Journal of Electrical Engineering and Electronic TechnologyISSN: 2325-9833

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Research Article, J Electr Eng Electron Technol Vol: 1 Issue: 1

Characteristics of Short-term LOLP Considering High Penetration of Wind Generation

Chenxi Lin*, Thordur Runolfsson and John Jiang
School of Electrical and Computer Engineering at the University of Oklahoma, USA
Corresponding author : Chenxi Lin
Student Member, School of Electrical and Computer Engineering at the University of Oklahoma, USA
E-mail: [email protected]
Received: July 30, 2012 Accepted: October 10, 2012 Published: October 12, 2012
Citation: Lin C, Runolfsson T, Jiang J (2012) Characteristics of Short-term LOLP Considering High Penetration of Wind Generation. J Electr Eng Electron Technol 1:1. doi:10.4172/2325-9833.1000103

Abstract

Characteristics of Short-term LOLP Considering High Penetration of Wind Generation

Loss of Load Probability (LOLP) is an important measure of generation adequacy. A good understanding of the new characteristics of LOLP exhibited after integration of variable renewable generation is essential to the power system reliability. This paper presents the results of a study on the impact of wind generation on short-term LOLP, which then becomes a fastchanging stochastic process, driven by the intermittent and variable wind. We firstly introduce a mathematical model for calculating short-term LOLP, and then a novel quantitative measure of its behavior when converging to its steady-state level is derived. In addition, the corresponding empirical formulas are offered which can be used in practice to estimate the convergence time of LOLP under different conditions. Finally, an application of the outcomes of the analytical work in estimation of the dynamic behavior of shortterm LOLP with an actual wind generation profile is presented to show the significance of the developed measures.

Keywords:

Introduction

Large-scale intermittent and variable wind generation in the future electric energy supply portfolio challenges the assessment of generation adequacy, especially when the penetration is high. Although the capacity provided by the wind power improves the system adequacy level to a certain extent in a long-term measure, the variability of its output can affect the system adequacy in a short-term, which imposes an immediate risk to system reliability operation [1,2].
Loss of Load Probability (LOLP) is an important measure of generation adequacy. By definition, for a power system with both wind generation and conventional generation, for any time t ∈ 0,T] and a given peak load l during this period of T, LOLP can be described as follows
where CT (t) is a variable representing the total available conventional generation at time t and CW (t) represents the wind generation at time t. Conventionally, controllable generation is usually modeled with a two-state process including an available state with planned capacity and an outage state with zero or reduced amount of output, while wind generation can be viewed as a stochastic process driven by the wind.
Incorporation of wind generation in LOLP calculation has been explored in several studies. For example, [3] and [4] used the Monte Carlo method to simulate hourly stochastic change of generation availability. An auto-regressive and moving average (ARMA) time series model is used in [5] to simulate hourly wind speed and available wind power in consideration of chronological characteristics, [6] and [7] presented some analytical methods for estimation of wind output with multi-state models of wind speed.
These existing techniques mainly focus on the long-term reliability evaluation ranging from several months to several years using the stationary LOLP. In long-term LOLP calculation, wind generation has been viewed as a multi-state power plant and probabilities for each output level are time-invariant, which cannot be directly used to describe the short-term change of LOLP. Lack of an appropriate techniques to assess the dynamics of short-term LOLP makes it difficult to quantify the impact of the wind on generation adequacy.
The concerns about the impact of variable wind generation on short-term generation adequacy are expressed in several recent articles, where some new methods are also proposed. For example, [8] developed an empirical sliding window to update LOLP on an hourly basis. This approach was extended and applied to assess the fraction of system reserve that can be allocated to wind farms [9]. In [10] and [11], the authors applied a Markov chain model to represent variable wind generation in operational risk evaluation. Based on our previous work in [12], it is found that the short-term LOLP converges to its steady-state value, i.e. its long-term level. This finding indicates that the information about variable and intermittent wind generation can be lost if the updating interval of LOLP is too long.
The goal of the study presented in this paper is to show that an appropriate time period can be found for estimation of the shortterm LOLP. More specifically, a novel analytical description of convergence time is developed and empirical formulas for calculation of convergence time are derived, which can be used to determine the appropriate period for updating LOLP and understanding its dynamic behavior. Finally, an application of the methods and measures are shown using the output profile of an actual wind farm. The discussion of the impact of short-term LOLP on generation adequacy under different wind penetration scenarios are also presented.
The remaining of the paper is organized as follows: the next pharagraph introduces the methodology for calculation of the shortterm LOLP based on Markov chains. The framework for estimation of the interval needed for updating LOLP and the study of short-term LOLP based on various actual wind generation profiles are presented in the following paragraphs and the conclusion is provided at the end.

Short-term LOLP by Markov chain

This section describes the framework for estimation of the shortterm LOLP for power system with wind generation, which consists of two parts: (1) mathematical description of instantaneous LOLP; (2) estimation formula of instantaneous state probabilities.
Formulation of Instantaneous LOLP
For a system with only conventional generation, the generation adequacy during a short time period can be evaluated by the PJM method [13]. This method is similar to other approaches in the static generation reserve studies except the time invariant forced outage rate (FOR) is replaced by time dependent outage replacement rate (ORR). Specifically, the ORR of a generation unit during period T is given by
(1)
where λ is the failure rate. In LOLP calculation, a capacity outage table is obtained by convoluting the generation outage replacement rate, where the risk value for a given load can be deduced directly.
Multi-state models have been used to model the variable wind generation in [6] and [14]. Although this model is effective in the long-term LOLP calculation with wind generation, it can not be directly used in the short-term study since the state probabilities in this model is time-invariant, which is inadequate for characterizing the inherent variable nature of the wind. To address this issue, an instantaneous multi-state model for short-term LOLP calculation using time dependent probabilities is derived as follows.
Assuming a k-state wind generation is added to conventional generation profiles, the instantaneous LOLP at time t∈ [0,T] can be described as follows
(2)
Where l is the peak load during this period. μt (j) are the instantaneous state probabilities of wind generation for state j at time t and are defined by
with the maximum output of wind generation. is the instantaneous cumulative probability of availability of conventional generation at time t and can be derived by the PJM method discussed previously. Since the generation outage replacement rate is constant once a certain period T is fixed, as described in Equation (1), we can replace with in equation (2).
To simplify the derivation, we rewrite μt (j), j = 0,...,k-1 in a probability row-vector form, i.e. , to represent the distribution for different levels of wind generation at time t. Equation (2) shows that the key to the computation of instantaneous LOLP is to calculate the instantaneous state probabilities of wind generation μt, which is presented next.

Probability estimation by Markov chain

In steady state LOLP calculation, the probability of the wind generation output staying at each output level can be computed using the power curve of wind turbines and the Weibull distribution of hourly wind speeds as discussed in [6] and [7]. However, since we want to compute the instantaneous transition probabilities between states, those computation methods for long-term LOLP cannot be used for short-term LOLP estimation because the state probabilities estimated by these methods are time-invariant, thus cannot characterize the impact of variable wind on LOLP.
A number of articles have adopted Markov chains to describe the variation of wind speed, where each state represents a discrete wind speed level. In some recent studies, such as [15], a first-order Markov chain is used to generate synthetic series of wind speed and the results show that short-term dynamics by the Markov chain are very close to the actual wind speed. In [16], a method for direct generation of synthetic time series of wind power output by Markov chain Monte Carlo is proposed. Another application of Markov chains in evaluation of the reliability of distribution networks containing embedded wind generation can be found in [17]. In addition, a study reported in [11] used a Markov process to model wind generation to evaluate operational risk in a power system with high wind penetration.
Note that for a process to be represented by a Markov chain, it need to be stationary. In other words, the transition rates between different states remain constant throughout the period of interest. Since the wind speed usually has seasonal patterns, the mean and standard deviation of wind speed cannot remain constant all the time. Therefore, the wind speed cannot be described as a stationary process. In some recent studies, such as [18], authors have proposed an approach to overcome this problem of non-stationary by partitioning the annual cycle into months and model monthly wind data.
Based on the idea proposed in previous studies, we use a Markov process to model wind generation and estimate the instantaneous state probabilities in Equation (2).
Let Δt be a very small time step compared to the minimum residence time in all states of wind generation. Then the time period of interest, [0,T], can be divided to N equal length intervals Δt = T / N. During each small time interval of Δt, the state probabilities, μt (j), are assumed constant. Let be the probability that wind generation is in state j during the (n+1)th time interval, and . Denote as the row vector representation of state probabilities μn (j) for all states. Thus, for a fixed Δt, the instantaneous state probabilities μt can be approximated with discrete time state probabilities for intervals n = 1,...,N-1 and
(3)
where is the floor function mapping a real number to the next smallest integer, is the stochastic transition matrix with entries pij characterizing the transition probability of wind generation from level I to level j, is the initial distribution of wind power during the first time interval [0,Δt]. The entries of the transition matrix are estimated by
(4)
where α ij the transition rate from state i to state j. Our detailed development of this transition matrix is presented in Appendix A.
For metered wind generation data over time period Tt, the maximum likelihood estimator for the transition rate α ij from state i to state j is given by
where nij is the number of occurrence of transition from state i to state j observed in the time period, Tt. Therefore, if the initial distribution and the transition probability matrix are given, the state distribution of the Markov chain for any step can be found.
Hence, once the time step Δt is determined and the transition matrix is constructed, the instantaneous LOLP during the time period [0,T] for a given initial wind generation condition can be obtained by equation 2.
In the next section, we will present a novel development of measures of dynamic behavior of short-term LOLP as well as a method for selection of the appropriate time period [0,T] for updating short-term LOLP.

Time Period For Updating Short-term LOLP

This section presents a method for updating the short-term LOLP in five parts: (1) investigation of its short-term convergence; (2) quantification of the convergence time; (3) estimation of convergence time; (4) simplifications of calculation using properties of Markov chains; (5) empirical formula for selecting the interval for updating LOLP in practice.
Convergence of LOLP
Figure 1 shows a short-term LOLP profile for a 6 hour time period with different initial conditions, based on the actual observations in northwest Oklahoma obtained in our previous study [12]. It is observed that different initial conditions can cause the different trajectories of short-term LOLP, since the initial conditions of wind power affect the instantaneous state probabilities according to Equation (3). Moreover, it is also observed that the convergences of instantaneous LOLP under both high and low initial wind speeds: LOLP(t) will converge to a steady state if the time period T for estimation is long enough. For example, as shown by the diamond marked line, if the estimation period is T =200 min, the LOLP(t) with a high initial wind speed will increase to its stationary level 0.054. On the other hand, as shown by the square marked line, LOLP(t) with a low initial wind speed will decrease to its steady state level in about T =180 min.
Figure 1: Short-term LOLP.
The reason for the convergence of short-term LOLP is due to the fact that the instantaneous state distribution converges. In particular, as t → ∞, it can be shown that the instantaneous state distribution μt will converge to the stationary distribution π of the Markov chain. The corresponding steady state LOLP is given by
(5)
Since the stationary distribution π characterizes the long-term behavior of a Markov chain, equation 5 also provides another way to look at long-term LOLP.
Once the LOLP converges, it will not change significantly as the time period further increases. This implies that the short-term effect of wind generation will be lost if the period of estimation is too long. For example, as shown in figure 1, for a given initial wind speed, instantaneous LOLP(t) with t∈ [0,T] for a period of T = 200 min is almost the same as the one estimated for T = 350 min. In order to have a better assessment, it is necessary to update the LOLP with a new initial wind generation condition before it converges to the steady-state.
We found that the appropriate updating frequency can be found by applying some properties of Markov chain. It is known that instantaneous state probabilities of an ergodic Markov chain converge to its steady state distribution as time increases, and the corresponding convergence rate is determined by the absolute value of the second largest eigenvalue of the transition matrix [19]. More specifically, if the absolute value of the second largest eigenvalue is small, the short-term LOLP for different initial wind power will quickly converge to the steady state LOLP. Therefore, choosing a suitable time period [0,T] is important to the calculation, since it will impact the accuracy of assessment of generation adequacy. The mathematical development of the time for the convergence of short-term LOLP will be presented in the following subsections.
Definition of convergence time of short-term LOLP
As discussed above, LOLP(t) converges to its steady state level as the time period of estimation T increases. In general, for short-term LOLP calculation, we have a maximum time period Tmax, say 12 hours. Moreover, we define the convergence time of instantaneous LOLP as the shortest time period T when the difference between LOLP(T) and LOLP(∞) stays within a certain tolerance level, which is the solution to the following optimization problem,
(6)
where Tmax is the maximum time period for the short-term LOLPcalculation and δ is the given tolerance level.
The optimal solution of (6), i.e. convergence time of instantaneous LOLP, is denoted as T*. The tolerance level used in this study is given by
(7)
where α∈ (0,1) is a coefficient.
The convergence time of LOLP(t) provides important information that can be used to determinate the appropriate updating interval. After the convergence time, the instantaneous LOLP reaches its steady state level. Therefore, if the estimation or forecast period T of LOLP(t) is longer than this convergence time T*, the result will be close to the long-term LOLP, which means that the information about the short-term contribution of wind generation is lost and the estimation is not accurate.
Estimation of convergence time
This section describes the relation between convergence time T* of LOLP and a given tolerance level δ. According to equations (2) and (5), we define the discrete form of convergence time N* with time step Δt
(8)
where is the cumulative probability of conventional generation, is the approximated state probability of wind generation at level j in step N, and π is the stationary distribution of the Markov chain. The discrete convergence time is the optimal solution N* to the problem described by Equation (8). The corresponding convergence time is
Definition 1: We refer the optimal solution N* as the time window or window.
In order to obtain a more accurate results in practical estimation, numerous states of wind generation might be needed. Therefore, it could be very challenging and time-consuming to solve for the window N* through iterative simulation. In order to address this issue, we will propose a simplified method for quick assessment of the convergence time or window using the properties of ergodic Markov chains.
Simplifications of calculation for empirical study
We found that the convergence properties of an ergodic Markov chain can be used to simplify the numerical problem described by equation 8. A Markov chain is ergodic if the chain is aperiodic and irreducible. Let be the transition matrix of a k-state Markov chain and be the eigenvalues of . Then, the stationary distribution π of the chain satisfies the equation [19].
The chain is ergodic if and only if π is unique which is equivalent to the condition that there is a unique eigenvalue equal to 1.
We notice that in this case the magnitude of all the remaining eigenvalues of are strictly less than one, i.e. if λ0=1 then for i =1,..,k−1. This property allows us to use the unique stationary distribution π to define a metric, so that the distance between probability distributions of the Markov chain at any time instant and the its stationary distribution can be characterized and then, used to simplify equation 8.
Definition 2: For two probability vectors μ,π on a finite state space X, the variation distance between these two probability vectors is defined as follows,
The proof of the following result can be found in [20].
Theorem 1: Let be an ergodic transition matrix on a finite state space X and let π be its stationary distribution. Then for any initial distribution μ0, the distribution of the chain at time n, denoted as μn, satisfies the following inequality:
where λ* is the absolute value of the second largest eigenvalue of .
Proposition 2: For an ergodic Markov chain, the stationary distribution is unique. Therefore, the variation distance in Definition 2 can be used to measure the convergence time between the distribution and the stationary distributions π, i.e., the two distributions used in computing the LOLP convergence time.
In many practical problems such as those associated with variable wind generation, the transition matrix has distinct eigenvalues and is therefore diagonalizable. For a diagonalizable transition matrix, we found that estimation of the variation distance by Theorem 1 can be further simplified to characterize the absolute difference between the current state distribution of Markov chain μn and the stationary distribution π for any point in the state spaces. This simplification is discussed in the following proposition, which is proven in Appendix B.
For an ergodic Markov chain defined on a finite state space X, assume the initial state distribution μ0 is given and transition matrix is diagonalizable with distinct eigenvalues , and define λ0 =1, Then, the absolute difference between the state distribution μn at time n and the stationary distribution π for point j ∈ X satisfies:
where is a basis of left eigenvectors of the transition matrix corresponding to , and are the unique coefficients such that
Next, the time window N* for the empirical estimation of the convergence time of the short-term LOLP will be derived using these analytical results.
Empirical formula for time window estimation
A faster computation is always desired in practice, especially in real time operation support. In this subsection, we derive further simplified formula for the estimation of time window using the boundary condition of the LOLP and the previous analytical results.
Note that the time period T affects the outage replacement rate ORRT and cumulative probability in equations 1 and 2. It is obvious that the following condition holds,
(9)
for any given x.
Using equation 9 and Theorem 1, we can have
(10)
The smallest N, i.e. window N*, is approximated by letting the upper bound of (10) equal to the boundary δ and the approximate window is given by
(11)
where is the floor function.
By Proposition 2, if the transition matrix is diagonalizable, we have
(12)
Since the transition matrix is diagonalizable, the left eigenvectors vi, i = 0,..,k−1 are independent and the coefficients can be obtained by
where is nonsingular.
The found in equations 11 and 12 is a suboptimal solution of 8 and the corresponding convergence time is slightly smaller than the actual optimal solution of 6. By equations 11 and 12, it is convenient to estimate the time window for updating LOLP in empirical analysis.
In the next paragraph, the results obtained above will be used to study the dynamic behavior of the short-term LOLP and its convergence time based on an actual wind profiles in different situations.

Results of Short-Term LOLP Study

This section presents three case studies on the characteristics of the short-term LOLP based on wind data measured at 10 min time interval at a wind farm located in northwest Oklahoma, with the concepts and the methodology developed in previous sections.
Case 1 only considers conventional generation in estimation of short-term LOLP. Case 2 studies the short-term LOLP with wind generation added. The wind generation profile in winter and in summer are considered separately to overcome the problem of non-stationary due to seasonal patterns, as suggested in [18]. Case 3 studies the impact of wind penetration level on the convergence time.
Case 1: LOLP with conventional generation only
Assume that the power system has 6 conventional generating units, each with capacity of 250 MW. The time horizon of interest, T, is set to be 6 hours. The generation failure rate is λ = 0.0139 h−1. Equation (1) gives an outage replacement rate ORRT =0.08. The convolution algorithm can be used to generate the capacity outage table by adding one generator at a time. The results are shown in table 1.
Table 1: LOLP with conventional generation.
If the peak load l is assumed to be 1000 MW, the LOLP with conventional generation is 0.077, which remains constant over time period T.
Case 2: short-term lolp with wind generation
In this case, 500 MW wind generation is added to the conventional generation portfolio described in Case 1. The wind generation consists of 100 wind turbines, each with capacity of 5 MW. The outage rate of wind turbine is ignored in this short-term performance study.
The time series of wind generation is based on a real wind profiles at a wind farm located in northwest Oklahoma measured at 10 min time interval. A simple cubic function below is assumed to describe the power curve, given by
where Pt is the power output of wind generation at time t, wt is wind speed at time t, wτ and Pτ are rated wind speed and power output, wci and wco are the cut-in and cut-out wind speeds. In the study, we choose wci = 3 m / s, =14wτ m / s, =25 wco m / s. The generated wind power output time series is shown in figure 2. Figure 2a shows that the wind generation output in winter (Dec, 2009), while figure 2b shows that in summer (Aug, 2009).
Figure 2: Time series of wind generation.
Wind generation in winter: In order to calculate the short-term LOLP numerically, we discretized the time series into 10 states with equal range between the adjacent states, i.e., . The time step chosen to be Δt = 1 min and we estimated the transition rate between different wind levels from the time series to construct a 10×10 transition matrix for the Markov chain.
In this study, we found that all eigenvalues of matrix are distinct, which means it is diagonalizable. Therefore, the assumption for Proposition 2 is valid for the study.
The variation of short-term LOLP depends on the initial state probability of wind generation, as described in Equation (3). In this study, three different initial state distributions are presented for low, medium and high initial wind generation levels, given by The result of the short-term LOLP with wind generation in winter is shown in figure 3.
Figure 3: Short-term LOLP in winter.
As shown in figure 3, the short-term LOLP converges to its stationary level regardless the initial levels, i.e., LOLP(∞) = 0.0378. Note that for 6 hours time period, LOLP has already been in its steady state for about 1 hours. This means that, if the time period for LOLP updating is 6 hours, it will have lost all short-term information. Therefore, LOLP has to be updated much more frequently than 6 hours.
The variation ranges for three different initial distributions are: δ1=0.0099, δ2=0.0073, δ3=0.0093, with the tolerance factor equal to 0.25 in equation 7. To simplify the problem, we assume Tmax = T = 6h. The corresponding window , , are also shown in figure 3. The approximated windows obtained by (12) are: =52, =87, = 110 for low, medium and high initial wind conditions, respectively. The corresponding approximated convergence times are: = 52 min, = 87 min, = 110 min.
Wind generation in summer: The result of the short-term LOLP with wind generation in summer is shown in figure 4.
Figure 4: Short-term LOLP in summer.
In this case, the states of Markov chain are the same as that for wind generation in winter while the transition matrix is constructed using time series in figure 2b. Compared to that in figure 3, the short-term LOLP in summer converges to its steady state level at a slower rate. The time windows for LOLP updates are shown in figure 4. Using the same tolerance level α, the approximated convergence times are: = 64 min, = 92 min, = 131 min.
The time window or convergence time for updating LOLP in summer is larger than that in winter due to the “smoother” time series of wind generation shown in figure 2b compared with figure 2a. It means that for a fast-changing intermittent wind generation, LOLP needs to be updated much more frequently to better reflect the actual system generation adequacy condition in short-term.
It is also shown that the steady state LOLP in figure 3 is better than that in figure 4 due to the fact that the wind generation in winter remains at a high level for longer time than in summer. In other words, the wind generation in winter has more contribution to generation adequacy than in summer for the time period of this case study.
Based on the observations in Case 2 study, it is also suggested that although the capacity benefit of wind generation is better in winter in a long-term, the LOLP needs to be updated more frequently to reflect the actual generation adequacy condition. In practice, the method and measures provided in Section II and III can be applied to many other weather situations.
Case 3: impact of wind penetration level
Case 3 studies the impact of wind penetration level on the convergence time of short-term LOLP by changing the number of wind turbines in wind farms. The wind penetration is defined as the percentage of total wind generation over total system generation. The results are shown in figure 5.
Figure 5: Impact of wind penetration level.
Figure 5 indicates that the impact of wind penetration level on the short-term LOLP is non-linear. By increasing the penetration, the time window decreases, which implies that we should update the short-term LOLP more frequently. Note that when the penetration level of wind generation is increased, the impact of variable wind generation on short-term LOLP becomes larger, thus more frequent update of LOLP is desired.
The finding and suggestion presented in above three case studies show that the proposed method and measures improve the understanding about the impact of variable wind generation on short-term generation adequacy, and provide quantitative support for power system reliability operation.

Conclusion

This paper presents quantitative measures and a method for understanding the short-term impact of wind generation on LOLP. The short-term LOLP is calculated using an instantaneous multi-state wind generation model. A discrete method based on a corresponding Markov chain is also proposed for estimation of short-term LOLP. Furthermore, by using properties of an ergodic Markov chains, several methods for determining the appropriate time interval for updating LOLP are provided. Finally, the methods are applied to a study of the short-term LOLP under different initial wind generation levels, different wind generation output profiles and different penetration rates.
The novel measures of LOLP convergence time presented in this paper can be used to better understand the impact of wind generation on system reliability and are very useful for short-term generation adequacy assessment. More specifically, the methods and measures enable a quantitative estimation of convergence characteristics of short-term LOLP, which provides scientific support for the system operator to understand the accountability of wind generation.
Additional enhancements could be made in our future research. For example, the failure rate of the wind turbines could be taken into consideration. Also, due to the daily and seasonality patterns of wind energy, different transition matrices for different time periods may be needed for more accurate results.

Appendix A

Derivation of transition matrix
The instantaneous state probabilities, μt(j), can be evaluated by solving the differential equations
If we rewrite the above equation with row-vector μt, we have
(13)
where matrix A is given by
The solution of Equation (13) is given by
(14)
for any t0[0,T]. Using discrete state probabilities , a discrete form of Equation (14) can be obtained for interval n=1,..., N-1,
Since Δt is small, first two terms in Taylor series are used as an approximation, that is
where is the transition matrix given by
which gives Equation (4).

Appendix B

Proof of proposition 2
Proof. For any eigenvalue λi of the transition matrix with the corresponding left eigenvector vi, we have
Furthermore, by iteration, for any n≥0, we also have
Assume λ0=1 and the initial distribution μ0 is given by
then, for any n≥0, we have
Since ,
For a point x ∈ X,
where

Acknowledgement

This research was supported by the National Science Foundation under grants ECCS-0955265.

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