#### Abstract

*Summary. *Generation planning is an important aspect of the probabilistic production simulation (PPS) process for power systems. The inclusion of nondispatchable energy sources in the conventional system needs efficient algorithms for reliability evaluation. Moreover, it is crucial to represent the availability of nondispatchable energy sources with a suitable probability distribution function (PDF) that can be easily integrated with the conventional system. The present work extends the generation reliability to include nondispatchable energy sources in the reliability evaluation methodology. The wind electrical system (WS) is represented using a simple wind speed model based on the normal distribution function. The multistate wind speed model based on the mean and standard deviation of wind speed at a particular site is integrated with the conventional system for reliability study. The proposed fast Fourier transform (FFT)-based method eliminates extensive calculations encountered in the conventional approach due to cumbersome time-domain convolution. The efficacy of the methodology is validated with the case studies using the IEEE RTS-wind system with loss of load probability (LOLP) and expected energy not served (EENS) reliability indices.

#### 1. Introduction

One of the goals of generation planning is to ensure a reliable power supply to the consumers. Recently, the global climatic concerns have given impetus to incorporate more renewable energy sources. Renewable energy sources are intermittent in nature. Particularly, wind energy generation is quite variable in nature due to its dependence on factors such as wind speed, and density of air. Such variable energy resources thus come under a category of nondispatchable energy sources [1, 2], and incorporation of these sources in generation planning is quite complex [3]. Wind energy has a significant share in nondispatchable energy generation due to its expansion in recent years [4, 5]. Therefore, generation planning should incorporate the stochastic behavior of wind along with the generator outages. Furthermore, the conventional dispatchable units (DUs) are represented by a model representing two states, but due to large variations, wind speed-based nondispatchable units (NDUs) require a multistate representation, which is complex. The uncertainties in nondispatchable generation can be modeled using PDF [6], which can be correlated with the generated power and integrated with PPS to determine the generation adequacy and reliability of the system.

##### 1.1. Motivation

For efficient generation planning, it is also necessary to model the uncertainties associated with the DUs, which can be done by representing them in terms of their probabilities [7]. The most significant uncertainty in generation planning is the forced outage of generators [8].

The interaction of load probabilities, with the uncertainties associated with all the connected generating units, forms the basis for the probabilistic load model (PLM) used for reliability analysis [9]. PLM is a continuous modification of the load duration curve (LDC) considering generational uncertainties. LDC determines the time duration of load demands arranged in descending order in a given time interval [10], which can also be represented by the probability of occurrence of a particular demand in a given time interval. In order to obtain PLM, the probabilistic model of each generator is convolved multiple times with the LDC. With the increase in generating units, the convolution process becomes cumbersome and time-consuming. The complexity and memory requirement for convolution further increases with the involvement of multistate renewable generation sources. With ever-increasing demand and generation requirements, an efficient algorithm with minimal computation time is required. The multiple steps required for convolution operation, performed in the time-domain conventionally, can be reduced to a single step with the application of the fast Fourier transform (FFT) algorithm [11]. This requires efficient probabilistic modeling of the generation sources to obtain the reliability indices.

The motivation of the present work is to integrate the uncertainty associated with wind farms with the probabilistic generation of the DUs and reduce the computational time and complexity of generation planning.

##### 1.2. Related Literature

In recent years, numerous multistate models for wind turbines and reliability assessments of wind integrated power systems have been suggested due to the integration of wind energy systems (WS) into the conventional power system. The proposed works have concentrated on the reliability of protection systems [12], reliability considering elements connected to the power system network [13, 14], reliability enhancement using control strategies [15, 16], and reliability considering reactive power optimization [17]. However, the basic requirement of a power system is to satisfy the load demands considering the existing and upcoming system facilities [18], which needs to be dealt with on a priority basis. Although significant work has been done related to adequacy of generation considering conventional generators, adequacy studies considering wind integrated power system demands attention [19]. The amount of energy generated by wind farms varies with wind speed, depending on the farms’ location. Hence, wind farms need to be represented by a common multistate wind speed model [20], representing uncertainty in wind generation. Also, the number of iterations for PLM computation increases with the increase of states, which needs to be simplified.

Reliability assessment of wind farms considering wind intermittency and parameters related to wind turbines have been reported in [21, 22]. The methods consider wind-related uncertainties, but the conventional generators have not been taken into account. Although combined reliability evaluation considering wind farms and conventional generators have been reported in [23], the methods do not consider common wind speed model considering wind farms at different locations. References [24, 25] have taken into account wind models that consider topological variations, but the reliability evaluation methodologies of wind energy-based power systems are complex. The reliability assessment of power systems with multistate energy sources requires complicated and time-consuming calculations. Therefore, it is imperative to use digital signal processing (DSP) tools, which can reduce the computational burden. A frequency-domain approach to calculate the reliability of the system considering solar irradiation intermittency has been proposed in [26]; however, the method has not been implemented for wind speed intermittency.

##### 1.3. Innovative Contributions

The power generation capacity of nondispatchable energy sources is dependent on nongovernable factors. The factors can be accounted for in reliability studies if they are represented by their occurrence probability. Wind speed, which is a nongovernable factor of WS, depends on the geographical conditions of the site, which can be represented by a PDF divided into multiple states. The reliability evaluation involves the interaction of this multistate model, the load behavior, and the other generation uncertainties. Hence, a simple algorithm is required to represent the interactions to minimize the time and memory requirements of the processor. The following are the innovative contributions of the current work in comparison to the existing literature:(i)Impact assessment of grid-integrated nondispatchable energy sources having variable generation on reliability in combination with PLM(ii)Extension of generation reliability evaluation to incorporate nondispatchable energy sources using multistate generation model(iii)Frequency-domain approach for simplification of the process involved in the convolution of probabilistic multistate generation model with probabilistic load model

#### 2. Reliability Modeling of Generating Sources

##### 2.1. Reliability Model for Dispatchable Unit

Generation planning is an essential part of power system probabilistic simulation. In order to achieve an appropriate planning scheme, the simulation has to be run multiple times, and hence, efficient functions are required to represent the generation-load uncertainty. This can be done through PLM, which is derived from transformed LDC (where the axes are interchanged to represent the time duration as a function of the load) that expresses the duration of varying loads, as shown in Figure 1. In the figure, *T* is the duration of time for the investigation, and *x*_{m} is the maximum connected load in the system (in MW). Also, (*x*,*t*) is the point on the curve, representing the time duration *t* for which load exceeds the value *x* and is expressed as *t* = *F*(*x*). PLM is a representation of load (in MW) with associated probability sampled with a load interval of MW as shown in Figure 2(a). The probability “” that the load is can be calculated from (1), which is used to obtain PLM from transformed LDC. To understand this consider a system having a base load of 1,800 MW operating for 52 weeks(*T*), a load of 1,881 MW operating for 44 weeks, a load of 2,280 MW operating for 31 weeks, and a load of 2,850 MW operating for 1 week. The corresponding probabilities of loads given values used to obtain PLM are (52/52 = 1), (44/52 = 0.84), (31/52 = 0.59), and (1/52 = 0.01), respectively. The total energy (*E*_{L}) under the transformed LDC is given by (2), which is equal to the energy under PLM.

**(a)**

**(b)**

The forced outage of DUs has a tremendous impact on the available power, which may occur due to aging of units, failure of turbine and boiler, noncompliance with the safety norms, and so on [27, 28]. Considering the outage of DU as an individual event, the probability of outage of each unit can be modeled into two states. The probability of the failure state is given as forced outage rate (FOR) “*q*” and the probability of normal state is represented as .” The PLM can be constructed using FOR, based on recursively adding one unit at a time by convolution process. The formulation of PLM can be represented in pictorial form, as shown in Figure 2.

The original load duration curve can be represented as the original PLM given in Figure 2, illustrating the load delivered by all generating units with associated probabilities. The PLM is sampled, with each sample having an interval . The curve is modified in accordance with the priority of the operation of DUs. The outage of dispatchable unit with a capacity of is represented by the shifting of the original curve by as shown in Figure 2, mathematically represented as a convolution of original load duration curve and probability distribution of dispatchable unit given by (3) [29]. The generalized equation to calculate the PLM for the outage of the unit is given by (4).

The PLM represents the outage of the dispatchable unit by an increase in load demand, which further signifies the increase in energy as shown by the colored region in Figure 2. If the system has “” dispatchable units with a total capacity , the PLM has a maximum capacity .

##### 2.2. Modeling of Nondispatchable Units for Reliability Analysis

The DU outage is represented by a two-state model; however, NDUs have uncertainties having multiple states. Therefore, to include uncertainties with the units for reliability evaluation, multistate models representing them should be taken into account. If the uncertainty related to a DU has states, then the probability of generating power states are and is shown in Figure 3.where represents the maximum capacity of the wind farm. All wind farms’ power states must fulfill the probability constraint given by

The convolution formula to obtain PLM after integrating WS to represent the variability of power can be obtained by the modification of the formula obtained for dual state generating units. If the outage of dispatchable units has been taken into account for generation planning and the convolution of PLM, is performed. For the addition of a multistate nondispatchable generating unit , having power state with a probability , the equivalent load shared by generating unit is . This is mathematically represented by shifting of PLM to the right by , and hence, the curve is modified to having a probability. Hence, the final PLM is a weighted sum of all the generating states, given by

When , equation (7) reduces to (4), and when , is taken as .

#### 3. Wind Speed Uncertainty Model

Several methodologies for calculating wind farm output power have been proposed [30, 31]. The reliability analysis of a wind energy-based system requires a simplistic wind speed model that is easy to integrate with the conventional generating system. The output of the wind farms is wind speed dependent, which is variable in nature. This section elaborates on the wind speed model described in [20], taking into account the mean of wind speed and the standard deviation of past available wind data. The correlation between the varying wind speed and forced outage of generating units is also discussed.

##### 3.1. Wind Speed Model for Multiple Sites

Wind speeds obtained from various locations very closely fit the Weibull PDF. The shaping parameters of the distribution vary with different location and are difficult to calculate. However, the probability distribution of wind speeds can be approximated to normal distribution [20] by calculating the mean and standard deviation of past wind speed data. To consider wind speeds during extreme weather conditions, the distribution is taken up to . For reliability evaluation purposes, the PDF is divided into number of states and the width of each state being . If is the midpoint of the states, then the midpoint of each state can be calculated from (8), and the probability of each step is obtained from (9).where represents the total wind speed data points for a particular state and is the wind speed data points obtained in a year. If multiple sites for wind farms are taken into account, the probabilities of individual sites can be combined to obtain a common wind speed probability for multiple locations. The common wind speed probability of individual states for different sites represented by “” is obtained by taking the average of probabilities for sites into consideration. A 10-step model is used for the present work based on [20] to represent the multistate wind as shown in Figure 4 for the San Francisco Bay Area, with the probability and midpoint of each state indicated. The negative value of wind speed is ignored as it has no significance. The wind speeds are obtained from [32], and each wind site indicates an individual farm.

##### 3.2. Common Wind Power Generation Model

The power generated by a particular wind speed at different sites can be obtained from the speed-power curve of the wind turbine generator. The wind turbine generator starts generating power at wind speed , and the generator is shut off after wind speed for safety reasons. Rated power is achieved between rated speed and the power curve for the wind generator, and multistate probability representation of power with respect to wind speed is shown in Figure 5. The power-speed relation between speeds and is nonlinear, and the generated power by a wind turbine for a given speed state is given by

**(a)**

**(b)**

The constants , , and have been calculated from [21].

The output of the wind turbine generator is zero when the wind speed is less than and when the wind speed is greater than . Wind speed states corresponding to these values can therefore be combined into a single combined state whose probability is given by (11). The output power is rated power for wind speeds between and . These states can also be combined into a single state, and the probability of the combined state is given by (12).

#### 4. Reliability Evaluation Methodology Using FFT

##### 4.1. FFT for PLM

The convolution process to obtain PLM is given by (4) and (7), which becomes tedious to perform when generating units increase due to a large number of convolutions involved. Also, the number of data points obtained after each convolution increases twofold, which increases the memory requirement. FFT converts the sampled-time domain signal to the frequency domain, and thus, the time-domain convolution process is transformed to term-by-term multiplication. If and are two discrete time-domain signals, the signals are converted to and after FFT and are shown in the following equation [11]:

The FFT algorithm minimizes the number of computations required for discrete Fourier transform (DFT) calculations. The number of data points used for computation of FFT algorithm should satisfy (14), where is an integer. A signal can be represented by impulse function with a scaling factor , shifted by and given by (15). This equation can be represented by a uniform shifting of pulses, with each pulse having a size given by (16). The signal obtained is transformed to the frequency domain using (17), where *n* = 0,1,2, …, *N* − 1; ; and . The signal after convolving in the frequency domain needs to be converted back to the time domain and is given by (18). The sampling points for the present work are calculated based on (19) [33], where as the present work performs reliability evaluation based on PLM and is the sampling interval that in the present work is chosen as 1 MW. The FFT method to obtain PLM is depicted in Figure 6. The FFT process reduces the computation time drastically and is highly efficient compared to the conventional method to obtain PLM.

##### 4.2. Reliability Indices Calculation Using PLM

The PLM obtained using the FFT algorithm involves a convolution process multiple times, which also results in continuous modification of PLM, thereby changing the equivalent peak load in the representation. If the number of generating units in the system is “,” having overall capacity , the PLM, after all the generating units have convolved, is represented by the curve . The peak load of the system is modified to a value of , illustrated in Figure 7. The conclusive modified PLM is used to obtain the loss of load probability (LOLP) and expected energy not served (EENS), which are the indices obtained for the assessment of the reliability of the system. The present work considers generation reliability assessment taking into account generation adequacy, which is accurately represented by LOLP and EENS [34, 35]. LOLP is the probability that the maximum daily load of the system is greater than the capacity of generation and is given by (20), where is the set of states that are associated with the loss of load, is the probability of the system state “*i*,” and is the duration related to loss of load for system state “*i*.” EENS is the total energy not served when the load of the system is greater than the generation capacity and is given by (21) [35, 36], where is the generation capacity at state “” and is the load curtailment. The LOLP and EENS can be obtained from PLM as given by (22) and (23). LOLP from PLM can be obtained in terms of the probability of loss of load for a given time duration, as shown in Figure 7. The PLM combines the probability of loss of load for given states and capacity of loss, as shown in Figure 7. It can be observed from Figure 7 and (22) and (23) that when the load is increased with generation fixed, the indices also increase with the increase in outage capacity.

##### 4.3. Reliability Evaluation Methodology

The reliability evaluation involves obtaining load data for the system considered to formulate the PLM. Wind data for different sites is obtained, and each wind site represents a single wind farm. To avoid the representation of different wind farms with different characteristics, a wind speed model that can represent all the wind sites having common characteristics has been used in the present work. FFT is performed on the initial PLM and probability states of generating units. All the generating units are convolved with the PLM in a single step to obtain the final PLM representing the generator outages and wind uncertainty. The reliability indices for reliability assessment are finally obtained. The proposed methodology eliminated the loop, which is run multiple times to convolve the initial PLM with all the generating units. The conventional methodology is shown in Figure 8, and the proposed methodology is shown in Figure 9.

#### 5. Results and Discussion

The proposed methodology is validated by application to IEEE RTS-wind [3]. The IEEE RTS system has been used to validate and benchmark various reliability algorithms. The conventional IEEE RTS comprises 32 DUs, with a total generation capacity of 3,405 MW, and the maximum connected load in the system is 2,850 MW. The conventional system is modified to RTS-wind by replacing a 350 MW generating unit with a wind farm, consisting of 763 wind turbines with a rated capacity of 2 MW. The forced outage rate (FOR) of conventional generators is listed in Table 1, along with their rated capacity.

The present work considers wind data from two sites in California, USA, where each site is considered a wind farm. The wind data is obtained for a duration of two years, and the locations chosen are San Francisco Bay Area and Contra Costa County from [32]. The mean speed and standard deviation of the San Francisco Bay Area are 6.7246 and 2.5857, respectively; also, the mean and standard deviation of the Contra Costa County are 7.0241 and 3.5491, respectively. The 10-step speed models for the 2 sites are shown in Tables 2 and 3, respectively. Combining the probabilities of the two sites, the common wind speed model is shown in Table 4; two wind farms are represented by a common power and probability model. The present work considers Enercon E82 wind turbine [37], with cut-in speed of 4 (m/s), cut-out speed of 28 (m/s), rated speed 15 (m/s), and the rated power 2 (MW).

The LDC for the considered IEEE RTS-wind system is shown in Figure 10, representing the variation of load with time in weeks. The LDC has a maximum load of 2,850 MW, which all the generating units cater to during normal operation. The PLM is obtained from the convolution of LDC, and probabilistic outages of all the connected DUs are shown in Figure 11. The PLM has a maximum load of 6,255 MW, considering the outage of all the generating units. The conclusive modified PLM plot considering the variability of power of wind farms is shown in Figure 12; the maximum load of the PLM shifts to 7,431 MW. The LOLP and EENS have been obtained from the curves.

The reliability indices for the outage of different percentage of generation with wind variation are presented in Table 5. It can be observed with the outage of a lesser percentage of generating units that the reliability of the system is very high and decreases with an increase in the outage of the generating units. The reliability indices calculated from FFT-based PLM for different cases representing different load percentages are shown in Table 6. It can be observed that the reliability has increased with the inclusion of wind farms, as a large capacity wind farm (763 × 2 MW) is included in place of a 350 MW generator. It can also be observed that as the load increases, the reliability of the system decreases. When the system is underloaded, the reliability of the system is high; as the load of the system increases by 10%, the system is reliable as compared to the conventional system. However, when the load increases by 20%, the reliability of the system slightly decreases. The reliability indices using the time-domain approach are also presented in Table 6; the indices are almost similar as compared to the reliability indices obtained using frequency-domain approach, which proves the precision of the proposed frequency-domain approach.

#### 6. Conclusion

The replacement of conventional energy sources with renewable energy resources demands efficient and simplistic algorithms for generation planning. PLM serves as an important tool in generation planning to determine the reliability of the system. However, integrating a multistate wind model is challenging and requires the simulation process to run multiple times. The present work has extended the conventional two-state generation model considering generator outages to include multistate NDUs in generation planning. Also, the complexity of computation associated with multistate models is alleviated by using the frequency-domain approach using FFT. The efficacy of the proposed approach has been validated through case studies using a standard test system such as IEEE RTS and IEEE RTS-wind. Since the NDUs have a lower capacity factor compared to DUs, therefore, more NDU capacity addition is required to maintain the same reliability indices compared to the DUs. The reliability indices such as LOLP and EENS of IEEE RTS-wind are comparable with those of the original IEEE RTS system when about four times wind energy generation is integrated into the system. Thus, proper generation planning can ensure reliability even when NDUs are integrated into the power system.

#### Data Availability

All data used to support the findings of the study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.