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Transmission Line Overload Risk Assessment for Power Systems With Wind and Load-Power Generation Correlation Xue Li, Member, IEEE, Xiong Zhang, Lei Wu, Senior Member, IEEE, Pan Lu, and Shaohua Zhang
Abstract—In the risk-based security assessment, probability and severity of events are the two main factors for measuring the security level of power systems. This paper presents a method for assessing line overload risk of wind-integrated power systems with the consideration of wind and load-power generation correlation. The established risk assessment model fully considers the probability and the consequence of wind uncertainties and line flow fluctuations. The point estimate method is employed to deal with the probability of line overload and the severity function is applied to quantify line flow fluctuations. Moreover, with the Cholesky decomposition, the correlation between loads and power generations are simulated by the spatial transformation of probability distributions of random variables. In addition, Nataf transformation is used to address wind resource correlation. Finally, the line overload risk index is obtained, which can be used as an indicator for quantifying power system security. Numerical results on the modified IEEE 30-bus system and the modified IEEE 118-bus system show that the types and the parameters of the wind speed distribution would affect the risk indices of line overload, and the risk indices obtained with the consideration of wind resource correlation and load correlation would reflect the system security more accurately. Index Terms—Load-power generation correlation, overload risk assessment, point estimate method (PEM), probabilistic load flow (PLF), severity function, wind resource correlation.
N OMENCLATURE Variables B c, k CM D(·), E(·) f (Zl ) Fi (·) M MT
Orthogonal matrix. Scale parameter and Shape parameter of the Weibull probability density function (PDF). Variance-covariance matrix of vector M. Variance and mean. PDF of Zl . Nonlinear function. Random vector, including load and generation powers. Transpose of vector M.
Manuscript received March 24, 2014; revised July 23, 2014, November 1, 2014, and December 15, 2014; accepted December 26, 2014. This work was supported in part by the National Science Foundation of China under Grant 51007052 and Grant 71201097, and in part by the Science and Technology Commission of Shanghai Municipality under Grant 14ZR1415300. Paper no. TSG-00263-2014. X. Li, X. Zhang, P. Lu, and S. Zhang are with the Key Laboratory of Power Station Automation Technology, Department of Automation, Shanghai University, Shanghai 200072, China. L. Wu is with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY 13699 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSG.2014.2387281
Pt (Zl ) Risk (Zl ) Se (Zl ) v yi Zl μ(·) , σ(·) ξi,k ωi,k λi,j
Probability of overload at line l. Risk index of overload at line l. Severity of overload at line l. Wind speed (m/s). Random variables including wind speeds, loads, and conventional power generations. Line flow of branch l. Mean and standard deviation. Standard location. Weight. jth standard central moment of random variable yi with PDF fyi .
Constants I Identity matrix. t Confidence level. Zl min , Zl max Lower/upper capacity limits of branch l. I. I NTRODUCTION ECURITY continues to be the most important aspect in power system operation and planning, especially with high penetration of renewable sources. The smart grid uses intelligent transmission and distribution networks for delivering electricity, which would enhance security, reliability, and efficiency of power systems. Transmission line capacity limit violations [1], voltage violations [2], and voltage stability and/or transient stability [3] would impact the security of power systems and reduce the efficiency of system operation. In power system planning and operation, security assessment [4]–[7] would assist system operators in maintaining the system security level within an acceptable range. Three strategies, i.e., deterministic evaluation [8], probabilistic evaluation [9], and risk assessment [10]–[13], are used to explore power system security. In traditional deterministic assessment, a power system is operated under significant security margins with respect to a set of most credible contingencies [14]. This would result in highly conservative decisions with high operation costs. The probabilistic evaluation considers the likelihood of events and uses probabilistic indices such as loss of load probability (LOLP) and expected unserved energy (EUE). LOLP reflects the likelihood but not the severity, while EUE captures both likelihood and severity. However, EUE only measures load interruptions, but not the costs associated with equipment damages or lost
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opportunities from equipment unavailability [8]. Capturing both the likelihood and the severity of events, risk-based security assessment (RBSA) has been used to determine the security level. In order to allow a power system to be operated closer to or even beyond its limits, RBSA is applied for evaluating power system security at planning and operating stages. Online RBSA [10] condensed the contingency likelihood and the severity into probabilistic risk indices, which can be used in control room decision making for understanding potential network problems, such as overloads, cascading overloads, low voltages, and voltage instabilities. A risk-based approach was discussed in [11] to monitor and manage the probability and the consequence of potential cascading outages for reliable operations. The risk-based approach to the security assessment [12] provided more information to guide secure operating decisions than the traditional N-1 security criterion. Ni et al. [13] developed a software implementation for online RBSA. It computed indices based on probabilistic risks, which describes the system security level as a function of existing and near-future network conditions. System risk assessment includes a wide range of analysis on generation, transmission, and distribution assets. In terms of the transmission security, the transmission line overload risk assessment is a crucial task, which includes assessing the cumulative risk associated with overload security for mid-term power system planning [15], optimally controlling system security levels associated with overloads [16], deriving a more robust overload risk index for describing the total vulnerability level of power systems [17], and exploring the overload risk with the consideration of probability and consequence of line flow fluctuations [18]–[20]. Wind energy has been widely deployed throughout the world and is playing more and more important roles in emerging electric energy systems. Volatility and uncertainty of wind energy poses new challenges for the grid security and potentially increases outage risks [21], [22]. A smart grid should accommodate new technologies of renewable generation and provide the high level of security. There has, however, been relatively little work done on assessing the risk-based security of wind-integrated power systems. Existing studies fall into the general category of operating risk analysis, which involves evaluating the contribution of wind power on the load carrying capability in power systems [23], assessing the impact of high wind penetration on the generation adequacy [24], and committing additional generation sources for maintaining the system reliability [25], [26]. Specifically, few researches focus on evaluating the impacts of random wind energy or load correlation on risk indices, and quantifying the likelihood and the severity of line overloads in wind-integrated power systems. This paper aims at evaluating the transmission line overload risk in wind-integrated power systems while considering wind and load-power generation correlation. Cholesky decomposition is used to simulate the correlation between loads and power generations by the spatial transformation of probability distributions of random variables. By considering random wind correlation and the correlation between conventional generations and loads, the risk assessment model of line overload
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for wind-integrated power systems is constructed to formulate the likelihood and the severity of line overloads. Finally, 2m + 1 point estimate scheme is applied for the overload risk evaluation. The severity value of line overload only describes the outcome of line overloads. The risk value adopted in this paper, product of the probability and the severity of transmission line overloads, reflects the security level of system. Interesting simulation results and informative conclusions have been derived. It is observed that types and parameters of uncertain wind speed distribution would affect risk values of line overload. In addition, wind resource correlation and load-power generation correlation would also impact the risk values of line overload. Consequently, the proposed model can be used to evaluate the impacts of wind resource correlation and load-power generation correlation on overload risk indices with various wind speed distributions. Once the risk values of individual lines or the entire system are obtained, system operator can evaluate the system operation condition and decide necessary actions to reduce the risks. The rest of this paper is organized as follows. Section II presents the overload risk assessment model for windintegrated power systems. With the consideration of wind and load-conventional generation correlation, the point estimate method (PEM) for the overload risk assessment is described in Section III. The simulation results on the modified IEEE 30-bus system and the modified IEEE 118-bus system are described in Section IV, and the conclusions are drawn in Section V. II. M ODEL D ESCRIPTION A. Probabilistic Model for Line Overload The line overload possibility can be measured by the probability distribution of line flows. The probability distribution can be derived from the probabilistic load flow (PLF) calculation [27]. The output line-flow vectors Z can be expressed as a function of input random wind power, load, and power generation, as shown in (1), where random vector y is composed of uncertain wind speed, loads, and power generations in a power system Z = F(y).
(1)
The likelihood index in (2) denotes the cumulative distribution function (CDF) of random variables whose samples do not satisfy the safety threshold with the confidence level t. With the assumption that line flows follow the normal distribution, t is the confidence level when random variable samples comply with the “3σ principle” as shown in (3) / [Zl min , Zl max ]) Lik(Zl ) = Pt (Zl ∈ t = P |Zl − E(Zl )| < 3 D(Zl ) .
(2) (3)
The CDF of random variables can be written as (4), where the PDF f (Zl ) can be obtained from the results of the PLF calculation / [Zl min , Zl max ]) = f (Zl )dZl . (4) Pt (Zl ∈ Zl ∈[Z / l min ,Zl max ]
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Fig. 1.
Severity function of line overload.
B. Severity Model of Line Overload Severity function is used to uniformly quantify the severity of line overload. Severity function for line overload is related to the real power flow of a transmission line and is specified for each transmission line. Mean values of the real power flow as a percentage of the power rating would determine the overload severity of a line [10]. The continuous severity function for line overload shown in Fig. 1 is used in this paper. For each line, when the real power flow is no larger than 90% of the rated value, the risk severity value is zero. When the line real power flow is larger than 90% of the rated value, the risk severity value is a linear function of the real power flow, and is equal to one when the line real power flow is equal to the rated value. Thus, for each line, the closer the risk severity value to one, the more severe the line would be overloaded. Other severity functions, such as discrete severity functions [10], may also be used, and the proposed line overload risk assessment method is still valid. C. Risk Index of Line Overload The overload risk index Risk (Zl ) of line l (5) is defined as the product of the probability Pt (Zl ) and the severity Se (Zl ) of the transmission line overload. Therefore, the line overload risk index of the entire system can be calculated as (6) +∞ Pt (Zl )Se (Zl )dZl (5) Risk (Zl ) = −∞ Rall = Risk (Zl ). (6) III. PEM FOR OVERLOAD R ISK A SSESSMENT W ITH W IND U NCERTAINTY AND L OAD -P OWER G ENERATION C ORRELATION A. Point Estimate Theory The PLF calculation for wind-integrated power system derives the probabilistic distribution of line flows, which could describe probabilities of line overload with the consideration of uncertain wind, loads, and power generations. Many methods have been used for solving the PLF, including Monte Carlo simulation (MCS) [28], first order second moment method (FOSMM) [29], cumulant method (CM) [30], and PEM [31]–[34]. MCS needs a significant computational effort and is usually used as the benchmark for evaluating the effectiveness of various algorithms. The main idea behind other methods, including FOSMM, CM, and PEM, is to use
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approximate formulas for calculating statistical moments of random output quantities. PEM requires less computational burden. However, most PLF studies have assumed that random variables are independent, which largely ignore the correlation between loads and power generations or spatial correlation among wind speeds. The correlation would have significant impacts on the results of PLF [36], [37] and, in turn, the risk assessment results. Therefore, it is necessary to consider the impacts of correlated parameters when quantifying risk indices of wind-integrated power systems. Saunders [35] extends the PEM approach to include both spatial and temporal correlations among wind generations and loads. Saunders [35] applies the singular value decomposition (SVD) approach to create spatially and temporally correlated wind and load random variables, and transform the correlated random variables into uncorrelated random variables for finding the concentrations of the PEM analysis. The locations derived from the concentrations, which relate to the uncorrelated truncated random variable set, are remapped back to the original variable space. The SVD approach in [35] has certain advantages. For example, using the SVD, the basic idea of principal component extraction is applied in order to truncate the input variable set. According to the singular value matrix, random variables with the greatest variance (i.e., the most prominent impact on probabilistic outputs) can be selected for a truncated random variable set. This set of truncated random input variables can be utilized in the PEM with reduced computational efforts. Authors have made the effort for the variable set truncation (i.e., the subset selection problem) of variant correlation [42]. In [42], a novel forward subset selection algorithm was proposed which can not only choose the important input variables but also integrate the key properties of the group ability into the modeling procedure. Moreover, the relation of solution and the correlation between the two variables are also proved. Except the SVD approach, PCA principal component analysis and ICA independent component analysis can also be used to truncate variable set. The main objective of these methods is to extract the principal components. However, when all random variables are equally important, the variable selection process would not be needed. Thus, all input data are employed and both Cholesky decomposition and SVD method can be applied. The two algorithms have been compared in [43]. It has been clearly clarified that Cholesky decomposition is the fastest method, and SVD is more computational expensive but is numerically stable. PEM focuses on the statistical information provided by the first few central moments of input random variables with K points for each variable. K is a parameter depending on the Hong’s method [30], named concentrations. This paper carries on a 2 × m + 1 type scheme of PEM for evaluating the line overload risk, which gives a good tradeoff between the solution accuracy and the computational efforts [33]. m denotes the number of random variables. The scheme requires 2m+1 evaluations of function F and uses the first three concentrations for each input random variable, i.e., K = 3. Generally speaking, in the ac power flow calculation, the output line-flow vectors Z can be expressed as (7). The random variable yi (i = 1, . . . , m) denotes wind speeds,
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loads, and power generations in a power system. Multivariate dependent random variables are generated by Cholesky decomposition [36], [38]. Each random variable yi has known mean μyi and variance σyi , which can be obtained from historical data, statistical analysis, or engineering judgment
N-dimension random vector including loads and power generations in power systems. Its variance-covariance matrix CM is given by (14) CM = cov M, MT .
Zi = Fi (y1 , y2 , . . . , ym ).
Matrix CM is symmetric by definition. As a result, there always exists an orthogonal matrix B, by which the correlated variable vector M can be transformed into a new uncorrelated set W as shown in (15). The variance-covariance matrix CW is computed via (16)
(7)
Suppose that random variable yi follows the PDF fyi and the goal is to find the PDF of Z. According to Hong’s point [32], [33] estimate methods, the kth concentration (yi,k , ωi,k ) of a random variable yi can be defined as a pair composed of a location yi,k and a weight ωi,k . Location yi,k is the kth value of variable yi at which the function F is evaluated. Weight ωi,k is the weighting factor which accounts for the relative importance of this evaluation in output random variables. The location yi,k to be determined at the point (μy1 , μy2 , . . . , yi,k , . . . , μym ) is yi,k = μyi + ξi,k σyi , i = 1, . . . , m, k = 1, . . . , K.
(8)
For random variable yi , the PDF fyi can be approximated by matching the first few moments of yi . Thus, the standard location ξi,k and the weight ωi,k can be obtained by solving (9), where λi,j is given in (10) K
j ωi,k ξi,k = λi,j , i = 1, . . . , m, j = 1, . . . , 2K − 1
k=1
λi,j = E
j j σyi . yi − μyi
(9) (10)
sum of all weights is equal to one, i.e., The m K i=1 k=1 ωi,k = 1. In addition, it is specified that the sums of the weights for each i satisfy K
ωi,k = 1/m, i = 1, . . . , m.
(11)
k=1
The standard locations ξi,k and the weights ωi,k can be obtained by the solution of the system equation [33]. Once all the concentrations (yi,k , ωi,k ) (i = 1, . . . , m, k = 1, . . . , 3) are obtained, the function F can be evaluated at the point (μy1 , μy2 , . . . , yi,k , . . . , μym ) to yield Z(i, k). By using ωi,k and Z(i, k), the jth raw moment of the output random vector Z can be obtained from (12). The mean and the standard deviation of Z can be calculated via (13)
j E(Zj ) ∼ = ω0 F μy1 , μy2 , . . . , yi,k , . . . , μym +
m K
j ωi,k F μy1 , μy2 , . . . , yi,k , . . . , μym (12)
i=1 k=1 m 3
μZ = E[Z] ∼ =
ωi,k (Z(i, k))
i=1 k=1
σz =
var(Z) = E(Z2 ) − μ2z .
(13)
B. Generation of Correlated Random Variables For random loads and power generations, this section shows how the Cholesky factorization method can be used to generate multivariate-dependent random variables. Let M denote the
W = BM (15) T T T T CW = cov W, W = cov BM, M B = BCM B = I. (16) In most engineering applications, matrix CM is positive definite and can be decomposed through the Cholesky factorization (17), where L is a lower triangular matrix whose inverse matrix turns out to be the orthogonal matrix B as required in (15). That is, B = L−1 as shown in (18). Thus, the relevance is set in random vector M (17) CM = LLT T T T −1 CW = (BM) M B = (BM)(BM) = I ⇒ B = L . (18) For normal random variables, such as loads and conventional power generations, Cholesky decomposition can be used to transform independent normal random vector into dependent normal random vector, which will not change the correlations of normal random variables. However, when the correlation among wind speeds is considered, as wind speed is simulated to follow Weibull distribution, Cholesky decomposition may not be able to guarantee that the correlation among random wind speed variables will be unchanged after the transformation. Therefore, for random wind speeds, Nataf transformation [39] may be used after Cholesky decomposition to address the transformation from dependent normal random variables to dependent nonnormal random variables. C. Procedure for Overload Risk Assessment As shown in Fig. 2, the procedure for computing the transmission line overload risk indices is summarized as follows. Step 1: Initialize and generate the constructed points of wind speed as well as the correlated loads and power generations. It contains the following substeps. 1) Set initial values for nodal voltages and determine the number of random loads and conventional power generations. m represents the number of all random variables, including loads, conventional power generations, and wind speeds. 2) According to the 2m + 1 scheme, the PDF of a random variable yi can be replaced by matching the first four central moments of random variable, namely mean, variance, and coefficients of skewness and kurtosis. In the space of the standard normal distribution, the first four central moments of random variables are 0, 1, 0, and 1, respectively. 3) With the first four central moments, standard locations ξi,k and weights ωi,k for random variables are
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Fig. 3.
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Wind farm connected to the IEEE 30-bus system.
respectively. Nataf transformation is used to transform constructed-dependent points of wind speeds obtained in substep 4) from the space of the standard normal distribution to the space of the Weibull distribution. Step 2: Compute output of wind turbine generators [27]. Step 3: Input variable yi is selected. Step 4: The 2m + 1 PLF scheme is executed. Step 5: According to obtained PLF results, PDF of line power flows can be obtained and the possibility of each line overflow can be calculated. Step 6: The severity of each line overflow is determined. Step 7: The risk index of each line overload is obtained, and the line overload risk of the entire system can be calculated according to (6). IV. C ASE S TUDY
Fig. 2.
Flow chart of computing procedure for overload risk assessment.
calculated according to (9). Then, the concentrations (yi,k , ωi,k ) can be obtained by (8) and the point (μy1 , μy2 , . . . , yi,k , . . . , μym ) is constructed. 4) The Cholesky decomposition is used to transform independent points (μy1 , μy2 , . . . , yi,k , . . . , μym ) into dependent points in the space of standard normal distribution. Correlation coefficients for loads and conventional generations are set as 0.9 and correlation coefficients for wind resources are given in Section IV-A5. 5) For loads and conventional generations, means of nodal power injections are set as the benchmark data, and standard deviations are set as 5% of the corresponding means. The constructed-dependent points (μy1 , μy2 , . . . , yi,k , . . . , μym1) in the space of the standard normal distribution are transformed into the points (μy1 , μy2 , . . . , yi,k , . . . , μym ) in the space of the normal 1 distribution. 6) Shape and scale parameters of Weibull distribution for random wind speeds are set as 10.7 and 6.2892,
Considering uncertainties and correlations of wind, loads, and power generation, the proposed risk assessment of line overload is validated using the modified IEEE 30-bus system and IEEE 118-bus system. The wind farm is composed of 20 identical wind turbines, each of which has the rated capacity of 600 kW and other characteristic parameters are given in [27]. The base value is 100 MVA. Upper and lower voltage bounds are 1.1 and 0.95 p.u. for photovoltaic nodes, and 1.06 and 0.94 p.u. for all other nodes. All case studies are carried out using MATLAB in an Advanced Micro Devices 64 Dual Core 2.71 GHz PC. A. Modified IEEE 30-Bus System In the modified IEEE 30-bus system shown in Fig. 3, a wind farm is connected to bus 30. All lines are monitored for assessing the overload risk. Given the limited space available, Table I only lists real power capacities of nine selected lines, which will be discussed in details throughout the studies. 1) Impacts of Different Types and Parameters of Wind Speed Distribution on Line Overload: MCS and the 2m + 1 scheme of PEM are used to perform the PLF calculation for exploring the possibility of line overload. Wind speed uncertainty is simulated via Weibull PDF with shape and scale parameters. PDF of the Weibull distribution is given as v k k v k−1 (k > 0, v > 0, c > 1). (19) exp − ϕ (v) = c c c
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TABLE I B OUND OF L INE R EAL P OWER
TABLE II C OMPARISON OF PLF R ESULTS FOR D IFFERENT C ASES W ITH C ORRELATED L OADS
The following three cases are studied to explore the impacts of different types and parameters of wind speed distribution on line overload. Case 1: Weibull distribution with the shape parameter k of 2 and the scale parameter c of 6.7703 is used for simulating the wind speed uncertainty. Case 2: Weibull distribution with the shape parameter k of 10.7 and the scale parameter c of 6.2892 is used for simulating the wind speed uncertainty. Case 3: Normal PDF is used for simulating the wind speed uncertainty. The mean of wind speed is set to be same as that in case 1, and the standard deviation is taken as 5% of the mean. Table II shows means and standard deviations of real power flows of the nine monitored lines in all three cases. Figs. 4–6 show the PDFs of real power flows on lines 3 and 4 in cases 1–3, respectively. It can be seen from Table II that the means and the standard deviations of line flows obtained from PEM are far from the results of MCS in case 1. The PDF of PEM in case 1 also deviates significantly from that of MCS as shown in Fig. 4. This is because that using PEM, the
Fig. 4.
PDF of real power flow on lines 3 and 4 in case 1.
Fig. 5.
PDF of real power flow on lines 3 and 4 in case 2.
Fig. 6.
PDF of real power flow on lines 3 and 4 in case 3.
concentrations (yi,k , ωi,k ) of the wind speeds are far from those means and move beyond the domain of the Weibull distribution with the shape parameter k = 2 and the scale parameter c = 6.7703. Consequently, the results of line power flows are not well performed by the PLF computation. This can even happen when m is as small as 3 [40]. The ill-performed PLF
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TABLE III C OMPARISON OF L INE OVERLOAD R ISK I NDICES IN D IFFERENT C ASES
results impact the probability of line overload occurrence and derive exceptional risk values of line overload. Thus, Weibull distribution with parameters identified in case 1 may not be suitable for simulating uncertain wind data and being adopted in 2m + 1 scheme of PEM for the overload risk assessment. It is also observed from Table II that the means and the standard deviations of line power flows obtained from PEM and MCS in cases 2 and 3 are similar, which indicates that PEM provides good results in cases 2 and 3. PEM and MCS yield almost the same PDF as shown in Figs. 5 and 6. The reason is that the shape of Weibull PDF with k = 10.7 and c = 6.2892 used in case 2 is very close to that of the PDF of normal distribution used in case 3. This comparison shows that the 2m + 1 scheme of PEM is suitable for analyzing the possibility of line overload with probabilistic wind data generated by the distributions of cases 2 and 3. 2) Impacts of Wind Speed Distribution Types on Line Overload Risk: The risk indices for cases 2 and 3 are shown in Table III. It is observed from Table III that the risk indices of line overload obtained from PEM and MCS are similar, which indicates that PEM provides good results. Table III also shows that the risk indices obtained in cases 2 and 3 are slightly different. For instance, using PEM for the PLF computation, the risk index of lines 6 and 28 overload is 0.09030 in case 2, whereas is 0.08997 in case 3, which is caused by the use of different distributions in cases 2 and 3 (i.e., Weibull distribution in case 2 and normal distribution in case 3). The overload risk indices of the entire system in cases 2 and 3 are different, i.e., 2.4469 and 1.9186 with the use of PEM. The difference is mainly caused by the fact that risk indices depend on the overload possibility obtained by the PLF results. The line overload probability is calculated by the PDF of line powers according to (4). In turn, different wind speed distributions, i.e., the Weibull distribution in case 2 and the normal distribution in case 3, exert different influences on PDFs of line power flows, which would lead to different overload risk index values of individual lines and ultimately result in the difference in overload risk index for the entire system. As the Weibull distribution is commonly used in the estimation of long-term wind speed, the Weibull distribution with the shape parameter of 10.7 and the scale parameter of 6.2892 used in case 2 is applied in the following case study Sections 3) and 4).
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TABLE IV C OMPARISON OF PLF R ESULTS W ITH I NDEPENDENT L OADS AND C ORRELATED L OADS
Fig. 7.
PDF of real power flow on lines 23 and 24 with independent loads.
3) Impacts of Load and Generation Correlation on Line Overload: Table IV shows means and standard deviations of line real power flows with and without considering load and generation correlation. It can be clearly seen from Table IV that although the means are very close in the two cases, the standard deviations are different from each other. Thus, loads and power generation correlations should be considered when assessing line overload risks. PDFs of the real power flow on lines 23 and 24 with and without considering load and generation correlation are shown in Figs. 7 and 8, respectively. For the PEM, the mean and the standard deviation of line power flow are 0.0289 and 0.0038 without the consideration of load and generation correlation, while are 0.0289 and 0.0036 with the consideration of load and generation correlation. Consequently, according to (2), the probabilities of the real power flow on
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Fig. 9. Fig. 8.
PDF of real power flow on lines 23 and 24 with correlated loads.
TABLE V C OMPARISON OF R ISK I NDICES W ITH I NDEPENDENT L OADS AND C ORRELATED L OADS FOR THE MODIFIED IEEE 30- BUS SYSTEM
lines 23 and 24 over its capacity limit are calculated as 0.0145 and 0.0154 for uncorrelated and correlated cases, respectively. The difference in the line overload probability would lead to different risk indices of line overload in uncorrelated and correlated cases. It shows that the effect of load and generation correlation on line overload possibility should be considered. 4) Impacts of Load and Generation Correlation on Line Overload Risk Indices: Using the probabilities of line overload shown in Table IV and Figs. 7 and 8, overload severity value can be derived using the severity function defined in Fig. 1, and the overload risk index can be quantified. Table V shows the overload risk assessment results with and without considering load and generation correlation. From Table V, it can be observed that the risk indices obtained by PEM and MCS are almost the same. Thus, PEM is an appropriate tool for estimating the line overload risk for wind-integrated power systems. Moreover, from Table V, the risk indices for uncorrelated and correlated cases are different. For the entire system, it can be seen that the risk index with the consideration of load and generation correlation is much bigger than otherwise.
Modified IEEE 30-bus system with four wind farms.
TABLE VI C OMPARISON OF R ISK I NDICES W ITH W IND S PEED C ORRELATION AS W ELL AS L OAD AND G ENERATION C ORRELATION
That is, the line overload is more obvious when considering the correlation. Thus, it can be concluded that the risk indices are seriously affected by the load and generation correlation. 5) Impacts of Wind Speed Correlation as Well as Load and Generation Correlation on Line Overload Risk Indices: Considering the wind speed correlation as well as load and generation correlation, the proposed method is tested on a modified IEEE 30-bus system as shown in Fig. 9. The four wind farms are connected to buses 30, 26, 19, and 3 of the IEEE 30-bus system. Each wind farm is composed of 20 identical wind turbines, and their parameters are given in [27]. The correlation matrix among wind farms is given in ⎡ ⎤ 1 0.88 0.87 0.91 ⎢ 0.88 1 0.85 0.87 ⎥ ⎥ R=⎢ (20) ⎣ 0.87 0.85 1 0.85 ⎦. 0.91 0.87 0.85 1 The risk indices obtained from PEM and MCS are shown in Table VI. It is observed from Table VI that the risk indices of line overload obtained from PEM and MCS are similar, which indicates that PEM provides good results. Compared with Table V, Table VI also shows that the risk indices for most of lines with wind source correlation as well as load and generation correlation are larger than those considering only load and generation correlation or neither. For instance, using PEM for the risk index computation, the risk index of lines
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Fig. 10.
Modified IEEE 118-bus system.
TABLE VII C OMPARISON OF R ISK I NDICES W ITH I NDEPENDENT L OADS AND C ORRELATED L OADS FOR THE MODIFIED IEEE 118- BUS SYSTEM
9
relative error of the system risk index results from the PEM with respect to the MCS results is 1.21% without considering load and generation correlation, while is 0.32% with the consideration of load and generation correlation. Thus, PEM is an appropriate tool for estimating the line overload risk for wind-integrated power systems. Moreover, from Table VII, risk indices for uncorrelated and correlated cases are different. For the entire system using the PEM, the risk index with the consideration of load and generation correlation is 92.9354, while it is 70.8997 without the consideration of load and generation correlation. That is, the line overload is more obvious when considering the correlation. Thus, it can be concluded that risk indices are seriously affected by the load and generation correlation. V. C ONCLUSION
6 and 28 overload is 0.0970 in Table VI, whereas the risk indices are 0.0903 (which only considers load and generation correlation) and 0.0705 (which ignores both correlations) in Table V. It can be observed that the risk index for the entire system in Table VI is far from that in Table V. The risk index for the entire system with wind resource correlation as well as load and generation correlation is about three times and seven times larger than those considering only load and generation correlation or neither. It shows that wind resource correlation has profound influence on line overload risk. B. Modified IEEE 118-Bus System The proposed risk assessment approach of line overload is further validated via the modified IEEE 118-bus system [41], which connects a wind farm to bus 118 as shown in Fig. 10. The impacts of load and generation correlation on line overload risk are studied for this system. Table VII shows the overload risk assessment results of certain lines with and without considering load and generation correlation. From Table VII, it can be observed that risk indices obtained by PEM and MCS are almost the same. The
This paper presents a method for assessing line overload risk for wind-integrated power systems with the consideration of wind and load-power generation correlations. Using PEM for PLF calculations, the possibility of line overload can be computed. Combining the possibility with the severity of line overload, the quantitative risk indices can be obtained. The effectiveness of the proposed method is verified via the modified IEEE 30-bus system and IEEE 118-bus system. Numerical results show that the proposed method is appropriate for assessing line overload risks of wind-integrated power systems. It is observed that the types and the parameters of uncertain wind speed distribution would affect the risk indices of line overload. Thus, appropriate shape and scale parameters of Weibull distribution for wind speeds should be carefully tuned. In addition, wind resource correlation and load-power generation correlation would also impact the risk indices of line overload, and needs to be considered when assessing the line overload risk. Risk indices with the consideration of random variable correlation would more accurately reflect the system security and provide more effective information for the decision-maker. R EFERENCES [1] C. Z. Karatekin and C. Ucak, “Sensitivity analysis based on transmission line susceptances for congestion management,” Elect. Power Syst. Res., vol. 78, no. 9, pp. 1485–1493, 2008. [2] B. Leonardi and V. Ajjarapu, “Investigation of various generator reactive power reserve (GRPR) definitions for online voltage stability/security assessment,” in Proc. IEEE Power Energy Soc. Gen. Meeting, Pittsburgh, PA, USA, 2008, pp. 1–7. [3] M. Esmaili, H. A. Shayanfar, and N. Amjady, “Multi-objective congestion management incorporating voltage and transient stabilities,” Energy, vol. 34, no. 9, pp. 1401–1412, 2009. [4] D. Niebur and A. J. Germond, “Power system static security assessment using the Kohonen neural network classifier,” IEEE Trans. Power Syst., vol. 7, no. 2, pp. 865–872, May 1992. [5] A. V. Machias, J. L. Souflis, and N. Evangelos, “A fuzzy transient stability index in power system security evaluation,” IEEE Trans. Autom. Control, vol. 34, no. 6, pp. 662–666, Jun. 1989. [6] D. Jayaweera and S. Islam, “Steady-state security in distribution networks with large wind farms,” J. Mod. Power Syst. Clean Energy, vol. 2, no. 2, pp. 134–142, Apr. 2014. [7] P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994. [8] J. McCalley, V. Vittal, and N. Abi-Samra, “An overview of risk based security assessment,” in Proc. IEEE Power Eng. Soc. Sum. Meeting, Edmonton, AB, Canada, 1999, pp. 173–178.
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Xue Li (M’13) received the B.S. and M.S. degrees from Zhengzhou University, Zhengzhou, China, in 2002 and 2006, respectively, and the Ph.D. degree from Shanghai University, Shanghai, China, in 2009, all in electrical engineering. She is currently an Associate Professor with the Department of Automation, Shanghai University. Her current research interests include wind-integration into grid and power system economics and security.
Xiong Zhang received the B.S. degree in electrical engineering from Xuchang University, Xuchang, China, in 2007. He is currently pursuing the M.S. degree in automation from Shanghai University, Shanghai, China. His current research interest include risk assessment of power system and probabilistic methods applied to power systems.
Lei Wu (SM’13) received the B.S. and M.S. degrees in electrical engineering and systems engineering from Xi’an Jiaotong University, Xi’an, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from the Illinois Institute of Technology, Chicago, IL, USA, in 2008. From 2008 to 2010, he was a Senior Research Associate at the Electric Power and Power Electronics Center, Illinois Institute of Technology. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY, USA. His current research interests include power systems operation and economics.
Pan Lu is currently pursuing the M.S. degree in automation from Shanghai University, Shanghai, China. His current research interest include risk control of power system.
Shaohua Zhang received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China; the M.Eng. degree from the Shanghai University of Technology, Shanghai, China; and the Ph.D. degree from Shanghai University, Shanghai, in 1988, 1991, and 2001, respectively, all in electrical engineering. He is currently a Professor with Shanghai University. His current research interests include power system restructuring, pricing, and reliability.
1
Transmission Line Overload Risk Assessment for Power Systems With Wind and Load-Power Generation Correlation Xue Li, Member, IEEE, Xiong Zhang, Lei Wu, Senior Member, IEEE, Pan Lu, and Shaohua Zhang
Abstract—In the risk-based security assessment, probability and severity of events are the two main factors for measuring the security level of power systems. This paper presents a method for assessing line overload risk of wind-integrated power systems with the consideration of wind and load-power generation correlation. The established risk assessment model fully considers the probability and the consequence of wind uncertainties and line flow fluctuations. The point estimate method is employed to deal with the probability of line overload and the severity function is applied to quantify line flow fluctuations. Moreover, with the Cholesky decomposition, the correlation between loads and power generations are simulated by the spatial transformation of probability distributions of random variables. In addition, Nataf transformation is used to address wind resource correlation. Finally, the line overload risk index is obtained, which can be used as an indicator for quantifying power system security. Numerical results on the modified IEEE 30-bus system and the modified IEEE 118-bus system show that the types and the parameters of the wind speed distribution would affect the risk indices of line overload, and the risk indices obtained with the consideration of wind resource correlation and load correlation would reflect the system security more accurately. Index Terms—Load-power generation correlation, overload risk assessment, point estimate method (PEM), probabilistic load flow (PLF), severity function, wind resource correlation.
N OMENCLATURE Variables B c, k CM D(·), E(·) f (Zl ) Fi (·) M MT
Orthogonal matrix. Scale parameter and Shape parameter of the Weibull probability density function (PDF). Variance-covariance matrix of vector M. Variance and mean. PDF of Zl . Nonlinear function. Random vector, including load and generation powers. Transpose of vector M.
Manuscript received March 24, 2014; revised July 23, 2014, November 1, 2014, and December 15, 2014; accepted December 26, 2014. This work was supported in part by the National Science Foundation of China under Grant 51007052 and Grant 71201097, and in part by the Science and Technology Commission of Shanghai Municipality under Grant 14ZR1415300. Paper no. TSG-00263-2014. X. Li, X. Zhang, P. Lu, and S. Zhang are with the Key Laboratory of Power Station Automation Technology, Department of Automation, Shanghai University, Shanghai 200072, China. L. Wu is with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY 13699 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSG.2014.2387281
Pt (Zl ) Risk (Zl ) Se (Zl ) v yi Zl μ(·) , σ(·) ξi,k ωi,k λi,j
Probability of overload at line l. Risk index of overload at line l. Severity of overload at line l. Wind speed (m/s). Random variables including wind speeds, loads, and conventional power generations. Line flow of branch l. Mean and standard deviation. Standard location. Weight. jth standard central moment of random variable yi with PDF fyi .
Constants I Identity matrix. t Confidence level. Zl min , Zl max Lower/upper capacity limits of branch l. I. I NTRODUCTION ECURITY continues to be the most important aspect in power system operation and planning, especially with high penetration of renewable sources. The smart grid uses intelligent transmission and distribution networks for delivering electricity, which would enhance security, reliability, and efficiency of power systems. Transmission line capacity limit violations [1], voltage violations [2], and voltage stability and/or transient stability [3] would impact the security of power systems and reduce the efficiency of system operation. In power system planning and operation, security assessment [4]–[7] would assist system operators in maintaining the system security level within an acceptable range. Three strategies, i.e., deterministic evaluation [8], probabilistic evaluation [9], and risk assessment [10]–[13], are used to explore power system security. In traditional deterministic assessment, a power system is operated under significant security margins with respect to a set of most credible contingencies [14]. This would result in highly conservative decisions with high operation costs. The probabilistic evaluation considers the likelihood of events and uses probabilistic indices such as loss of load probability (LOLP) and expected unserved energy (EUE). LOLP reflects the likelihood but not the severity, while EUE captures both likelihood and severity. However, EUE only measures load interruptions, but not the costs associated with equipment damages or lost
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opportunities from equipment unavailability [8]. Capturing both the likelihood and the severity of events, risk-based security assessment (RBSA) has been used to determine the security level. In order to allow a power system to be operated closer to or even beyond its limits, RBSA is applied for evaluating power system security at planning and operating stages. Online RBSA [10] condensed the contingency likelihood and the severity into probabilistic risk indices, which can be used in control room decision making for understanding potential network problems, such as overloads, cascading overloads, low voltages, and voltage instabilities. A risk-based approach was discussed in [11] to monitor and manage the probability and the consequence of potential cascading outages for reliable operations. The risk-based approach to the security assessment [12] provided more information to guide secure operating decisions than the traditional N-1 security criterion. Ni et al. [13] developed a software implementation for online RBSA. It computed indices based on probabilistic risks, which describes the system security level as a function of existing and near-future network conditions. System risk assessment includes a wide range of analysis on generation, transmission, and distribution assets. In terms of the transmission security, the transmission line overload risk assessment is a crucial task, which includes assessing the cumulative risk associated with overload security for mid-term power system planning [15], optimally controlling system security levels associated with overloads [16], deriving a more robust overload risk index for describing the total vulnerability level of power systems [17], and exploring the overload risk with the consideration of probability and consequence of line flow fluctuations [18]–[20]. Wind energy has been widely deployed throughout the world and is playing more and more important roles in emerging electric energy systems. Volatility and uncertainty of wind energy poses new challenges for the grid security and potentially increases outage risks [21], [22]. A smart grid should accommodate new technologies of renewable generation and provide the high level of security. There has, however, been relatively little work done on assessing the risk-based security of wind-integrated power systems. Existing studies fall into the general category of operating risk analysis, which involves evaluating the contribution of wind power on the load carrying capability in power systems [23], assessing the impact of high wind penetration on the generation adequacy [24], and committing additional generation sources for maintaining the system reliability [25], [26]. Specifically, few researches focus on evaluating the impacts of random wind energy or load correlation on risk indices, and quantifying the likelihood and the severity of line overloads in wind-integrated power systems. This paper aims at evaluating the transmission line overload risk in wind-integrated power systems while considering wind and load-power generation correlation. Cholesky decomposition is used to simulate the correlation between loads and power generations by the spatial transformation of probability distributions of random variables. By considering random wind correlation and the correlation between conventional generations and loads, the risk assessment model of line overload
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for wind-integrated power systems is constructed to formulate the likelihood and the severity of line overloads. Finally, 2m + 1 point estimate scheme is applied for the overload risk evaluation. The severity value of line overload only describes the outcome of line overloads. The risk value adopted in this paper, product of the probability and the severity of transmission line overloads, reflects the security level of system. Interesting simulation results and informative conclusions have been derived. It is observed that types and parameters of uncertain wind speed distribution would affect risk values of line overload. In addition, wind resource correlation and load-power generation correlation would also impact the risk values of line overload. Consequently, the proposed model can be used to evaluate the impacts of wind resource correlation and load-power generation correlation on overload risk indices with various wind speed distributions. Once the risk values of individual lines or the entire system are obtained, system operator can evaluate the system operation condition and decide necessary actions to reduce the risks. The rest of this paper is organized as follows. Section II presents the overload risk assessment model for windintegrated power systems. With the consideration of wind and load-conventional generation correlation, the point estimate method (PEM) for the overload risk assessment is described in Section III. The simulation results on the modified IEEE 30-bus system and the modified IEEE 118-bus system are described in Section IV, and the conclusions are drawn in Section V. II. M ODEL D ESCRIPTION A. Probabilistic Model for Line Overload The line overload possibility can be measured by the probability distribution of line flows. The probability distribution can be derived from the probabilistic load flow (PLF) calculation [27]. The output line-flow vectors Z can be expressed as a function of input random wind power, load, and power generation, as shown in (1), where random vector y is composed of uncertain wind speed, loads, and power generations in a power system Z = F(y).
(1)
The likelihood index in (2) denotes the cumulative distribution function (CDF) of random variables whose samples do not satisfy the safety threshold with the confidence level t. With the assumption that line flows follow the normal distribution, t is the confidence level when random variable samples comply with the “3σ principle” as shown in (3) / [Zl min , Zl max ]) Lik(Zl ) = Pt (Zl ∈ t = P |Zl − E(Zl )| < 3 D(Zl ) .
(2) (3)
The CDF of random variables can be written as (4), where the PDF f (Zl ) can be obtained from the results of the PLF calculation / [Zl min , Zl max ]) = f (Zl )dZl . (4) Pt (Zl ∈ Zl ∈[Z / l min ,Zl max ]
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Fig. 1.
Severity function of line overload.
B. Severity Model of Line Overload Severity function is used to uniformly quantify the severity of line overload. Severity function for line overload is related to the real power flow of a transmission line and is specified for each transmission line. Mean values of the real power flow as a percentage of the power rating would determine the overload severity of a line [10]. The continuous severity function for line overload shown in Fig. 1 is used in this paper. For each line, when the real power flow is no larger than 90% of the rated value, the risk severity value is zero. When the line real power flow is larger than 90% of the rated value, the risk severity value is a linear function of the real power flow, and is equal to one when the line real power flow is equal to the rated value. Thus, for each line, the closer the risk severity value to one, the more severe the line would be overloaded. Other severity functions, such as discrete severity functions [10], may also be used, and the proposed line overload risk assessment method is still valid. C. Risk Index of Line Overload The overload risk index Risk (Zl ) of line l (5) is defined as the product of the probability Pt (Zl ) and the severity Se (Zl ) of the transmission line overload. Therefore, the line overload risk index of the entire system can be calculated as (6) +∞ Pt (Zl )Se (Zl )dZl (5) Risk (Zl ) = −∞ Rall = Risk (Zl ). (6) III. PEM FOR OVERLOAD R ISK A SSESSMENT W ITH W IND U NCERTAINTY AND L OAD -P OWER G ENERATION C ORRELATION A. Point Estimate Theory The PLF calculation for wind-integrated power system derives the probabilistic distribution of line flows, which could describe probabilities of line overload with the consideration of uncertain wind, loads, and power generations. Many methods have been used for solving the PLF, including Monte Carlo simulation (MCS) [28], first order second moment method (FOSMM) [29], cumulant method (CM) [30], and PEM [31]–[34]. MCS needs a significant computational effort and is usually used as the benchmark for evaluating the effectiveness of various algorithms. The main idea behind other methods, including FOSMM, CM, and PEM, is to use
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approximate formulas for calculating statistical moments of random output quantities. PEM requires less computational burden. However, most PLF studies have assumed that random variables are independent, which largely ignore the correlation between loads and power generations or spatial correlation among wind speeds. The correlation would have significant impacts on the results of PLF [36], [37] and, in turn, the risk assessment results. Therefore, it is necessary to consider the impacts of correlated parameters when quantifying risk indices of wind-integrated power systems. Saunders [35] extends the PEM approach to include both spatial and temporal correlations among wind generations and loads. Saunders [35] applies the singular value decomposition (SVD) approach to create spatially and temporally correlated wind and load random variables, and transform the correlated random variables into uncorrelated random variables for finding the concentrations of the PEM analysis. The locations derived from the concentrations, which relate to the uncorrelated truncated random variable set, are remapped back to the original variable space. The SVD approach in [35] has certain advantages. For example, using the SVD, the basic idea of principal component extraction is applied in order to truncate the input variable set. According to the singular value matrix, random variables with the greatest variance (i.e., the most prominent impact on probabilistic outputs) can be selected for a truncated random variable set. This set of truncated random input variables can be utilized in the PEM with reduced computational efforts. Authors have made the effort for the variable set truncation (i.e., the subset selection problem) of variant correlation [42]. In [42], a novel forward subset selection algorithm was proposed which can not only choose the important input variables but also integrate the key properties of the group ability into the modeling procedure. Moreover, the relation of solution and the correlation between the two variables are also proved. Except the SVD approach, PCA principal component analysis and ICA independent component analysis can also be used to truncate variable set. The main objective of these methods is to extract the principal components. However, when all random variables are equally important, the variable selection process would not be needed. Thus, all input data are employed and both Cholesky decomposition and SVD method can be applied. The two algorithms have been compared in [43]. It has been clearly clarified that Cholesky decomposition is the fastest method, and SVD is more computational expensive but is numerically stable. PEM focuses on the statistical information provided by the first few central moments of input random variables with K points for each variable. K is a parameter depending on the Hong’s method [30], named concentrations. This paper carries on a 2 × m + 1 type scheme of PEM for evaluating the line overload risk, which gives a good tradeoff between the solution accuracy and the computational efforts [33]. m denotes the number of random variables. The scheme requires 2m+1 evaluations of function F and uses the first three concentrations for each input random variable, i.e., K = 3. Generally speaking, in the ac power flow calculation, the output line-flow vectors Z can be expressed as (7). The random variable yi (i = 1, . . . , m) denotes wind speeds,
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loads, and power generations in a power system. Multivariate dependent random variables are generated by Cholesky decomposition [36], [38]. Each random variable yi has known mean μyi and variance σyi , which can be obtained from historical data, statistical analysis, or engineering judgment
N-dimension random vector including loads and power generations in power systems. Its variance-covariance matrix CM is given by (14) CM = cov M, MT .
Zi = Fi (y1 , y2 , . . . , ym ).
Matrix CM is symmetric by definition. As a result, there always exists an orthogonal matrix B, by which the correlated variable vector M can be transformed into a new uncorrelated set W as shown in (15). The variance-covariance matrix CW is computed via (16)
(7)
Suppose that random variable yi follows the PDF fyi and the goal is to find the PDF of Z. According to Hong’s point [32], [33] estimate methods, the kth concentration (yi,k , ωi,k ) of a random variable yi can be defined as a pair composed of a location yi,k and a weight ωi,k . Location yi,k is the kth value of variable yi at which the function F is evaluated. Weight ωi,k is the weighting factor which accounts for the relative importance of this evaluation in output random variables. The location yi,k to be determined at the point (μy1 , μy2 , . . . , yi,k , . . . , μym ) is yi,k = μyi + ξi,k σyi , i = 1, . . . , m, k = 1, . . . , K.
(8)
For random variable yi , the PDF fyi can be approximated by matching the first few moments of yi . Thus, the standard location ξi,k and the weight ωi,k can be obtained by solving (9), where λi,j is given in (10) K
j ωi,k ξi,k = λi,j , i = 1, . . . , m, j = 1, . . . , 2K − 1
k=1
λi,j = E
j j σyi . yi − μyi
(9) (10)
sum of all weights is equal to one, i.e., The m K i=1 k=1 ωi,k = 1. In addition, it is specified that the sums of the weights for each i satisfy K
ωi,k = 1/m, i = 1, . . . , m.
(11)
k=1
The standard locations ξi,k and the weights ωi,k can be obtained by the solution of the system equation [33]. Once all the concentrations (yi,k , ωi,k ) (i = 1, . . . , m, k = 1, . . . , 3) are obtained, the function F can be evaluated at the point (μy1 , μy2 , . . . , yi,k , . . . , μym ) to yield Z(i, k). By using ωi,k and Z(i, k), the jth raw moment of the output random vector Z can be obtained from (12). The mean and the standard deviation of Z can be calculated via (13)
j E(Zj ) ∼ = ω0 F μy1 , μy2 , . . . , yi,k , . . . , μym +
m K
j ωi,k F μy1 , μy2 , . . . , yi,k , . . . , μym (12)
i=1 k=1 m 3
μZ = E[Z] ∼ =
ωi,k (Z(i, k))
i=1 k=1
σz =
var(Z) = E(Z2 ) − μ2z .
(13)
B. Generation of Correlated Random Variables For random loads and power generations, this section shows how the Cholesky factorization method can be used to generate multivariate-dependent random variables. Let M denote the
W = BM (15) T T T T CW = cov W, W = cov BM, M B = BCM B = I. (16) In most engineering applications, matrix CM is positive definite and can be decomposed through the Cholesky factorization (17), where L is a lower triangular matrix whose inverse matrix turns out to be the orthogonal matrix B as required in (15). That is, B = L−1 as shown in (18). Thus, the relevance is set in random vector M (17) CM = LLT T T T −1 CW = (BM) M B = (BM)(BM) = I ⇒ B = L . (18) For normal random variables, such as loads and conventional power generations, Cholesky decomposition can be used to transform independent normal random vector into dependent normal random vector, which will not change the correlations of normal random variables. However, when the correlation among wind speeds is considered, as wind speed is simulated to follow Weibull distribution, Cholesky decomposition may not be able to guarantee that the correlation among random wind speed variables will be unchanged after the transformation. Therefore, for random wind speeds, Nataf transformation [39] may be used after Cholesky decomposition to address the transformation from dependent normal random variables to dependent nonnormal random variables. C. Procedure for Overload Risk Assessment As shown in Fig. 2, the procedure for computing the transmission line overload risk indices is summarized as follows. Step 1: Initialize and generate the constructed points of wind speed as well as the correlated loads and power generations. It contains the following substeps. 1) Set initial values for nodal voltages and determine the number of random loads and conventional power generations. m represents the number of all random variables, including loads, conventional power generations, and wind speeds. 2) According to the 2m + 1 scheme, the PDF of a random variable yi can be replaced by matching the first four central moments of random variable, namely mean, variance, and coefficients of skewness and kurtosis. In the space of the standard normal distribution, the first four central moments of random variables are 0, 1, 0, and 1, respectively. 3) With the first four central moments, standard locations ξi,k and weights ωi,k for random variables are
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Fig. 3.
5
Wind farm connected to the IEEE 30-bus system.
respectively. Nataf transformation is used to transform constructed-dependent points of wind speeds obtained in substep 4) from the space of the standard normal distribution to the space of the Weibull distribution. Step 2: Compute output of wind turbine generators [27]. Step 3: Input variable yi is selected. Step 4: The 2m + 1 PLF scheme is executed. Step 5: According to obtained PLF results, PDF of line power flows can be obtained and the possibility of each line overflow can be calculated. Step 6: The severity of each line overflow is determined. Step 7: The risk index of each line overload is obtained, and the line overload risk of the entire system can be calculated according to (6). IV. C ASE S TUDY
Fig. 2.
Flow chart of computing procedure for overload risk assessment.
calculated according to (9). Then, the concentrations (yi,k , ωi,k ) can be obtained by (8) and the point (μy1 , μy2 , . . . , yi,k , . . . , μym ) is constructed. 4) The Cholesky decomposition is used to transform independent points (μy1 , μy2 , . . . , yi,k , . . . , μym ) into dependent points in the space of standard normal distribution. Correlation coefficients for loads and conventional generations are set as 0.9 and correlation coefficients for wind resources are given in Section IV-A5. 5) For loads and conventional generations, means of nodal power injections are set as the benchmark data, and standard deviations are set as 5% of the corresponding means. The constructed-dependent points (μy1 , μy2 , . . . , yi,k , . . . , μym1) in the space of the standard normal distribution are transformed into the points (μy1 , μy2 , . . . , yi,k , . . . , μym ) in the space of the normal 1 distribution. 6) Shape and scale parameters of Weibull distribution for random wind speeds are set as 10.7 and 6.2892,
Considering uncertainties and correlations of wind, loads, and power generation, the proposed risk assessment of line overload is validated using the modified IEEE 30-bus system and IEEE 118-bus system. The wind farm is composed of 20 identical wind turbines, each of which has the rated capacity of 600 kW and other characteristic parameters are given in [27]. The base value is 100 MVA. Upper and lower voltage bounds are 1.1 and 0.95 p.u. for photovoltaic nodes, and 1.06 and 0.94 p.u. for all other nodes. All case studies are carried out using MATLAB in an Advanced Micro Devices 64 Dual Core 2.71 GHz PC. A. Modified IEEE 30-Bus System In the modified IEEE 30-bus system shown in Fig. 3, a wind farm is connected to bus 30. All lines are monitored for assessing the overload risk. Given the limited space available, Table I only lists real power capacities of nine selected lines, which will be discussed in details throughout the studies. 1) Impacts of Different Types and Parameters of Wind Speed Distribution on Line Overload: MCS and the 2m + 1 scheme of PEM are used to perform the PLF calculation for exploring the possibility of line overload. Wind speed uncertainty is simulated via Weibull PDF with shape and scale parameters. PDF of the Weibull distribution is given as v k k v k−1 (k > 0, v > 0, c > 1). (19) exp − ϕ (v) = c c c
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TABLE I B OUND OF L INE R EAL P OWER
TABLE II C OMPARISON OF PLF R ESULTS FOR D IFFERENT C ASES W ITH C ORRELATED L OADS
The following three cases are studied to explore the impacts of different types and parameters of wind speed distribution on line overload. Case 1: Weibull distribution with the shape parameter k of 2 and the scale parameter c of 6.7703 is used for simulating the wind speed uncertainty. Case 2: Weibull distribution with the shape parameter k of 10.7 and the scale parameter c of 6.2892 is used for simulating the wind speed uncertainty. Case 3: Normal PDF is used for simulating the wind speed uncertainty. The mean of wind speed is set to be same as that in case 1, and the standard deviation is taken as 5% of the mean. Table II shows means and standard deviations of real power flows of the nine monitored lines in all three cases. Figs. 4–6 show the PDFs of real power flows on lines 3 and 4 in cases 1–3, respectively. It can be seen from Table II that the means and the standard deviations of line flows obtained from PEM are far from the results of MCS in case 1. The PDF of PEM in case 1 also deviates significantly from that of MCS as shown in Fig. 4. This is because that using PEM, the
Fig. 4.
PDF of real power flow on lines 3 and 4 in case 1.
Fig. 5.
PDF of real power flow on lines 3 and 4 in case 2.
Fig. 6.
PDF of real power flow on lines 3 and 4 in case 3.
concentrations (yi,k , ωi,k ) of the wind speeds are far from those means and move beyond the domain of the Weibull distribution with the shape parameter k = 2 and the scale parameter c = 6.7703. Consequently, the results of line power flows are not well performed by the PLF computation. This can even happen when m is as small as 3 [40]. The ill-performed PLF
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TABLE III C OMPARISON OF L INE OVERLOAD R ISK I NDICES IN D IFFERENT C ASES
results impact the probability of line overload occurrence and derive exceptional risk values of line overload. Thus, Weibull distribution with parameters identified in case 1 may not be suitable for simulating uncertain wind data and being adopted in 2m + 1 scheme of PEM for the overload risk assessment. It is also observed from Table II that the means and the standard deviations of line power flows obtained from PEM and MCS in cases 2 and 3 are similar, which indicates that PEM provides good results in cases 2 and 3. PEM and MCS yield almost the same PDF as shown in Figs. 5 and 6. The reason is that the shape of Weibull PDF with k = 10.7 and c = 6.2892 used in case 2 is very close to that of the PDF of normal distribution used in case 3. This comparison shows that the 2m + 1 scheme of PEM is suitable for analyzing the possibility of line overload with probabilistic wind data generated by the distributions of cases 2 and 3. 2) Impacts of Wind Speed Distribution Types on Line Overload Risk: The risk indices for cases 2 and 3 are shown in Table III. It is observed from Table III that the risk indices of line overload obtained from PEM and MCS are similar, which indicates that PEM provides good results. Table III also shows that the risk indices obtained in cases 2 and 3 are slightly different. For instance, using PEM for the PLF computation, the risk index of lines 6 and 28 overload is 0.09030 in case 2, whereas is 0.08997 in case 3, which is caused by the use of different distributions in cases 2 and 3 (i.e., Weibull distribution in case 2 and normal distribution in case 3). The overload risk indices of the entire system in cases 2 and 3 are different, i.e., 2.4469 and 1.9186 with the use of PEM. The difference is mainly caused by the fact that risk indices depend on the overload possibility obtained by the PLF results. The line overload probability is calculated by the PDF of line powers according to (4). In turn, different wind speed distributions, i.e., the Weibull distribution in case 2 and the normal distribution in case 3, exert different influences on PDFs of line power flows, which would lead to different overload risk index values of individual lines and ultimately result in the difference in overload risk index for the entire system. As the Weibull distribution is commonly used in the estimation of long-term wind speed, the Weibull distribution with the shape parameter of 10.7 and the scale parameter of 6.2892 used in case 2 is applied in the following case study Sections 3) and 4).
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TABLE IV C OMPARISON OF PLF R ESULTS W ITH I NDEPENDENT L OADS AND C ORRELATED L OADS
Fig. 7.
PDF of real power flow on lines 23 and 24 with independent loads.
3) Impacts of Load and Generation Correlation on Line Overload: Table IV shows means and standard deviations of line real power flows with and without considering load and generation correlation. It can be clearly seen from Table IV that although the means are very close in the two cases, the standard deviations are different from each other. Thus, loads and power generation correlations should be considered when assessing line overload risks. PDFs of the real power flow on lines 23 and 24 with and without considering load and generation correlation are shown in Figs. 7 and 8, respectively. For the PEM, the mean and the standard deviation of line power flow are 0.0289 and 0.0038 without the consideration of load and generation correlation, while are 0.0289 and 0.0036 with the consideration of load and generation correlation. Consequently, according to (2), the probabilities of the real power flow on
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Fig. 9. Fig. 8.
PDF of real power flow on lines 23 and 24 with correlated loads.
TABLE V C OMPARISON OF R ISK I NDICES W ITH I NDEPENDENT L OADS AND C ORRELATED L OADS FOR THE MODIFIED IEEE 30- BUS SYSTEM
lines 23 and 24 over its capacity limit are calculated as 0.0145 and 0.0154 for uncorrelated and correlated cases, respectively. The difference in the line overload probability would lead to different risk indices of line overload in uncorrelated and correlated cases. It shows that the effect of load and generation correlation on line overload possibility should be considered. 4) Impacts of Load and Generation Correlation on Line Overload Risk Indices: Using the probabilities of line overload shown in Table IV and Figs. 7 and 8, overload severity value can be derived using the severity function defined in Fig. 1, and the overload risk index can be quantified. Table V shows the overload risk assessment results with and without considering load and generation correlation. From Table V, it can be observed that the risk indices obtained by PEM and MCS are almost the same. Thus, PEM is an appropriate tool for estimating the line overload risk for wind-integrated power systems. Moreover, from Table V, the risk indices for uncorrelated and correlated cases are different. For the entire system, it can be seen that the risk index with the consideration of load and generation correlation is much bigger than otherwise.
Modified IEEE 30-bus system with four wind farms.
TABLE VI C OMPARISON OF R ISK I NDICES W ITH W IND S PEED C ORRELATION AS W ELL AS L OAD AND G ENERATION C ORRELATION
That is, the line overload is more obvious when considering the correlation. Thus, it can be concluded that the risk indices are seriously affected by the load and generation correlation. 5) Impacts of Wind Speed Correlation as Well as Load and Generation Correlation on Line Overload Risk Indices: Considering the wind speed correlation as well as load and generation correlation, the proposed method is tested on a modified IEEE 30-bus system as shown in Fig. 9. The four wind farms are connected to buses 30, 26, 19, and 3 of the IEEE 30-bus system. Each wind farm is composed of 20 identical wind turbines, and their parameters are given in [27]. The correlation matrix among wind farms is given in ⎡ ⎤ 1 0.88 0.87 0.91 ⎢ 0.88 1 0.85 0.87 ⎥ ⎥ R=⎢ (20) ⎣ 0.87 0.85 1 0.85 ⎦. 0.91 0.87 0.85 1 The risk indices obtained from PEM and MCS are shown in Table VI. It is observed from Table VI that the risk indices of line overload obtained from PEM and MCS are similar, which indicates that PEM provides good results. Compared with Table V, Table VI also shows that the risk indices for most of lines with wind source correlation as well as load and generation correlation are larger than those considering only load and generation correlation or neither. For instance, using PEM for the risk index computation, the risk index of lines
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Fig. 10.
Modified IEEE 118-bus system.
TABLE VII C OMPARISON OF R ISK I NDICES W ITH I NDEPENDENT L OADS AND C ORRELATED L OADS FOR THE MODIFIED IEEE 118- BUS SYSTEM
9
relative error of the system risk index results from the PEM with respect to the MCS results is 1.21% without considering load and generation correlation, while is 0.32% with the consideration of load and generation correlation. Thus, PEM is an appropriate tool for estimating the line overload risk for wind-integrated power systems. Moreover, from Table VII, risk indices for uncorrelated and correlated cases are different. For the entire system using the PEM, the risk index with the consideration of load and generation correlation is 92.9354, while it is 70.8997 without the consideration of load and generation correlation. That is, the line overload is more obvious when considering the correlation. Thus, it can be concluded that risk indices are seriously affected by the load and generation correlation. V. C ONCLUSION
6 and 28 overload is 0.0970 in Table VI, whereas the risk indices are 0.0903 (which only considers load and generation correlation) and 0.0705 (which ignores both correlations) in Table V. It can be observed that the risk index for the entire system in Table VI is far from that in Table V. The risk index for the entire system with wind resource correlation as well as load and generation correlation is about three times and seven times larger than those considering only load and generation correlation or neither. It shows that wind resource correlation has profound influence on line overload risk. B. Modified IEEE 118-Bus System The proposed risk assessment approach of line overload is further validated via the modified IEEE 118-bus system [41], which connects a wind farm to bus 118 as shown in Fig. 10. The impacts of load and generation correlation on line overload risk are studied for this system. Table VII shows the overload risk assessment results of certain lines with and without considering load and generation correlation. From Table VII, it can be observed that risk indices obtained by PEM and MCS are almost the same. The
This paper presents a method for assessing line overload risk for wind-integrated power systems with the consideration of wind and load-power generation correlations. Using PEM for PLF calculations, the possibility of line overload can be computed. Combining the possibility with the severity of line overload, the quantitative risk indices can be obtained. The effectiveness of the proposed method is verified via the modified IEEE 30-bus system and IEEE 118-bus system. Numerical results show that the proposed method is appropriate for assessing line overload risks of wind-integrated power systems. It is observed that the types and the parameters of uncertain wind speed distribution would affect the risk indices of line overload. Thus, appropriate shape and scale parameters of Weibull distribution for wind speeds should be carefully tuned. In addition, wind resource correlation and load-power generation correlation would also impact the risk indices of line overload, and needs to be considered when assessing the line overload risk. Risk indices with the consideration of random variable correlation would more accurately reflect the system security and provide more effective information for the decision-maker. R EFERENCES [1] C. Z. Karatekin and C. Ucak, “Sensitivity analysis based on transmission line susceptances for congestion management,” Elect. Power Syst. Res., vol. 78, no. 9, pp. 1485–1493, 2008. [2] B. Leonardi and V. Ajjarapu, “Investigation of various generator reactive power reserve (GRPR) definitions for online voltage stability/security assessment,” in Proc. IEEE Power Energy Soc. Gen. Meeting, Pittsburgh, PA, USA, 2008, pp. 1–7. [3] M. Esmaili, H. A. Shayanfar, and N. Amjady, “Multi-objective congestion management incorporating voltage and transient stabilities,” Energy, vol. 34, no. 9, pp. 1401–1412, 2009. [4] D. Niebur and A. J. Germond, “Power system static security assessment using the Kohonen neural network classifier,” IEEE Trans. Power Syst., vol. 7, no. 2, pp. 865–872, May 1992. [5] A. V. Machias, J. L. Souflis, and N. Evangelos, “A fuzzy transient stability index in power system security evaluation,” IEEE Trans. Autom. Control, vol. 34, no. 6, pp. 662–666, Jun. 1989. [6] D. Jayaweera and S. Islam, “Steady-state security in distribution networks with large wind farms,” J. Mod. Power Syst. Clean Energy, vol. 2, no. 2, pp. 134–142, Apr. 2014. [7] P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994. [8] J. McCalley, V. Vittal, and N. Abi-Samra, “An overview of risk based security assessment,” in Proc. IEEE Power Eng. Soc. Sum. Meeting, Edmonton, AB, Canada, 1999, pp. 173–178.
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Xue Li (M’13) received the B.S. and M.S. degrees from Zhengzhou University, Zhengzhou, China, in 2002 and 2006, respectively, and the Ph.D. degree from Shanghai University, Shanghai, China, in 2009, all in electrical engineering. She is currently an Associate Professor with the Department of Automation, Shanghai University. Her current research interests include wind-integration into grid and power system economics and security.
Xiong Zhang received the B.S. degree in electrical engineering from Xuchang University, Xuchang, China, in 2007. He is currently pursuing the M.S. degree in automation from Shanghai University, Shanghai, China. His current research interest include risk assessment of power system and probabilistic methods applied to power systems.
Lei Wu (SM’13) received the B.S. and M.S. degrees in electrical engineering and systems engineering from Xi’an Jiaotong University, Xi’an, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from the Illinois Institute of Technology, Chicago, IL, USA, in 2008. From 2008 to 2010, he was a Senior Research Associate at the Electric Power and Power Electronics Center, Illinois Institute of Technology. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY, USA. His current research interests include power systems operation and economics.
Pan Lu is currently pursuing the M.S. degree in automation from Shanghai University, Shanghai, China. His current research interest include risk control of power system.
Shaohua Zhang received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China; the M.Eng. degree from the Shanghai University of Technology, Shanghai, China; and the Ph.D. degree from Shanghai University, Shanghai, in 1988, 1991, and 2001, respectively, all in electrical engineering. He is currently a Professor with Shanghai University. His current research interests include power system restructuring, pricing, and reliability.
