Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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An Adaptive Many-objective Robust Optimization Model of Dynamic Reactive Power Sources for Voltage Stability Enhancement  PDF

  • Yuan Chi (Member, IEEE)
  • Anqi Tao
  • Xiaolong Xu
  • Qianggang Wang (Member, IEEE)
  • Niancheng Zhou (Member, IEEE)
State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing 400044, China

Updated:2023-11-15

DOI:10.35833/MPCE.2022.000431

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Abstract

The deployment of dynamic reactive power source can effectively improve the voltage performance after a disturbance for a power system with increasing wind power penetration level and ubiquitous induction loads. To improve the voltage stability of the power system, this paper proposes an adaptive many-objective robust optimization model to deal with the deployment issue of dynamic reactive power sources. Firstly, two metrics are adopted to assess the voltage stability of the system at two different stages, and one metric is proposed to assess the tie-line reactive power flow. Then, a robustness index is developed to assess the sensitivity of a solution when subjected to operational uncertainties, using the estimation of acceptable sensitivity region (ASR) and D-vine Copula. Five objectives are optimized simultaneously

total equipment investment; adaptive short-term voltage stability evaluation; tie-line power flow evaluation; prioritized steady-state voltage stability evaluation; and robustness evaluation. Finally, an angle-based adaptive many-objective evolutionary algorithm (MaOEA) is developed with two improvements designed for the application in a practical engineering problem: adaptive mutation rate; and elimination procedure without a requirement for a threshold value. The proposed model is verified on a modified Nordic 74-bus system and a real-world power system. Numerical results demonstrate the effectiveness and efficiency of the proposed model.

I. Introduction

GROWING integration of renewable energy reduces the inertia of a power system and makes it more vulnerable to disturbance caused by power equipment failure or severe weather conditions [

1], [2]. Besides, a proliferation of industrial and residential motor loads will be detrimental to the dynamic voltage performance of the wind-penetrated power system since these loads consume significant reactive power after a major voltage disturbance [3]. As a prevalent conventional countermeasure to address the reactive power deficiency issue, the capacitor banks are ineffective in alleviating the short-term voltage performance due to their static and discrete nature [4].

Dynamic reactive power compensation devices such as static synchronous compensator (STATCOM) are better candidates to alleviate the voltage stability issue because of their ability to provide dynamic and continuous reactive power in a short time (usually in the millisecond scale) [

5], [6]. The effectiveness of STATCOMs in mitigating the reactive power deficiency is mainly limited by the location of the deployment and the installed capacities [7], [8]. Various metrics are proposed for the steady-state and short-term voltage stability assessment separately or collectively [8]-[13]. For the steady-state voltage stability assessment, load margin of network branches and steady-state bus voltage deviation are prevalent metrics for the reactive power planning problems [9], [10]. On the other hand, voltage trajectory based metrics are used for assessment of short-term voltage stability during the recovery stage [8], [11], [12]. However, these studies do not consider the delayed voltage recovery in the metrics, resulting in an underestimation of voltage disturbance [14]. In [7], [13], voltage deviation time is used to evaluate the voltage recovery speed, but a fixed priority is used for the buses. It should be noted that, during the capacity optimization, the buses with slow voltage recovery speed and poor voltage response at the initial stage of the disturbance might be adequately compensated, so the fixed weighting factors may lead to inefficient reactive power compensation. Furthermore, since all these indices are actually designed as an explicit indicator of the voltage stability, not for the capacity optimization of reactive power source, the non-linearity between different buses and the varying impacts of different capacities throughout the whole optimization process are ignored.

As for the uncertainties, since extreme weather condition is major direct or indirect causes for some of the large scale blackout events [

15], it is important to fully consider the uncertainty of weather-sensitive wind farms before and after a disturbance, along with other operational uncertainties such as load demands. In previous literature, these uncertainties of generation side and demand side are usually modeled in a deterministic approach. For a practical engineering problem, the optimal solutions obtained from the model might be less effective and too sensitive to the varied operational conditions. In other words, the robustness of these solution should be further improved. In [16], [17], a robust reactive power source planning model is proposed for the enhancement of steady-state voltage stability, incorporating the uncertainty of loads. In [18], several uncertainties are implemented in a multi-objective reactive power source planning model, with a requirement for the probability distribution function (PDF) of wind power generation and the probability of load demands. These probabilities, for a long-term engineering planning problem, are difficult and impractical to obtain. An alternative approach to model the stochasticity of wind power generation is through scenario generation [19]. Instead of the PDF, parameters for load and capacity are still needed. In a word, currently operational uncertainties are not systematically incorporated in the field of reactive power planning. This leads to an inferior robustness of the obtained planning decisions. How to effectively and quantitatively address the sensitivity of the optimal solutions when they are subjected to uncertainties is still an open question.

For a reactive power source planning problem considering various and conflicting objectives such as planning and operational cost, voltage stability, optimal reactive power flow, and power loss, a multi-objective (3 or fewer objectives) optimization model [

17], [18] might be inadequate. Prevalent approaches to address this issue include grouping multiple objectives through weighted factors or converting an objective into a constraint [20], [21]. For instance, although in practice a power system is usually divided into different operating regions to limit the reactive power transmission between regions, the tie-line reactive power flow is not optimized in the reactive power source planning problem.

In this paper, an adaptive many-objective robust optimization model of dynamic reactive power sources is proposed to enhance the voltage stability of the power system with a high wind penetration level. The major contributions are as follows.

1) Tie-line power flow index (TPFI) is proposed to assess the reactive power transmission for the minimization of cross-region reactive power transmission.

2) Based on the estimation of acceptable sensitivity region (ASR), a robustness index (RI) is developed to assess the robust optimality of the solutions quantitatively when subjected to correlated uncertainties. There is no requirement for probability distributions of any operational uncertainty considered in the robust optimality analysis. D-vine Copula is adopted to represent the correlation between operational uncertainties and identify the low-probability correlated scenarios that might lead to a conservative estimation of ASR.

3) An index that can effectively differentiate the common voltage deviations and the delayed voltage recovery is proposed to evaluate the short-term voltage stability. This index is with a self-adaptive strategy to assign weighting factors to different buses, enabling a high-efficient reactive power compensation.

One other minor contribution of this study is to improve the applicability of a many-objective evolutionary algorithm (MaOEA) for a practical reactive power source planning problem in the following aspects: ① an adaptive mutation rate to increase the diversity; and ② an angle-based elimination procedure to avoid unrealistic parameter tuning for a complicated practical engineering problem.

II. Adaptive Transient Voltage Severity Index and TPFI

A. Discussion on Voltage Stability

Voltage stability is defined as the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition [

3]. Generally, the voltage stability depends on the ability of power systems to maintain (or restore to) an equilibrium between load and generation. From the perspective of the time scale, the voltage stability problems can be categorized into a long-term phenomenon group or a short-term phenomenon group. Short-term voltage stability is mainly affected by the dynamics of fast-acting load components like induction motors, electronically controlled loads, and inverter-based generators. The time scale of the study is in the order of a few seconds. Differential equations of power systems are required for the analysis and the accurate dynamic modeling of loads is also necessary. A typical example for short-term voltage instability is the stalling of induction motors following a large disturbance. As for the long-term voltage severity, it is mainly affected by equipment with slow response (like thermal loads and tap-changing transformers). The time scale of the study can extend to several minutes.

In this study, two indices are employed to assess the voltage stability of a wind-penetrated power system. An improved voltage collapse proximity index (VCPI) VCPIp considering the priorities of power lines is used to assess the voltage stability in the pre-contingency stage. An adaptive transient voltage severity index (TVSI) TVSIa is used to evaluate voltage stability after a contingency. Tie-line reactive power flow is assessed using TPFI.

B. Adaptive Short-term Voltage Stability Evaluation

TVSI proposed in [

22] has been tested in several short-term voltage stability studies [7], [23]. It is a quantitative evaluation of the power system transient voltage performance of the buses after a disturbance and its mathematical expression is as follows.

TVSIk=it[tc,tend]TVDIi,t,kNb (1)
TVDIi,t,k=vi,t-vi,0vi,0    vi,t-vi,0vi,0δ0                      otherwise    t[tc,tend] (2)

With the growing integration of wind turbines in the power system and the inherent weather-sensitive nature, the extent of the voltage deviation and the time used to restore to an acceptable voltage level, which are not fully revealed using TVSI, should be considered in the evaluation of voltage stability. Therefore, an adaptive index TVSIia, which is based on the original TVSI, is used to describe the immediate voltage behavior of buses, as proposed in our previous work [

14]. TVSIia is defined as:

TVSIia=S1+αlS2+S3+αuS4=tftf+tdl(vdl-v)dt+αltf+tdltl2(vdl-v)dt+tu1tu1+tdu(v-vdu)dt+αutu1+tdutu2(v-vdu)dt (3)

An illustration for the calculation of the index is shown in Fig. 1. S1, S2, S3, and S4 are the voltage deviation areas formulated by the post-contingency voltage trajectory, time-varying lower and upper voltage thresholds, and time spans used to indicate different stages after a disturbance. αu and αl are used to evaluate the negative impact of the prolonged voltage restoration.

Fig. 1  Illustration for calculation of TVSIia.

Using a fixed weighting factor for a bus throughout the whole optimization process is not cost-efficient, since the marginal cost is high to increase installed capacity of a reactive power source at the bus with slow recovery speed and poor voltage response at the beginning stage when it has been compensated adequately. For the case of a fixed weight factor, the focus of the optimization process will be distracted because the importance of the buses with poor voltage response initially should not be prioritized at a later stage of the optimization. Therefore, a self-adaptive approach is used to determine the weighting factors as proposed in our previous work [

24]. Instead of using a fixed weighting factor, high priority always goes to the buses with the worst voltage response currently. Therefore, the overall short-term voltage stability can be quantitatively evaluated as (4). σ serves as a threshold parameter to determine the relative importance of a specific bus, compared with the current average response of others during a disturbance.

TVSIa=iTVSIiaβi/Nb (4)
βi=plow     TVSIia(1-σ)iTVSIia/Nbphigh    TVSIia(1+σ)iTVSIia/Nbpave     otherwise (5)

C. Tie-line Power Flow Evaluation

For a practical utility grid, it is usually divided into different operating regions to limit the reactive power transmission between regions and make it easier to regulate the voltage. However, in existing studies on the planning problem of reactive power resources, the tie-line reactive power flow is not optimized. Instead, it is only considered as a steady-state constraint. If the reactive power flow is not optimized, it is possible that there is a large reactive power flow, but still within the constraint, through the tie-lines, which is detrimental to the voltage stability of the power system. This is because long-distance or regional transmission of reactive power is unfavorable, particularly large tie-line power flow usually exacerbates the severity of the contingency (at or near tie-lines). So, the implementation of the tie-line reactive power flow as an objective can optimize the reactive power flow to alleviate the situation.

Although not as effective as reactive power source deployment, optimization of tie-line power flow will be an economic option to alleviate the reactive power deficiency problem during a contingency. Therefore, TPFI is proposed in this paper for the purpose of optimization.

TPFI=ltQlt (6)

D. Prioritized Steady-state Voltage Stability Evaluation

Line-based VCPI, of which mathematical details can be found in [

25], is a widely-used metric to assess steady-state voltage stability. Both the reliability and the accuracy of the VCPI are high. In this paper, to better reveal the different importance and the different impacts of power lines in the studied power system, an improved VCPI, named VCPIp as (7), is used to assess steady-state voltage stability. For a practical engineering problem, geographical location of the lines can also be incorporated as factors to determine ql.

VCPIp=lqlVCPIl-lVCPIl/Nl2/Nl (7)

III. Many-objective Robust Optimization Model

The proposed model uses electromagnetic transient simulation in Power System Simulator/Engineering (PSS/E) to obtain the post-contingency voltage trajectories for the dynamic analysis and voltage stability evaluation. The electromagnetic transient model cannot be solved by prevalent commercial solvers like CPLEX. Besides, the studied reactive power source deployment problem relates to many planning and operating criteria such as device investment, steady-state and short-term voltage stabilities, and operational costs. These non-linear and sometimes conflicting (such as high stability versus low cost) objectives should be optimized simultaneously instead of being formulated as a single-objective model. Specifically, combining five objectives as one objective through the weighting-sum method has the following issues: ① it is difficult to subjectively determine the weighting factors (for a weighted-sum single-objective model) for each sub-objective; ② the units of each sub-objective might be inherently different; and ③ sensitivities of each sub-objective are different.

Pareto optimality theory [

26] is widely used to find compromise solutions for optimization problems with several objectives that cannot be simultaneously minimized or maximized. But the robustness of these trade-offs is not quantitatively considered in Pareto optimality. This section addresses the issue by employing the concept of sensitivity region to assess sensitivity of the solutions when they are subjected to multi-dimensional uncertainties. To obtain a quantitative assessment result, an RI with the consideration of correlated uncertainties is developed to estimate the robust optimality. A 5-objective optimization model is proposed considering comprehensive economic and technical goals.

A. Compact Mathematical Model

A compact mathematical model, presented as (8)-(10), is used to explain the concept of sensitivity region for a many-objective optimization problem. Uncertainties incorporated in the robust optimality analysis include outputs of wind turbines and load levels.

minuF[f1(x,u,w˜),f2(x,u,w˜),,fNf(x,u,w˜)] (8)

s.t.

G(x,u,w˜)=0 (9)
H(x,u,w˜)0 (10)

B. ASR

Since the optimal solution is obtained under a specific operational status, the random variations of the uncertain variables will result in the change of objective values. A robust solution means the solution is less sensitive to variations of operating conditions. To describe this robustness, the concept of ASR [

27] is used in this paper. For a solution u0 to be analysed for its robustness, ASR is the area in the space of objective value circling the point fi(x0,u0,w˜0), as illustrated in Fig. 2.

Fig. 2  Illustration of ASR.

Subscript “0” denotes the original variable and state. Edge of ASR is formulated by the maximum acceptable change of the objective function when a specific uncertain variable w˜ changes. An example of two uncertainties, namely Δw˜1 and Δw˜2, are shown in Fig. 2. Point A is the case when Δfi, the change of fi, is smaller than the maximum acceptable change Δfimax; point B indicates that Δfi equals Δfimax; and point C means that fi experiences a larger change than Δfimax. The sensitivity of fi(x0,u0,w˜0) against different uncertainties is different. For instance, in the direction s1, fi changes slowly, which means that the objective is relatively insensitive to the change of uncertainty in s1; in direction s2, fi experiences the fastest changes, which means that the objective is much more sensitive to the changes of uncertainties along s2. It should also be noted that the directions discussed are usually related to the changes of more than one uncertainty.

C. Robust Optimality and Robust Pareto Front (RPF)

Conventionally, the optimality of a solution is actually evaluated with a group of specific scenarios. However, for an engineering problem, the changes of different uncertainties might deteriorate the optimality to some extent. Figure 3 shows the negative impact of 3 typical operational uncertainties (load profile, wind power penetration, initial status of STATCOMs) on the voltage response after a fault. In comparison with the original scenarios (considered in the original optimization planning model), the planning decisions might experience poorer steady-state voltage when the voltage restores to a stable level, or slower voltage restoration process, or constant and considerable fluctuation of voltage profiles in post-contingency stage. In order to address this problem, the concept of robust optimality is employed to describe sensitivity of the solutions when they are subjected to different operational uncertainties.

Fig. 3  Negative impact of 3 typical operational uncertainties on voltage response after a fault.

Define nominal Pareto front (NPF) as the Pareto front generated with a fixed w˜0. Then, RPF, all of which are inferior to NPF, can be illustrated in Fig. 4 in 3 typical situations. ① RPF is completely less optimal to NPF. ② There is one or several overlaps between RPF and NPF. But other robust Pareto solutions are sub-optimal. ③ RPF is a subset of NPF. Obviously, a robust solution from RPF might be less optimal compared with the solution from NPF, but its robustness is higher than its counterpart when it is subjected to changes of w˜.

Fig. 4  RPF in 3 typical situations.

Besides, unlike other robust analysis methods, probability distribution of the studied variables is not needed in the following robust analysis. Therefore, given the difficulty in narrowing the gap between planned voltage performance and the actual performance of a power system with high wind penetration level and high percentages of motor loads, the implementation of robust optimality is acceptable.

D. Estimation of ASR

As shown in Fig. 2, the robustness of a solution is directly proportional to the area of ASR. In other words, the solution can cover a larger area under a specific maximum acceptable change of the objective function. Therefore, the area of ASR can be compared with that of other solutions and a larger ASR indicates a better robustness.

In practice, the actual contour of ASR is usually far from symmetric and regular. Therefore, evaluating the robustness of a solution using an accurate calculation of the shadowed area is very difficult and cannot reveal the sensitivity of the solution in the worst scenario. As discussed in Fig. 2, the solution is much more sensitive to the changes of uncertainties in the direction s2 than in the direction s1. It indicates that a smaller change in this direction will cause a relatively larger change in the objective functions. It should be noted that for the more sensitive direction, like s2 in Fig. 2, the length from the original point to the edge of the ASR along this direction is shorter. In other words, a longer length is always favored. In this paper, the most sensitive direction (MSD) is proposed to quantitatively assess the sensitivity of a solution against an operational scenario with multiple uncertainties.

MSD of ASR is defined as the MSD within the ASR, in terms of the change rate of fi when subjected to n uncertainties. Figure 5 illustrates the example of MSD for ASR and feasibility sensitivity region (FSR). The length Rs, from the original point to the edge of ASR along this direction, is employed to indicate the multi-dimensional robust optimality of a solution, instead of the actual area of ASR that is computationally difficult to obtain. Specifically, Rs is defined as Lp-norm in (11) and p in (11) can be 1, 2, or .

minΔw˜Rs(Δw˜)=j=1NuΔw˜jp1p (11)

Fig. 5  Illustrations of MSD for ASR and FSR. (a) MSD for ASR. (b) MSD for FSR.

s.t.

maxi=1,2,,MΔfi-Δf0,iτif (12)

where Δf0,i is an acceptable change caused by Δw˜; and Δfi=fi(x0,u0,w˜0+Δw˜)-fi(x0,u0,w˜0).

Although Rs is efficient, in terms of computation burden, in estimating the ASR, it might lead to a conservative result if the shape of the ASR, which of course cannot be known in advance, has several points that are much closer to the origin of the coordinates than other points on the edge of ASR. As illustrated in Fig. 6, compared with Rs that is dominated by few low-probability cases, Rs' is a better alternative to estimate ASR. The realization of Rs' can be implemented into (11) through an additional constraint expressed as (13).

Fig. 6  Illustration of improved estimation of ASR to avoid conservativeness.

𝒮R/𝒮CpsR (13)

where 𝒮R is the set of correlated scenarios that are within the area surrounded by arc determined by Rs and the x-y coordinates; 𝒮C is the set of all the scenarios studied; and psR is the minimum acceptable ratio of 𝒮R/𝒮C. Instead of using Monte Carlo method or other conventional random sampling methods that consider the operational uncertainty separately, to generate the scenarios for (13), the correlated 𝒮C, considering the dependent uncertainties of outputs of wind turbines and load demands, is generated based on D-vine Copula [

28] and the method proposed in our previous work [24], which is originally proposed for the candidate bus selection, to correctly reveal correlation between uncertainties. The corresponding mathematical details and the methodology can be found in [24].

E. Estimation of FSR

Similar to ASR, FSR is defined as the maximum variations of uncertainties within the constraints. It is used to estimate the feasibility of the solution. An example is shown in Fig. 5(b), and any Δw˜ inside FSR does not affect the feasibility of the solutions (in other words, does not violate any constraint). Similarly to Rs, the length Rf, which is from the original point to the edge of ASR along this direction, can be used to assess the feasibility of the solution against uncertainties as (14). Similar to constraint (13), constraint (17) is also added to avoid a conservative estimation of FSR and 𝒮Rf/𝒮Cf are the corresponding variables like the ones used in (13).

minΔw˜Rf(Δw˜)=j=1NuΔw˜jp1p (14)

s.t.

maxi=1,2,,Mhi(x0,u0,w˜0+Δw˜)0 (15)
maxi=1,2,,Mgi(x0,u0,w˜0+Δw˜)τig (16)
𝒮Rf/𝒮CfpfR (17)

F. Objectives

A 5-objective robust optimization model of dynamic reactive power sources is formulated with following sub-objectives: ① total equipment investment; ② adaptive short-term voltage stability evaluation; ③ tie-line power flow evaluation; ④ prioritized steady-state voltage stability evaluation; and ⑤ robustness evaluation.

1) Total Equipment Investment f1

The total equipment investment f1 is calculated as:

f1=Cinvest=isCiinvest (18)
Ciinvest=ciinstall+ccapcapi    capi00                                capi=0 (19)

2) Adaptive Short-term Voltage Stability Evaluation f2

A risk-based assessment for contingency analysis is used in this paper. It is carried out in following four steps: ① select the index for assessment; ② select a critical contingency (with probability); ③ evaluate the contingency; and ④ calculate the risk-based evaluation results. Specifically, the 2nd objective function is calculated as:

f2=k𝒦ETVSIa(k)P(k) (20)

3) Tie-line Power Flow Evaluation f3

Like f2, a risk-based assessment is also used for the 3rd objective function as follows.

f3=k𝒦ETPFI(k)P(k) (21)

4) Prioritized Steady-state Voltage Stability Evaluation f4

The 4th objective function is calculated as:

f4=k𝒦EVCPIp(k)P(k) (22)

5) Robustness Evaluation f5

For a many-objective optimization problem, it is hard to strike a trade-off between the robustness and the voltage performance only based on Rs and Rf, because they just provide a reference for robustness comparison. For instance, with Rs and Rf, the planners and the operators cannot tell whether this installation plan solution is robust. Therefore, an RI is developed as (23) to represent the robust optimality of the planned reactive power source installation decisions when subjected to multiple uncertainties quantitatively. Rref serves as a reference and it can be selected based on an acceptable Δfi.

f5=Rref/min(Rs,Rf) (23)

G. Constraints

Constraint (24) is for power flow balance. The steady-state operational limits are expressed as (25). Constraint (26) is for the rotor angle stability. It is used to make sure that the rotor angle deviation between 2 generators during a period of T (Δδgg'T) is smaller or equals ω. Constraint (27) is the time-varying voltage trajectory constraint of low-voltage ride through (LVHT) or high-voltage ride through (HVRT).

Pg+P˜w-VijVj(Gijcos θij+Bijsin θij)=0Qg+Q˜w+Qstat-VijVj(Gijsin θij-Bijcos θij)=0 (24)
ViminViVimaxLlminLlLlmaxPgminPgPgmaxQgminQgQgmax    i,l,g𝒢 (25)
max(Δδgg'T)kω    g,g'𝒢s,k𝒦 (26)
Vi,t-LVRT(t)0    t𝒯, iVi,t-HVRT(t)0    t𝒯, i (27)

IV. Solution Method

A. MaOEAs

Generally, there are 5 steps in MaOEAs: initialization, fitness assignment, offspring selection, crossover and mutation, and environmental selection. Fitness assignment is essentially the focus of most of MaOEAs such as non-dominated sorting genetic algorithm III (NSGA-III). However, few algorithms can perform well in convergence and diversity simultaneously. MaOEA with coordinated selection strategy (MaOEA-CSS) is proposed in [

29] to find a balance between convergence and diversity. Its superiority over other MaOEAs is because of an advanced selection strategy with novel convergence and diversity measurements. Although it outperforms prevalent MaOEAs in benchmark problems, it still faces challenges when implemented to solve practical many-objective engineering problems. Specifically, 2 major limitations of MaOEA-CSS are summarized as follows.

1) Inflexible Mutation Rate

It is favourable to have a smaller mutation rate for the first few population generations since their diversity is relatively good. However, if the diversity becomes poor as the generation number increases, it is more suitable to have a higher mutation rate for the improvement of diversity. An inflexible mutation rate, which is insensitive to the diversity, might generate unsatisfactory offspring.

2) Threshold Value for Environmental Selection

For environmental selection step, Euclidean distance is used in MaOEA-CSS to eliminate inferior solutions. A threshold value is needed to compare the Euclidean distances, which plays a critical role in convergence and diversity performance of the whole population. However, there is no universal method to determine the threshold for a specific problem. It varies significantly even for the benchmark problems, let alone a practical engineering problem that is much more complicated.

B. Adaptive Angle-based MaOEA for Reactive Power Source Deployment Problem

To further improve MaOEA-CSS, an adaptive angle-based MaOEA is developed with 2 improvements designed specifically for the reactive power source deployment problem.

1) Self-adaptive Mutation Rate

A self-adaptive mutation rate is proposed to address the complexity of a practical engineering problem based on the crowdedness of the individuals and the generation number. First, a crowdedness index (CI), which is exclusively determined by the population crowdedness, is defined as (28). Equation (29) is used to make sure that the CI is ranging from 0 to 1 for the nth generation.

CIn*=ind=1Nind(uind,n-unavg)/(unmax-unmin) (28)
CIn=(CIn*-CInmin)/(CInmax-CInmin) (29)

A self-adaptive mutation rate pm is defined as (30) and n is the current generation.

pm=p0+CIn(1-p0)(n-1)/(nmax-1) (30)

For the first few generations, (1-p0)(n-1)/(nmax-1) is relatively small. As the algorithm proceeds, (1-p0)(n-1)/(nmax-1) is increasing with the generation number. Instead of a fixed mutation rate or a mutation rate straightforward in proportion to the generation number, pm in this paper will be more adaptive and practical since CIn is only determined by the population crowdedness. In this case, both crowdedness of individuals and the generation number are considered in the adaptive mutation rate.

2) Angle-based Elimination Procedure

Instead of using the threshold value, which is very difficult to be determined for a practical engineering problem, an angle-based elimination procedure is employed to replace the original elimination procedure in [

29], as shown in Algorithm 1. It consists of two steps: handling extreme solutions and eliminating non-extreme solutions.

For the kth objective, a unit vector αk is defined as αk=[0,0,0,1,0,,0]. Only the kth element of αk is 1 and the other elements are all 0. Then, an extreme solution xk can be defined as the solution that has the smallest angle with αk. m is the number of objectives in Algorithm 1. It should be noted that before the elimination procedure, all acute angles between any two individuals in S are calculated and stored for the elimination procedure.

Algorithm 1  : elimination procedure

Input: S (combination of parents and offspring)

Output: new population P'

1. Assign to P'

2. Remove m extreme solutions from S and add them to P'

3. while P'+S>N do

4.  Find the individual pair (xr,xt) which has the smallest angle in S

5.  Find the worst individual x' and remove it from S

6. end while

7. Assign P'S to P'

8. Return P'

C. Computation Steps

The details of each step are described as follows and the illustration of computation flowchart is also given in Fig. 7.

Fig. 7  Illustration of computation flowchart.

Step 1:   initialization. First, algorithm parameters are configured. Then, based on the methodology proposed in our previous work [

14], the candidate buses for the reactive power source installations are selected. The 1st generation of adaptive angle-based MaOEA is generated based on Latin hypercube sampling (LHS) [30].

Step 2:   evaluation of steady-state voltage stability and calculation of tie-line power flow before contingencies. Steady-state voltage stability evaluation is conducted based on VCPIp. Tie-line power flow before contingencies is also calculated in this stage. The base case evaluation result is stored and will be used for the robust optimality analysis in later stages.

Step 3:   evaluation of short-term voltage stability after contingency and tie-line power flow evaluation. Short-term voltage stability of the system is evaluated based on the post-contingency voltage trajectories obtained from time-domain simulations under a set of critical contingencies. Tie-line power flow after contingencies is also calculated in this stage. The base case evaluation result is stored and will be used for the robust optimality analysis in a later stage.

Step 4:   robust optimality analysis. As an estimation of ASR, RI is calculated according to (11) and (14) for each solution, with the consideration of correlated uncertainties.

Step 5:   calculation of objectives. Firstly, total equipment investment, TVSIa, VCPIp, TPFI, and RI are calculated for each solution under each contingency. Then, five objectives are calculated accordingly based on (18), (20)-(23).

Step 6:   update of offspring. The best offspring of the current generation is identified and updated. This procedure includes individual comparison, elimination of inferior solutions (using the proposed angle-based elimination procedure) and selection of the best Pareto solution. Then the selected solution is regarded as the best offspring in the generation.

Step 7:   termination. If any of the following two criteria is satisfied, terminate the whole procedure and generate an RPF: ① the iteration number reaches the pre-determined maximum number; or ② there is a smaller change in fitness function compared with the minimum tolerance.

Step 8:   update of mutation rate and generation of offspring. Update the mutation rate based on the self-adaptive strategy proposed in Section IV-B if the termination does not occur, followed by generation of offspring. Then, go back to Step 2.

Step 9:   judgement of stop criteria. All the evaluations and cost calculations of this solution will be stopped if there is an unsatisfied constraint during the time-domain simulation, and the corresponding objective functions are computed with a high penalty factor. Then, go back to Step 4.

V. Case Studies

A. Studied System and Parameter Setting

A Nordic 74-bus test system [

31] is modified to test the effectiveness of the proposed model, as shown in Fig. 8.

Fig. 8  Nordic 74-bus test system.

Candidate buses are marked in red circles. Doubly-fed induction generators (DFIGs) are located at buses 4021, 4042, and 4062. Time-varying load dynamics are represented by a complex load model [

32], consisting of large motor, small motor, discharging load, transformer saturation, constant power load, and voltage-based load. Six parameters (LM, SM, DL, TS, CP, Kp) listed in Table I are used to represent the percentages of load types. Contingency is selected according to our previous work [7], namely lines 4031-4041, 4032-4044, and 4042-4044. Uncertainties of load demands (0.8-1.2 p.u.) and outputs of wind power (penetration level 20%-30%) are considered in the simulations. Euclidean norm is used in both (11) and (14), namely p=2.

TABLE I  Critical Parameter Setting for Case Studies
ParameterValueParameterValue
cinstall ($) 1500000 αu 2.0
ccap ($) 50000 psR (%) 1
p0 0.15 pfR (%) 1
ω π LM (%) 25
δ 0.1 SM (%) 15
tdl  (s) 0.5 DL (%) 10
tdu (s) 0.5 TS (%) 10
vdl (p.u.) 0.95 CP (%) 10
vdu (p.u.) 1.05 Kp 2
αl 2.0

B. Numerical Results and Discussions

Figure 9 shows an RPF consisting of 80 robust optimal solutions. Fuzzy membership function [

33] is used to find the compromise solution (marked in Fig. 9), of which capacities are listed in Table II.

Fig. 9  RPF consisting of 80 robust optimal solutions.

TABLE II  Capacities of Planned STATCOMs
Bus No.Capacity (Mvar)Bus No.Capacity (Mvar)
41 38.0 g11 37.5
42 105.9 g14 62.9
46 108.4 g17 45.6
2031 48.2

Five additional models are compared to show the advantages of the proposed model (Model A), as shown in Table III. The following aspects are considered and compared: overall performance of the proposed model, effectiveness of the proposed voltage stability indices, effectiveness of the proposed TPFI, effectiveness of the proposed RI, and diversity of the Pareto fronts.

TABLE III  Comparison of Different Models
ModelObjectiveSolution algorithmSelection strategy (from Pareto front)
A f1, f2, f3, f4, f5 Proposed Fuzzy membership [33]
B f1, f3, f5, TVSIa, VCPIp Proposed Fuzzy membership
C f1, f3, f5, TVSIa, VCPIp Proposed Best TVSIa results
D f1, f2, f3, f4, f5 NSGA-III [34] Fuzzy membership
E f1, f2, f4 NSGA-II [35] Fuzzy membership
F f1, f2, f3, f5, and RI calculated without (13) and (17) Proposed Fuzzy membership

1) The comparison between Model A and Model B aims to demonstrate the superiority of the evaluation metrics (TVSIa and VCPIp).

2) The comparison between Model A and Model C also aims to demonstrate the superiority of the evaluation metrics (TVSIa and VCPIp), even with a more aggressive selection strategy.

3) The comparison between Model A and Model D aims to demonstrate the superiority of the proposed solution algorithm.

4) The comparison between Model A and Model E aims to demonstrate the effectiveness of the implementation of proposed TPFI and RI as objectives.

5) The comparison between Model A and Model F aims to demonstrate the effectiveness of constraints (13) and (17). The constraints are introduced to avoid a conservative estimation of ASR and FSR, respectively.

1) Overall Performance of Proposed Model

The simulation results are summarized in Table IV. Model A outperforms all other models in all aspects, with an exception for the total cost f1 (Model E is with the least cost). However, the robustness of the solution from Model E is inferior to that from Model A since Model E ignores the quantitative robustness analysis of the solutions. So, when the solution of Model E is subjected to operational uncertainties, the voltage response of the power system in the post-contingency stage is expected to significantly deteriorate, as illustrated in Fig. 3.

TABLE IV  Simulation Results
Modelf1f2f3f4f5Computation time (s)
A 32.83 0.3024 0.6068 31.475 1.3598 249580
B 32.68 0.4300 0.6213 31.553 1.3759 266076
C 35.14 0.3553 0.6161 31.504 1.4027 266076
D 32.86 0.3199 0.6131 31.544 1.4457 272863
E 31.20 0.3561 0.6267 31.479 1.4682 267459
F 32.96 0.4199 0.6144 31.540 1.4569 251114

2) Effectiveness of Proposed Voltage Stability Indices

The comparison of numerical results between Model A and Model B demonstrates that the proposed voltage stability indices make the investment more cost-efficient: a slight investment increase (0.46%) can make a significant performance improvement in all five objectives. A significant improvement (29.6%) is observed in the evaluation result of the short-term voltage stability (TVSIa).

Furthermore, even with 7.0% more investment, the voltage performance of Model C (based on original TVSIa and VCPIp) is still much worse than that of the Model A. It should be noted that the decision-making is solely based on TVSIa results.

Figure 10 shows the superiority of TVSIa compared with the original TVSI [

22].

Fig. 10  Superiority of TVSIa compared with original TVSI.

For the original TVSI only considering the voltage violation in the post-contingency stage, the delayed voltage recovery after 6 s has very little impact on the final evaluation results. Based on the definition of original TVSI, which does not differentiate the common voltage deviations (within the required time window) and the delayed voltage recovery, the evaluation result (0.302 after normalized) for short-term voltage stability is calculated by integrating all the voltage deviations. Nonetheless, the inferior voltage recovery process will put wind turbines in danger of automatically triggering and frequent voltage violations. Alternatively, TVSIa can penalize and reflect the prolonged voltage restoration. For the case of Fig. 10, a significant increase (49.67%) in the evaluation result (0.452 after normalized) can clearly and correctly reveal the potential danger imposed on wind turbines. Besides, a larger variation can also make the proposed solution algorithm more efficient in finding optimal solutions.

As shown in Fig. 11, thanks to the adaptive strategy adopted in (4) and (5), TVSIa can also help the proposed solution algorithm focus on the buses with the worst voltage performance. If the original TVSI is used, the increase in its value is proportional to the installed STATCOM capacity at bus 42. This indifference leads to inefficient decisions because as the capacity increases, bus 42 might not be the worst one among the load buses. In contrast, the change rate of TVSIa adaptively varies: at the beginning, TVSIa is very sensitive to the capacity increase at bus 42 and is becoming less sensitive as the installed capacity reaches a certain value. This index can correctly reflect the marginal cost of improving the voltage performance of a bus with adequate reactive power support.

Fig. 11  Sensitivity analysis of TVSIa and TVSI subjected to various capacities.

3) Effectiveness of Proposed TPFI

Since the proposed TPFI is implemented in the optimization model as an objective (f3), Model A has the smallest reactive power flow. This is particularly beneficial for a scenario when a major voltage disturbance or fault occurs on the tie-lines, since a fault on a heavy-loaded tie-line is usually a trigger for cascading contingencies. For Model E, of which tie-line power flow is not optimized, there is a large reactive power flow compared with all other models. Since reactive power source deployment in a real-world power system is usually carried out in an area-based way instead of taking the whole system as one entity, the implementation of the proposed TPFI in the optimization model is helpful in minimizing the reactive power flow, compared with considering the reactive power flow as a steady-state constraint.

4) Effectiveness of Proposed RI

Among all the models with RI as an objective, Model F (using a conservative way to calculate RI) has the worst robustness. It is exactly the conservative case illustrated in Fig. 6: the estimated Rs is dominated by few low-probability cases. The only difference between Model A and Model F is that constraints (13) and (17) are not considered in the calculation of RI. It is clear that the proposed RI can effectively avoid conservativeness and accurately estimate ASR without compromising the voltage stability performance.

5) Diversity of Pareto Fronts

Figure 12 illustrates Pareto front of solutions from Model D (solved by NSGA-III).

Fig. 12  Pareto front of solutions from Model D.

Compared with Fig. 9 (Model A, solved by the proposed adaptive angle-based MaOEA), it can be observed that the values of objectives (Model D) cover a relatively smaller space. Besides, the average distance between decision variables of Model D (0.8808) is also smaller than that of Model A (0.9271). The diversity of the solutions from Model A is improved by the proposed adaptive angle-based MaOEA. Furthermore, thanks to the LHS adopted for the first generation and the adaptive strategy used for the mutation rates, the total computation of Model A is less than that of Model D, even with the same objectives and constraints.

C. Scalability and Computation Efficiency Discussion

Due to the highly non-linear and non-convex features, as well as the time-consuming time-domain simulations, the reactive power source deployment problem considering short-term voltage stability issues is naturally with high computational complexity. The total execution time T can be estimated based on the parameters listed in (31).

T(Nc,Ngen,Npop,Ndec,τf/g,Tsim,TOPF) (31)

where τf/g represents the tolerance degree for constraint (12) or (16). In practice, reactive power source deployment of a large-scale power system is usually studied in an area-based method rather than in a way that takes the whole system as one entity. By employing the area-based method, the self-sufficiency of each area can be maximized and the reactive power transmission between different areas can be minimized [

36]. Specifically, a large-scale power system can be divided into different areas and then the reactive power source planning can be conducted within one area at a time while the other areas are modelled as equivalent sub-systems.

In the proposed planning model, with the help of proposed TPFI implemented as an objective in the optimization model, the computation complexity is alleviated by dividing the studied system into several smaller ones. When the size of the power system increases, the corresponding increase in computation time is relatively moderate since the number of candidate installation sites in a larger system is not necessary to increase proportionally.

A modified real-world power system in China, with 446 buses and 27 power plants (including wind farms), is adopted to show the efficiency and effectiveness of the proposed model. To better reflect the impact of the increasing penetration level of wind generation, seven thermal power plants in the original power system are replaced with wind farms. As shown in Table V, there is only a 54.6% increase in the total computation time, while the number of total buses and candidate buses significantly increase (from 74 to 446 total buses, an increase of 502.7%, and from 7 to 14 candidate buses, an increase of 100%).

TABLE V  Comparison of Computation Costs
Test systemTotal computation time (s)
Nordic 74-bus (74 buses and 7 candidate buses) 249580
Real-world power system (446 buses and 14 candidate buses) 386017

So, the proposed model is applicable for reactive power source deployment of a real-world transmission network as long as the network can be divided into areas, which is also the operational practice for most electric utility corporations. However, among the factors that affect the computation cost as listed in (31), due to the expensive time-domain simulation (a single run costs a few seconds for a large-scale power system), Tsim is the dominant factor in the total computation cost of the proposed model. Therefore, for a modern power system with over thousands of buses, a distributed computing platform [

37] can be adopted to speed up the computation, enabling parallel simulations for the contingency and robustness evaluation.

VI. Conclusion

In this paper, an adaptive 5-objective robust optimization model of dynamic reactive power sources is proposed to enhance the voltage stability of a wind-penetrated power system and improve the robust optimality of the planned STATCOMs simultaneously. The key conclusions are listed as follows.

1) Compared with other models, robust optimal solutions from the RPF are robust to the changes of operational uncertainties. And a balance between the robustness optimality and voltage performance of a solution is achieved. Specifically, the proposed RI not only considers the correlation between uncertainties, but also effectively avoids a conservative estimation of the robustness without a compromise in voltage stability performance.

2) The proposed TPFI implemented in the optimization model as an objective can effectively minimize the reactive power flow, compared with a steady-state constraint approach.

3) The proposed short-term voltage stability index can effectively differentiate the common voltage deviations (within the required time window) and the delayed voltage recovery. Furthermore, this index can correctly reflect the marginal cost of improving the voltage performance of a bus with adequate reactive power support by adaptively changing the priorities of buses.

4) Compared with the original MaOEA-CSS, the proposed adaptive angle-based MaOEA does not need the parameter tuning for the threshold value. The feature is favorable for reactive power source deployment since it is usually difficult to determine such parameters for a practical engineering problem.

Nomenclature

Symbol —— Definition
A. —— Indices
g, g' —— Indices for generators
i, j —— Indices for buses, objectives, or constraints
k —— Index for contingency
l —— Index for lines
n —— Index for uncertainties or generations
o —— Index for an original state
p —— Index for Pareto solutions
t —— Index for time
B. —— Sets
—— Set of all buses
s —— Set of buses of static synchronous compensators (STATCOMs)
w —— Set of wind turbine buses
𝒢 —— Set of all generators
𝒢s —— Set of synchronous generators
—— Set of lines
t —— Set of tie-lines
𝒦 —— Set of contingencies
𝒯 —— Set of time
C. —— Parameters
αl, αu —— Lower and upper penalty parameters for voltage violation
τif —— Tolerance for an acceptable deviation Δf0,i
τig —— Tolerance for constraint gi
ω —— Rotor deviation under an extreme condition
σ —— A threshold for the importance of buses
δ —— A threshold for voltage deviation
cinstall —— Installation cost of STATCOM
ccap —— Unit cost of STATCOM
CP —— Percentage of constant power loads in complex load model
CInmin, CInmax —— The minimum and maximum crowdedness of the nth generation
DL —— Percentage of discharging lighting loads in complex load model
ETVSIa(k) —— Adaptive transient voltage severity index (TVSI) result of contingency k
ETPFI(k) —— Tie-line power flow index (TPFI) result of contingency k
ETVSIp(k) —— Prioritized voltage collapse proximity index (VCPI) result of contingency k
HVRT(t) —— Time-varying voltage trajectory constraints for high-voltage ride through (HVRT)
Kp —— Parameter for voltage-dependent real power load in complex load model
LM —— Percentage of large motor loads in complex load model
LVRT(t) —— Time-varying voltage trajectory constraints for low-voltage ride through (LVRT)
Llmin, Llmax —— The minimum and maximum apparent power flows on line l
Nb —— Number of total buses
Nc —— Number of contingencies considered in time-domain simulations
Ngen, Npop, Ndec —— Number of generations, population size, and number of decision variables for non-dominated sorting genetic algorithm II (NSGA-II)
Nf —— Number of objective functions
Ng —— Number of equality constraints
Nh —— Number of inequality constraints
Nind —— Number of individual (in a generation)
Nl —— Number of total power lines
Nu —— Number of uncertainties
nmax —— The maximum value of generations
p —— Parameter for Lp-norm
plow, phigh, pave —— Adaptive weighting factors
p0 —— Mutation rate of the first generation
Pgmin, Pgmax —— The minimum and maximum active power outputs of generator g
P(k) —— Probability of contingency k
ql —— Priority of line l
Qgmin, Qgmax —— The minimum and maximum reactive power outputs of generator g
Rref —— Reference robust radius
SM —— Percentage of small motor loads in complex load model
tc —— Fault clearing time
tdl —— The maximum voltage deviation time (lower)
tdu —— The maximum voltage deviation time (upper)
tend —— End of time-domain simulation time
tf —— Fault time
tl1, tl2 —— End of voltage deviation time within and beyond tdl (lower)
tu1, tu2 —— End of voltage deviation time within and beyond tdu (upper)
Tsim —— Computation time for a single time-domain simulation run
TOPF —— Computation time for a single optimal power flow run
TS —— Percentage of transformer saturation loads in complex load model
vdl, vdu —— Lower and upper voltage thresholds
Vimin, Vimax —— The minimum and maximum voltages of bus i
D. —— Variables
w˜ —— Uncertainties
Δfi —— Variations of objective function i
Δf0,i —— Acceptable deviation of objective function i
Δw˜ —— Variation of uncertainty
P˜w —— Active power output of wind turbine w
Q˜w —— Reactive power output of wind turbine w
Δδgg'T —— Rotor angle deviation between two generators during a period of T
θij —— Voltage angle difference between buses i and j
βi —— Adaptive weighting factor for bus i
Bij —— Susceptance between buses i and j
capi —— Reactive power source capacities at bus i
CIn, CIn* —— Normalized and un-normalized crowdedness of the nth generation
Gij —— Conductance between buses i and j
Ll —— Apparent power flow on line l
pk —— Probability of contingency k
pm —— Adaptive mutation rate
Pg —— Active power output of generator g
Plr —— Active power transferred to the receiving end through line l
Qg —— Reactive power output of generator g
Qlr —— Reactive power transferred to the receiving end through line l
Qlt —— Reactive power in tie-line l
Qstat —— Reactive power output of STATCOM
Rf —— Radius of estimated feasibility sensitivity region (FSR)
Rs —— Radius of estimated acceptable sensitivity region (ASR)
s1,s2 —— Sensitive and insensitive directions
TVSIa —— Adaptive TVSI
u —— Decision variables
uind,n —— Value of fitness function of individual d in the nth generation
unavg, unmax, unmin —— Average, the maximum, and the minimum values of fitness functions in the nth generation
v —— Voltage magnitude
vi,t —— Voltage magnitude of bus i at time t
vi,0 —— Voltage magnitude of bus i at original state
VCPIp —— Prioritized VCPI
VCPIl —— VCPI of line l
x —— Power system states
E. —— Functions
Cinvest —— Total investment cost
fi() —— Sub-objective function
F —— Overall objective function
gi —— Equality constraint i
G —— Equality constraints
hi —— Inequality constraint i
H —— Inequality constraints

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