1. Introduction
The increasing penetration of distributed generations (DGs) significantly alleviates the greenhouse gas emission and energy supply shortage issue. However, the high penetration of DGs brings big impacts on the secure and stable operation of the regional power grid. For example, the unidirectional power flow pattern of the regional power grid is changing to the complicated bidirectional power flow pattern [
1,
2,
3,
4]. Moreover, the large-scale integration of DGs influences the power quality of regional power grids [
5,
6], especially the stability of voltage [
7,
8]. For example, in [
8], the impacts of different types of DGs, such as wind turbine (WT) and photovoltaic (PV), on the voltage profile of the regional power grid was studied. The results showed that different types, locations, and sizes of DGs have different influences on regional power grids. Therefore, it is imperative to perform voltage management in regional power grids.
Different means have been proposed to improve the voltage profile of power grids. In [
9], the author proposed a decision-making algorithm that can determine the optimal location and size for DG based on the improvement of the voltage profile and the reduction of total reactive power losses. The proposed algorithm was tested on the IEEE 33-bus radial regional power grid and simulation results demonstrated that the proposed algorithm has an acceptable accuracy. In [
10], the impacts of the integration of energy storage systems (ESSs) of different capacity on the power network were analyzed. The results showed that the integration of ESSs has positive an effect on the voltage profile of the power grid integrated with DGs. In addition to the above means, reactive power planning (RPP) also plays an important role in improving the voltage profile by optimizing the allocation of VAR compensators in terms of location and size. In [
11], a voltage stability index-based method was proposed to find the best placement of static VAR compensators (SVCs) to avoid the voltage collapse. The proposed method first identifies the critical path that experiences the maximum voltage drop and then determines the best location for placing SVC. In [
12], the author addressed the optimal placement of SVC devices by formulating a nonlinear programming problem that maximizes system loading margin and constrains voltage deviations. The proposed optimization was formulated based on the multi-scenario framework and was solved by the bender decomposition technique incorporating multiple restarts. In [
13], the author presented an application of Cuckoo search algorithm to determine the location and size of SVC device. The objective is to minimize the energy losses, voltage deviations and operational cost of SVC. The results demonstrate that the Cuckoo search algorithm always gives the better solution with the high performance.
Although the above methods are effective in determining the optimal placement of SVC, the influence of the integration of DGs on reactive power planning is not considered. In [
14], an RPP model for the regional power grid integrated with DGs was proposed, in which the active power output of DG was considered as its expected value that is calculated using the probability distribution function (PDF) of DG actual power output. The objective function of the proposed model is to minimize the SVC investment cost, system line losses and voltage deviations, and NSGA-II algorithm was used to solve the proposed model. In [
15], the scenario analysis method was adopted to deal with the uncertainty of DG. According to the PDF of DG power output, multiple typical scenarios were generated. Based on these scenarios, an RPP model was formulated to minimize the expected SVC investment cost and expected system energy loss cost. In [
16], a chance-constrained RPP model was proposed for the distribution system integrated with wind farm. The author used the point estimate method (PEM) as the probability power flow calculation methodology and formulated a probabilistic model of wind turbine. In the RPP model, nodal voltage and branch power constraints are formulated as chance-constrained constraints and the voltage stability index is considered to be one of the multiple objectives. In [
17], a cumulant-based stochastic RPP model considering stochastic nature of wind power was proposed. Firstly, power outputs of DGs and load forecasts are modelled using the PDF. Then, a stochastic optimization is proposed to minimize the cost of capacitors and annual energy loss and is solved using the Logarithmic Barrier Interior Point (LBIP) method, which can offer a linear relationship between the cumulants of input variables and output variables. Therefore, the output variables, e.g., capacitor size, have their PDFs that can be reconstructed using the Gram-Charlier Expansion theory.
However, the above-mentioned RPP models do not consider the reactive power adjustments of DGs that can also be used to improve the voltage profile. The author in [
18] pointed out that the PV, WT, and hydropower plant can output reactive power within their capacity such that they can participate in reactive power dispatch in power systems. Therefore, taking advantage of the reactive power adjustments of DGs can reduce the VAR compensators investment cost in regional power grids. The authors in [
19] formulated a fuzzy stochastic RPP model considering reactive power supply from wind generation. The proposed optimization minimizes the total cost of capacitor investment and the annual energy loss while constrains voltages within limits. Moreover, the fuzzy optimization models were used to represent bus voltage constraints in the proposed model. In [
20], the RPP problem is formulated as a two-stage programming model, in which the reactive power adjustment of DG is considered. The model first optimizes the location and size of VAR compensators in one stage, and then minimizes the fuel cost in other stage and, eventually, finds the global optimal RPP results iteratively.
In some extreme conditions, extreme overvoltage problem may occur in the real power system. To cope with this situation, even if the reactive power adjustment of DG is considered in the RPP model, it still requires huge VAR compensators investment due to the limited capacity of reactive power adjustments. Obviously, such a decision is not economic because the occurrence probability of the extreme overvoltage event is very small. It is not reasonable to require a huge VAR compensators investment to deal with the event with very small occurrence probability. To deal with such a problem, the active power adjustment of DG should also be considered in the proposed RPP model. It is expected that the VAR compensator investment cost can be reduced if the active power adjustment of DG can be used to regulate overvoltage under extreme overvoltage scenarios. To the best knowledge of the authors, considering the active power adjustment of DG in the RPP model has not be studied. Therefore, this paper proposes an RPP model considering the active and reactive power adjustments of DGs to obtain an optimal decision for the allocation of VAR compensators.
Firstly, the Latin hypercube sampling (LHS) method [
21] is used to generate scenarios of power outputs of DGs and load consumption, and the number of generated scenarios is reduced using the simultaneous backward reduction technique [
22]. Secondly, based on the typical scenarios, an RPP model considering the active and reactive power adjustments of DGs is proposed to determine the optimal allocation of VAR compensators [
23,
24,
25]. Finally, the proposed RPP model is solved by the proposed primal dual interior point (PDIP) method-based particle swarm optimization (PSO) algorithm.
2. The Proposed RPP Model Based on the Active and Reactive Power Adjustments of DGs
2.1. Scenario Generation and Reduction
In the study, the uncertainties of DGs and load consumption are dealt with using the scenario analysis method. The LHS method is used to generate scenarios of active power outputs of DGs and load consumption and the simultaneous backward reduction technique is used to reduce the number of generated scenarios.
2.1.1. Latin Hypercube Sampling
The LHS method is a stratified sampling method, which ensures that the sampling points can be uniformly and completely covered in the distribution range of the variable. The LHS method consists of two steps, namely sampling and permutation.
Defining
Fk (
Xk) as the cumulative distribution function (CDF) of the input variable
Xk (
k = 1, 2,…,
m). Then, the scale of CDF is divided into
N equal intervals. A value is extracted in each interval and the sampling value is obtained through the transformation of the inverse function of
Fk. The
n-th sampling value is as follows:
The sampling procedure is illustrated in
Figure 1.
Since the correlation of derived sampling values does not reflect the practical correlation between input variables, the permutation procedure is required. In the study, the PSO algorithm is used to obtain the desired correlation between sampling values. The detailed procedures are as follows:
Step (1) Generating a population including a number (Np) of particles:
and (k = 1, 2,…, Np) are the position and updating velocity, respectively, of the k-th particle in the j-th iteration. Here, a particle represents a permutation operation.
Step (2) Defining a fitness function
F(
) to represent the quality of
. The fitness function is as follows:
where
f(
) is the correlation coefficient between sampling values after performing the permutation operation denoted by
on the sampling values;
is the desired correlation coefficient. The
is better when its fitness function is smaller. Accordingly, let
be the best position of the
k-th particle among all positions where it has passed, and let
be the best position among all
(
k = 1,2,…,
Np).
Step (3) Updating
and
as follows:
where,
are the learning factors;
e is the inertial weight; and
are random numbers between 0 and 1.
Step (4) Calculating the fitness functions for all particles in the j + 1 iteration. Comparing the current position of the k-th particle with , and set to be the better one of them. Then, set be the best one of all . Then, if the permutation operation denoted by can make f() closer to the desired correlation coefficient, the corresponding permutation operation is made on the sampling values; otherwise, no permutation operation is performed.
Step (5) Iterative process stops when the correlation coefficient between samples reaches a desired value; otherwise, back to step 3 and the iteration number increases by one.
2.1.2. Simultaneous Backward Reduction Technique
In the study, the simultaneous backward reduction technique is used to reduce the number of generated scenarios. A scenario can be defined as follows:
where
represents
i-th scenario;
Ns is the number of generated scenarios;
is the
s-th element of the
i-th scenario;
l is the length of the scenario. Accordingly, the Kantorovich distance between scenario
i and scenario
j is as follows:
According to above definitions, the procedures of using the simultaneous backward reduction technique to reduce the number of generated scenarios are as follows:
Step (1) Deleting the scenario
that is closest to all the other scenarios. The scenario
satisfies the following equation:
where
is the occurrence probability of
s-th scenario.
Step (2) The number of scenarios decreases by one and selecting the scenario
that is closest to the deleted scenario
using the following equation:
Step (3) Changing the occurrence probability of the scenario
using the following equation:
Step (4) Back to step 1 if the number of remaining scenarios is larger than the specified number; otherwise, the reduction process stops.
2.2. Verification of the Effectiveness of the Active Power Adjustments of DGs under Extreme Overvoltage Scenarios
Based on the generated scenarios of active power outputs of DGs and load consumption, the PDF of bus voltages in regional power grids can be obtained and an example of the PDF of bus voltages is shown in
Figure 2. It can be seen that the area I represents the qualified voltage area while area IV is the extreme overvoltage area. Area II and III are the low voltage area and overvoltage area, respectively, in which the reactive power adjustments of DGs and VAR compensators can meet the demand of voltage management. In the extreme overvoltage area (area IV), the active power adjustments of DGs can be used to manage voltages. The effectiveness of such a means for voltage management is validated in this section.
The modified IEEE 30-bus system is selected as the simulation system, as shown in
Figure 3, in which four DFIG-based wind turbines (WTs) are connected to bus 2 (WT2), bus 5 (WT5), bus 31 (WT31) and bus 33 (WT33), two small hydropower plants (HPs) are connected to bus 11 (HP11) and bus 13 (HP13). Detailed parameters of the system and DGs are given in the
Section 4. It is assumed that the required range of the bus voltage is 0.94 p.u.–1.06 p.u.
Three cases studies were conducted under the extreme overvoltage scenario. The power outputs of DGs, which are connected in the distribution network, under the extreme overvoltage scenario are listed in
Table 1. In case 1, voltage management is not performed; in case 2, the reactive power adjustments of DGs and VAR compensators are used to manage voltages; in case 3, the active and reactive power adjustments of DGs and VAR compensators are used to manage voltages. The power flow results of three cases are listed in
Table A1 in
Appendix and illustrated in
Figure 4. It can be seen that, in case 1, there are extreme overvoltage problems, e.g., bus 25, bus 26 and bus 31, and low voltage problem, e.g., bus 24. In case 2, power outputs of DGs after adjustments are listed in
Table 1 and the location and size of VAR compensators are listed in
Table A2 in
Appendix. It can be seen that, in this case, WT31 absorbs reactive power to alleviate extreme overvoltage at bus 31; however, as shown in
Figure 4, an overvoltage problem still occurs at bus 31. In such a case, an additional investment of VAR compensators is required to regulate voltages due to the limited capabilities of reactive power adjustments of DGs. However, in case 3, the active power curtailment of DGs can help alleviate overvoltage. The power outputs of DGs in case 3 are listed in
Table 1, it can be seen that the active power output of WT31 is reduced and overvoltage at bus 31 is alleviated. Therefore, it can be concluded that the active power adjustments of DGs can be used to effectively improve the voltage profile under extreme overvoltage scenarios. Therefore, the active power adjustment of DG can be considered in the RPP model to deal with extreme overvoltage scenarios and obtain an optimal allocation of VAR compensators. As a result, the cost of VAR compensator investment can be reduced.
2.3. Proposed RPP Model Based on the Active and Reactive Power Adjustments of DGs
Based on typical scenarios of active power outputs of DGs and loads, a novel RPP model considering the active and reactive power adjustments of DGs is proposed. Since the active power curtailment of DGs will reduce the profits of power generation companies, the objective of the proposed RPP model is to minimize the cost of VAR compensator investment and the lost profits of power generation companies for active power curtailment of DGs. Moreover, to accurately characterize the VAR compensator investment cost, the life cycle cost (LCC) of VAR compensator is used to represent the total cost over its entire life cycle.
Objective 1: Minimize the equivalent annual cost of VAR compensator investment.
where
Li is the life cycle cost of the VAR compensator at
i-th bus; if a VAR compensator is installed at
i-th bus,
ki = 1; otherwise,
ki = 0;
Nb is the number of buses in the system;
r is annual discount rate;
n is the number of usable years of the VAR compensator;
CIi,
CMi and
CDi are, respectively, the initial investment cost, operating maintenance cost and the scrap cost of the VAR compensator.
Objective 2: Minimize the expected cost of active power curtailment under typical scenarios
where
Ns is the number of generated typical scenarios;
G is the number of DGs that are used to regulate voltages by adjusting their active power outputs;
c3 is the retail tariff;
T is the number of annual operating hours;
is the expected power output of
i-th DG under
s-th scenario;
is the actual power output of
i-th DG under
s-th scenario after the active power curtailment.
According to the objective 1 and objective 2, the objective function of the proposed RPP model can be described as follows:
Moreover, a penalty term associated with voltage magnitudes can be added in the objective Equation (12) to ensure bus voltages within the acceptable range as follows:
where the third part of the objective Equation (13) is the penalty term;
is the penalty factor;
is the voltage of
i-th bus under
s-th scenario;
and
are, respectively, the lower and upper limits of bus voltages.
Finally, the proposed RPP model can be described as follows:
where
and
are, respectively, active and reactive demands at node
i under
s-th scenario;
is the reactive power output of
i-th VAR compensators under
s-th scenario;
is the reactive power output of
i-th DG under
s-th scenario;
is the bus voltage angle difference;
and
are, respectively, the upper and lower limits of active power output of
i-th DG;
and
are, respectively, the upper and lower limits of reactive power output of
i-th DG;
and
are, respectively, the upper and lower limits of reactive power output of
i-th VAR compensators. Equation (16) represents active and reactive power balance constraints under
s-th scenario; the Equation (17) represents the limits of active and reactive power adjustments of
i-th DG under
s-th scenario; the Equation (18) represents the limit of reactive power output of
i-th VAR compensator under
s-th scenario.
It can be found that the minimum of the sum of the second part and third part in the objective Equation (13) is zero when all bus voltages are within the acceptable range and active power adjustments of DGs are not required. In such a case, the bus voltages can be effectively managed by the installed VAR compensators and reactive power adjustments of DGs. However, when the VAR compensators and reactive power adjustments of DGs cannot meet the requirement of voltage management under extreme overvoltage scenarios, the active power adjustments of DGs are used in the proposed RPP model.
The model Equation (15)–(18) is formulated as a mixed integer nonlinear programming (MINLP) problem, the discrete variables are the location (k1, k2,…, ki), size (, ) and reactive power output () of VAR compensator that discretely provides reactive power. Continuous variables are the active and reactive power outputs of DGs, voltage magnitudes, voltage angles and reactive power output of the VAR compensator that can output reactive power continuously.
3. Proposed Primal-Dual Interior Point Based Particle Swarm Optimization Algorithm
Since the proposed RPP model is formulated as a MINLP problem, a PDIP-based PSO algorithm is developed to effectively solve the proposed model.
3.1. Primal-Dual Interior Point Method
The brief descriptions about PDIP are given in this subsection. An optimization problem can be formulated in the following compact form:
where
X is the vector of decision variables;
f(
X) is the objective function;
H(
X) represents equality constraints;
G(
X) represents inequality constraints. The inequality constraints can be transformed into equality constraints by adding slack variable
Z and a logarithmic barrier function can be added in the objective function to penalize the slack variable. The corresponding model is as follows:
where
γ is the penalty coefficient. Based on Equation (20), the augmented Lagrangian function can be formulated as follows:
where
λ and
μ are Lagrangian multipliers of equality constraints. Corresponding first order Karush-Kuhn-Tucker (KKT) optimality conditions are as follows:
where
.
Then, applying the newton method to the KKT condition Equation (22), the following newton equations can be derived:
where
. In each iteration of the newton method, the variables can be updated using the following equations:
where
is the step size. In addition, in each iteration, the coefficient
γ can be update using the following equation:
where
σ is the centering parameter between 0 and 1;
n is the number of slack variables. Repeating Equations (23)–(25) until the convergence conditions are satisfied.
3.2. Proposed PDIP Based PSO Algorithm
The proposed PDIP-based PSO algorithm has similar procedures as step 1-step 5 in
Section 2.1.1 with the following differences.
Step (1) Generating a population including a number (Np) of particles: and (k = 1, 2,…, Np). Here, a particle represents the location (ki) and size (, ) of a VAR compensator.
Step (2) Defining a fitness function F(), namely objective Equation (15), to represent the quality of . For a given particle (), the fitness function can be obtained by solving the model (15)–(18) with fixed variables (ki, , ) using the PDIP method. Moreover, it should be mentioned that the discrete variables () are firstly assumed to be continuous and the PDIP method is used to search the optimal solution. After obtaining the optimal solution, the derived continuous variables () are rounded and fixed. Then, the optimization model is solved again. The larger the fitness function, the better is the .
Step (3) Updating and using (3).
Step (4) Calculating the fitness functions for all particles in the j + 1 iteration and resetting and .
Step (5) Iterative process stops when the number of iterations reaches a preset maximum or other termination conditions are satisfied, and output as the result; otherwise, go to step (3) and the iteration number increases by one.
The flow chart of the proposed algorithm is illustrated in
Figure 5.
5. Discussion
The comparison results of the four types of RPP models are summarized in this section.
From the perspective of the VAR compensators investment cost, case 3 has the minimum VAR compensator investment cost because overvoltage is regulated by the active power and reactive power adjustments of DGs under extreme overvoltage scenarios. Case 4 has the maximum investment cost because only VAR compensators are used to regulate overvoltage under extreme overvoltage scenarios. Compared with case 4, the VAR compensator investment cost can be reduced in case 3 because reactive power adjustment of DG can be used to help alleviate overvoltage under extreme overvoltage scenarios. In case 1, the VAR compensator investment cost can be further reduced because active power adjustment of DG can be used as the additional means to alleviate overvoltage under extreme overvoltage scenarios.
From the perspective of the active power curtailment cost of DG, there is no active power curtailment cost in case 2 and case 4 because the active power adjustment of DG is not used to regulate overvoltage under extreme overvoltage scenarios in these two cases. In case 3, the active power curtailment cost is the largest because only the active and reactive power adjustments of DGs are used to alleviate overvoltage under the extreme overvoltage scenario. However, in case 1, the VAR compensators can also be used to help alleviate overvoltage; therefore, the active power curtailment cost can be reduced.
From the perspective of the total cost, the proposed RPP model in case 1 has the minimum total cost. For case 3, although case 3 has the minimum VAR compensator investment cost, it has the largest active power curtailment cost, resulting in the largest total cost. For case 2 and case 4, although there is no active power curtailment cost, they have higher VAR compensator investment cost than case 1. Consequently, they have higher total costs than case 1.
Above analyses demonstrate the economic efficiency of the proposed RPP model. In addition, the feasibility of the proposed RPP model is also analyzed. As shown in
Table 5, for the proposed RPP model in case 1, the generation companies lose 0.43% of profits while the percentage of VAR compensator investment saving at power gird side reaches 29.9%. Therefore, the power grid can offer appropriate compensation to power generation companies so that generation companies can accept the proposed RPP model.