliminated, so the vaccination strategy based on the diversity of concentration mechanism can efficiently ensure all fitness levels by maintaining a certain concentration, the i-th concentration value of particles is defined as follows: where x i and f (x i ), i = 1,2, . . .,num, refer to the i-th particle and the fitness value of the i-th particle, respectively.num refers to particle number.The probability of the particle concentration is defined as follows: We can see from the above equation that, if particle j is similar to particle i, then Prob(x i ) is less, which means the chance of particle i to be chosen is less.If the particle j is more different from particle i, then Prob(x i ) is larger.Therefore, the particles have higher diversity.
Immune selection is inspired by the biological immune system.The biological immune system is an intricate network of specialized tissues, organs, cells, and chemicals with the capability of distinguishing entities within the body as "self" or "non-self" and eliminating those that are non-self.If the fitness value of particles is inferior to those of their parent, then the vaccination process is canceled, otherwise, the particles are retained.
The specific flow chart for the LMI-based PSO algorithm for ATC assessment is shown in Figure 1.The specific procedures are as follows: (1) Input grid original parameters, the first and second order moments of the wind power, PSO control parameters and control variables.The branch power and node voltage constraints in Equations ( 10) and ( 11) can be converted into LMI form as shown in Equations ( 19) and ( 20).(2) Initialization: set the initial solution P 0 G and Q 0 G , calculate the fitness value, and let the iteration counter K = 0.
(3) Parameter optimization: substitute 19) and (20).Solve the resulting LMI problem and obtain the optimal solution τ * k1,l 2 τ * k2,l 2 τ * k3,l 2 , and set 21), solve the matrix positive condition and power flow calculation to obtain the updated optimal solution P * G and Q * G and the corresponding objective value ∑ P * a→b .Vaccine selection and immune selection are defined as shown in Equations ( 22) and ( 23), the particle P Gi , Q G,i with high Prob(P Gi , Q G,i ) value would be selected, and set

Numerical Example
The modified IEEE 30-bus transmission system used for simulation calculation is shown in Figure 2. It is divided into three regions, and the revised system includes six conventional generators, the specific parameters of which are shown in Table 1.There are 22 P-Q buses (where bus 11 and bus 24 are reactive power compensation points, the compensation step is 0.048), one balance bus (bus 1), and the rest are P-V buses.Upper and lower limits of node voltage are 1.1 pu (per unit) and 0.95 pu.There are 41 branches in the transmission system.The transformer parameters are shown in Table 2.There are six wind farms integrated to the transmission system, which are connected to the grid at bus 3, 16, 23, 26, 27 and 28.
To illustrate the validity of the proposed strategy, we compare the results of the DRCC optimal ATC assessment (DRCC-ATC) model and the TCC optimal ATC assessment (TCC-ATC) model.Wind power is assumed to follow the normal distribution in the TCC model.In order to analyze the impact of wind power on the optimal ATC assessment, we test the ATC assessment strategy under different confidence levels and covariance values.

Numerical Example
The modified IEEE 30-bus transmission system used for simulation calculation is shown in Figure 2. It is divided into three regions, and the revised system includes six conventional generators, the specific parameters of which are shown in Table 1.There are 22 P-Q buses (where bus 11 and bus 24 are reactive power compensation points, the compensation step is 0.048), one balance bus (bus 1), and the rest are P-V buses.Upper and lower limits of node voltage are 1.1 pu (per unit) and 0.95 pu.There are 41 branches in the transmission system.The transformer parameters are shown in Table 2.There are six wind farms integrated to the transmission system, which are connected to the grid at bus 3, 16, 23, 26, 27 and 28.
To illustrate the validity of the proposed strategy, we compare the results of the DRCC optimal ATC assessment (DRCC-ATC) model and the TCC optimal ATC assessment (TCC-ATC) model.Wind power is assumed to follow the normal distribution in the TCC model.In order to analyze the impact of wind power on the optimal ATC assessment, we test the ATC assessment strategy under different confidence levels and covariance values.

Comparison of the Distributional Robust Chance Constrained ATC (DRCC-ATC) Model and the Traditional Chance Constrained ATC (TCC-ATC) Model
To compare the distributional robust chance constrained ATC (DRCC-ATC) model and the traditional chance constrained ATC (TCC-ATC) model, we set the expectation of wind power probability distribution to 2.0 MW, and the covariance of the wind power probability distribution to 0.4 MW 2 .The optimal values of the DRCC-ATC model and the TCC-ATC model under different confidence levels are shown in Figure 3.

Comparison of the Distributional Robust Chance Constrained ATC (DRCC-ATC) Model and the Traditional Chance Constrained ATC (TCC-ATC) Model
To compare the distributional robust chance constrained ATC (DRCC-ATC) model and the traditional chance constrained ATC (TCC-ATC) model, we set the expectation of wind power probability distribution to 2.0 MW, and the covariance of the wind power probability distribution to 0.4 MW 2 .The optimal values of the DRCC-ATC model and the TCC-ATC model under different confidence levels are shown in Figure 3. Figure 3 illustrates the effect of the confidence level on the ATC.It is shown in Figure 3 that with increasing confidence level, the ATC obtained by the DRCC and TCC both gradually decrease to satisfy the security constraints caused by the wind power uncertainty.For a given confidence level, the ATC under the DRCC model is smaller than that of the TCC model.In the DRCC model, the node voltages and branch power should meet the constraints at a given confidence level under any given wind power probability distribution.In the TCC model, the node voltage and branch power should meet the constraint at a given confidence level for only one kind of wind power probability distribution.Therefore, the DRCC model argues for a more rigid security constraint for the windintegrated transmission system, which results in a smaller ATC when this model is employed.

Available Transfer Capability (ATC) under Different Expectations of Wind Power Probability Distribution, Where Covariance = 0.4 MW 2
Figure 4 illustrates the relationship between ATC and the wind power probability distribution expectation.Given the same covariance of wind power PD and confidence level, we compare ATC values under different expectations of wind power probability.We can find in Figure 4 that with the increase of the confidence level, ATC gradually decreases.When the confidence level increases, ATC decreases to avoid breaking the security constraints.Figure 3 illustrates the effect of the confidence level on the ATC.It is shown in Figure 3 that with increasing confidence level, the ATC obtained by the DRCC and TCC both gradually decrease to satisfy the security constraints caused by the wind power uncertainty.For a given confidence level, the ATC under the DRCC model is smaller than that of the TCC model.In the DRCC model, the node voltages and branch power should meet the constraints at a given confidence level under any given wind power probability distribution.In the TCC model, the node voltage and branch power should meet the constraint at a given confidence level for only one kind of wind power probability distribution.Therefore, the DRCC model argues for a more rigid security constraint for the wind-integrated transmission system, which results in a smaller ATC when this model is employed.

Available Transfer Capability (ATC) under Different Expectations of Wind Power Probability Distribution,
Where Covariance = 0.4 MW 2 Figure 4 illustrates the relationship between ATC and the wind power probability distribution expectation.Given the same covariance of wind power PD and confidence level, we compare ATC values under different expectations of wind power probability.We can find in Figure 4 that with the increase of the confidence level, ATC gradually decreases.When the confidence level increases, ATC decreases to avoid breaking the security constraints.
Figure 4 illustrates the relationship between ATC and the wind power probability distribution expectation.Given the same covariance of wind power PD and confidence level, we compare ATC values under different expectations of wind power probability.We can find in Figure 4 that with the increase of the confidence level, ATC gradually decreases.When the confidence level increases, ATC decreases to avoid breaking the security constraints.In the actual wind integrated transmission system operation, the transmission system should set a reasonable confidence level according to the actual situation with the security satisfied to thus obtain an effective ATC assessment scheme.

Summary
This paper proposes an ATC evaluation strategy considering the uncertainty on the wind power due to the fact that the exact probability distribution of wind power is not given.With respect to the fact that the partial information such as the first and second order moments of wind power probability distribution can be collected from historical data, a DRCC model is presented for ATC assessment.The In the actual wind integrated transmission system operation, the transmission system should set a reasonable confidence level according to the actual situation with the security satisfied to thus obtain an effective ATC assessment scheme.

Summary
This paper proposes an ATC evaluation strategy considering the uncertainty on the wind power due to the fact that the exact probability distribution of wind power is not given.With respect to the fact that the partial information such as the first and second order moments of wind power probability distribution can be collected from historical data, a DRCC model is presented for ATC assessment.The proposed method can meet the system requirements for security operation under any possible wind power probability distribution.The DRCC model is converted into a deterministic model.For solving the deterministic model, the PSO algorithm based on the LMI is employed.The effectiveness and feasibility of the optimal ATC evaluation strategy method is verified through numerical examples.In modeling the ATC assessment strategy for a wind integrated transmission system, it is more in accord with the actual engineering situation using the apparent power in the branch power and node voltage model, which we will concentrate on in future research work.
) Termination: If the iteration counter K meets the predefined value, then the final optima P K+1 G , Q K+1 G and ∑ P K+1 a→b are obtained, and the algorithm ends.Otherwise, let K = K + 1, and repeat Steps (3) and (4).

Energies 2016, 9 , 704 11 of 17 Figure 3 .
Figure 3. Optimal ATC obtained by the DRCC and TCC models for the modified IEEE 30-bus system.

Figure 3 .
Figure 3. Optimal ATC obtained by the DRCC and TCC models for the modified IEEE 30-bus system.

Figure 4 .
Figure 4. ATC under different expectations of wind power PD for the modified IEEE 30-bus system, where the covariance = 0.4 MW 2 .Figure 4. ATC under different expectations of wind power PD for the modified IEEE 30-bus system, the covariance = 0.4 MW 2 .

Figure 4 .Figure 5 Figure 5
Figure 4. ATC under different expectations of wind power PD for the modified IEEE 30-bus system, where the covariance = 0.4 MW 2 .Figure 4. ATC under different expectations of wind power PD for the modified IEEE 30-bus system, the covariance = 0.4 MW 2 .

Figure 5 .
Figure 5. ATC under different wind power covariances for the modified IEEE 30-bus system, where expectation = 2.0 MW.

Figure 5 .
Figure 5. ATC under different wind power covariances for the modified IEEE 30-bus system, where expectation = 2.0 MW.

Table 1 .
Generator parameters and limits.

Table 2 .
Transformer parameters and limits.

Table 1 .
Generator parameters and limits.

Table 2 .
Transformer parameters and limits.