Stochastic Modeling of Renewable Energy Sources for Capacity Credit Evaluation

Abstract

In power system planning, the growth of renewable energy generation leads to several challenges including system reliability due to its intermittency and uncertainty. To quantify the relatively reliable capacity of this generation, capacity credit is usually adopted for long-term power system planning. This paper proposes an evaluation of the capacity credit of renewable energy generation using stochastic models for resource availability. Six renewable energy generation types including wind, solar PV, small hydro, biomass, biogas, and waste were considered. The proposed models are based on the stochastic process using the Wiener process and other probability distribution functions to explain the randomness of the intermittency. Moreover, for solar PV—the generation of which depends on two key random variables, namely irradiance and temperature—a copula function is used to model their joint probabilistic behavior. These proposed models are used to simulate power outputs of renewable energy generations and then determine the capacity credit which is defined as the capacity of conventional generation that can maintain a similar level of system reliability. The proposed method is tested with Thailand’s power system and the results show that the capacity credit depends on the time of day and the size of installed capacity of the considered renewable energy generation.

1. Introduction

As climate change becomes a critical issue around the world and the power sector is the most carbon-intensive sector, renewable energy (RE) has then increasingly played an important role in power generation. Global RE capacity is forecasted to increase significantly in the coming years as many countries have planned to raise their RE share for power generation [1,2,3,4,5]. For instance, the European Union has established the ‘European Green Deal’ policy initiative and ‘Fit for 55′ legislative package, committing to cutting emissions by at least 55% by 2030 [6]. Under this target, European countries aim for more than 32% of power generation coming from renewable sources by 2030 [6]. China is another country that is committed to achieving the net zero target by 2060 with a capacity target of 1200 GW in solar and wind power by 2030 [7,8]. In addition, the adoption of RE technologies for power generation has been continuously increasing due to the declining costs of RE technologies, particularly utility-scale solar and onshore wind [9,10,11,12]. The levelized costs of electricity from solar and wind mean that they can compete with conventional energy sources [12,13,14]. Moreover, the costs of other RE technologies, i.e., biomass, biogas, waste, and small hydro, have tended to decrease, as recent studies have shown the potential of decreasing biomass feedstock costs in Europe [15] and a decline in equipment costs in China [16] and worldwide [17]. Additionally, economic and environmental externalities are considered to reflect other co-benefits to support RE technology adoption [18,19,20,21,22,23,24,25,26].

As the capacity of RE power generation—particularly variable RE such as solar and wind—has been increasing, dealing with the uncertainties associated with supply availability is one of the difficult tasks in power system planning and operation [19,27,28,29,30,31]. Due to this availability issue, unsuitable RE integration or operations can lead to a decrease in system reliability or even lead to power outages. For instance, the RE integration can result in difficulties in setting both operating and long-term reserves [32]. Different levels of RE penetration and operational behaviors also have impacts on the reliability and vulnerability of electrical power networks [27]. It was also found that the levels of RE generation mix prior to an outage event are associated with the degree of outages [13].

Due to the intermittency and uncertainty of RE power output, the reliability evaluation with RE resources is challenging [33,34]. To measure the system adequacy with RE resources, capacity credit, also known as capacity value or effective capacity, is usually used. It is a concept used to quantify the relatively reliable capacity of generating resources during a considered period or peak load hours [35,36,37,38]. In Thailand, the capacity credit is called dependable capacity. It is used to evaluate the reserve margin, which is a reliability index for generation expansion planning [39]. The capacity credit can be calculated by approximation and reliability-based methods [40,41,42,43]. The approximation methods, e.g., the Z method [44] and capacity factor-based method [45,46], require less data and computational time since they usually focus on critical periods of inadequate supply.

Table 1. Capacity credit of RE resources and its approximation methods adopted by some countries.

Country	Capacity Credit (% with Respect to Its Installed Capacity)						Method
	Wind	Solar PV	Small Hydro	Biomass	Biogas	MSW
United States (MISO) [49]	16.8	50.7	-	-			Based on the average power output of resources over a defined number of summer peak load hours
United States (PJM) [49]	15.0	46.2	-	-
United States (Southwest Power Pool) [49]	21.1	18.8	-	-
United States (Texas RE-ERCOT) [49]	24.3	55.0	-	-
Spain [40,50]	14.0	7.0	77.0 (Run-of-river)	55.0			Based on power productions with a probability to be exceeded
S. Korea [51]	3.1	13.9	-	44.7	44.7		NA
Thailand [52]	14.0	42.0	29.0	52.0	28.0	47.0	Based on a specific percentile of historical data of resource’s power output

shows the adoption of the approximation methods in some countries. These adopted methods depend on data availability in that country and are acceptable to evaluate capacity credits of RE resources such as wind, solar PV, small hydro, biomass, biogas, and municipal solid waste (MSW). However, a main drawback of these approximation methods is its estimation accuracy and sometimes depending on the discretion of the system planner. It will later be shown in the discussion section that capacity credits used in Thailand’s Power Development Plan 2010 revision 3 [47] and Thailand’s Power Development Plan 2018 [48] created by a deterministic method but different criteria are extremely diverse.

As other distributed energy resources such as energy storage systems (ESSs) and demand response (DR) are increasingly integrated in a power system to manage the demand side, the capacity credit evaluation must have been directed toward techniques that can provide more accurate values [49,53,54]. One of those techniques is the reliability-based method. Several studies have been presented, usually dealing with the capacity credit of variable RE such as wind and solar power [37,38,41,42,43]. Although these reliability-based methods can provide an accurate estimation of capacity credit, it requires the detailed reliability data of systems along with a huge computational effort. Different reliability-based methods of capacity credit evaluation include the effective load carrying capability (ELCC), equivalent firm capacity (EFC), and equivalent conventional power (ECP) [55,56,57,58]. All three methods usually deploy the loss of load expectation (LOLE) as the measurement of system reliability. The ELCC method is the method to evaluate additional loads that can be served by the system with an additional generating unit while maintaining the LOLE. The EFC and ECP method are the methods to determine the capacity of a fully and not fully reliable generator, respectively, where that capacity provides the same LOLE as the considered generator does [55,56]. Although the ELCC method is more widely adopted in system expansion planning, the ECP method is more suitable for determining the capacity credit of an RE generator used in both planning and operation since the capacity credit is measured in terms of a dispatchable generator [58]. Therefore, in this study, the ECP-based method was adopted to estimate the capacity credits for different RE sources, not only highly intermittent RE sources, but also bio-based power and MSW.

To determine capacity credit using the ECP-based method, a generation model of the RE power plant plays an important part in calculating the LOLE of the system. A power output of RE resources, especially highly intermittent RE such as wind and solar PV, is usually represented by a probabilistic model since it can better explain the uncertainty of RE with higher credibility and a wider range of applications [32]. In addition, for highly intermittent and continuous time-varying RE resources, such as wind or solar, the stochastic differential equation (SDE) is usually used to simulate RE resources’ availability such as wind speed [59,60,61,62,63,64] and solar irradiance [65]. For wind speed, the Ornstein–Uhlenbeck (OU) process [59] and geometric Brownian motion (GBM) [62] are often deployed to simulate wind speed with the assumption that the wind speed has a time-varying trend. For solar irradiance, a hidden Markov process (HMM) [66] which is a more complicated model combined with a GBM and semi-Markov process [67] which is a discrete time model can be applied.

Thus, in this study, a probabilistic model is adopted to represent a generation model of each RE resources including wind, solar PV, small hydro, biomass, biogas, and waste. For wind and solar PV, the random variables of RE resources’ availability are modeled in a slightly different way where these random variables fluctuate around their mean during a short time period. Thus, rather than modeling the random variable itself, the noise is modeled instead. In addition, the generation model of solar PV in this study considers a correlation of two uncertainties of solar irradiance and ambient temperature in detail [68]. According to solar PV characteristics, the ambient temperature somehow correlates with solar irradiance. These two variables are not independent from each other; thus, the joint probability between these two random variables needs to be considered [31] and is represented by copula function. Different copula functions will be used during different periods or seasons.

For small hydro power plant, which is usually a run-of-river power plant without a reservoir [69], the uncertainty of water flow is incorporated into its generation model using normal (Gaussian) distribution model. Finally, for bio-based RE generations (biomass, biogas, and waste), its power output mainly depends on fuel availability and fuel quality [70] since these fuels are residues from residential, agricultural, and forestry activities which are highly inconsistent. Moreover, the a system’s availability cannot be negligible since the generation equipment of these RE power plants is similar to those of conventional power plants. Thus, in bio-based RE generation models, uncertainties associated with feedstock availability, fuel conversion process, and the system’ availability are focused upon.

This paper proposes an evaluation of the capacity credit considering the uncertainties of RE resource availability and power output. Six RE resources including wind, solar PV, small hydro, biomass, biogas, and waste, are considered in this paper. The proposed method takes uncertainties due to intermittent wind speed, solar irradiance, ambient temperature, and the unavailability of their corresponding generators into consideration. The proposed models are based on the stochastic process using the Wiener process and the probability distribution functions to explain the randomness of the intermittency around their historical average data. Moreover, for solar PV—the generation of which depends on two random variables, namely solar irradiance and ambient temperature—a copula function is used to model their joint probabilistic behavior. For hydropower and bio-based power, uncertainties associated with their power generation process, i.e., in terms of resource availability, conversion process, and machine availability, are introduced. The capacity credit of RE generating units are determined by the ECP method, which is based on the principle of generation system reliability evaluation using LOLE as an indicator for system reliability.

The rest of the paper is organized as follows. Section 2 describes the evaluating framework of the capacity credit and proposed probabilistic models of six RE technologies. In addition, the data and assumptions for simulations are provided. Section 3 and Section 4 present the simulation results with a discussion and conclusion of the study, respectively.

2. Methodology and Data

In this section, the proposed capacity credit evaluation method and probabilistic models of RE power plants are presented. Moreover, the data and assumptions used as a case study are also presented in this section.

2.1. Capacity Credit Evaluation of RE Power Plants

In this work, a capacity credit of RE power plants is defined as an equivalent installed capacity of conventional power plants that provide the same reliability level of generation system, which is measured by LOLE [71]. In other words, the capacity credit is a ratio of the conventional capacity to the RE capacity where both of these capacities yield similar reliability levels, as illustrated in Figure 1. This definition is similar to the definition of the ECP method where the reliability data of any conventional power plant types can be used to represent the conventional generation.

The procedure to determine the capacity credit can be summarized as shown in Figure 2. Once the input data are obtained, the first process generates the stochastic operation models of the considered RE technology. By performing the simulation of RE generation using the generated models, RE resources’ availability and power output can be determined. Details of RE generation models including wind, solar PV, small hydro, biomass, biogas, and MSW power generations are explained in Section 2.2.

Then, to evaluate the system reliability measured by LOLE, Monte Carlo simulation is used. The operating statuses of all generating units including the RE power plants and other conventional power plants in the system are also simulated to create a generation operating cycle. The operating statuses of each generating unit are represented by the two-state Markov model, as shown in Figure 3. The status of each generator is either available (UP) during time to failure (TTF) or unavailable (DOWN) during time to repair (TTR) [72]. Typically, the TTF can be described by exponential distribution, as shown in Equation (1). TTR can also be described by either the exponential or normal distribution. However, the normal distribution is used in this paper as shown in Equation (2). The operating cycle of each generating unit during a considered period can be obtained by the Monte Carlo simulation technique using Equations (1) and (2), as shown in Figure 4. These obtained generating cycles of all generation units will be used to evaluate the LOLE of the system.

$$TTF=\frac{1}{\lambda }ln\left(1-U\right)$$

(1)

$$TTR=r+{\sigma }_{r}Z$$

(2)

where:

• $TTF$ is the time to failure or the duration of the generating unit availability;
• $TTR$ is the time to repair or the duration of the generating unit unavailability;
• $U$ is a random value with a uniform distribution in [0, 1];
• $Z$ is a random value with a standard normal distribution;
• $\lambda$ is the failure rate of each generating unit in a two-state Markov model;
• $r$ is the average repair time of each generating unit and equals 1/$\mu$;
• $\mu$ is the repair rate of each generating unit in a two-state Markov model;
• ${\sigma }_r$ is the variance of the repair time of each generating unit, modeled by 0.1r [71].

A system load is also generated using a load model expressed by Equation (3). Since the uncertainties of load data are caused by errors of load forecast, the load model is then built from the forecasted load values with their errors.

$$L_t=L_t^{forecast}+{\sigma }_{L,t}Z$$

(3)

where:

• $L_t$ is the load at time t;
• $L_t^{forecast}$ is the load forecast at time t;
• ${\sigma }_{L,t}$ is the variance of load forecast error, modeled by ${\sigma }_{L,t}=\alpha \sqrt{L_t}$ [73];
• $Z$ is a standard normally distributed random number, $Z~N\left(0,1\right)$.

Once generation and load cycles are generated, LOLE is calculated using Equation (4) [74]. This process are iteratively performed until the calculated LOLE converges.

$$LOLE=\frac{{\sum }_t^{T}d\left(P_t<L_t\right)}{T}\times 8760$$

(4)

where:

• $d\left(P_t<L_t\right)$ is the duration that the total power output of all units is less than load $L_t$;
• $T$ is the total considering period.

To determine the capacity credit, firstly, LOLE of the system with the considered RE generation unit is evaluated. Then, the system without that RE generation unit is considered. Then, a conventional generation unit is added into the system and the LOLE is evaluated. The capacity of the added conventional generation unit that yields the same LOLE of the system with the considered RE generation unit will be used to determine the capacity credit using the concept shown in Figure 1.

2.2. Stochastic Model of RE Generation

This section explains the proposed probabilistic models of the RE generating unit, i.e., wind, solar PV, small hydro, biomass, biogas, and MSW. These models are used to generate the power output of each RE generation unit during the simulation period, which is used to evaluate the LOLE of the system, and then its capacity credit, as explained in previous section. In the proposed model, each RE generation is divided into two parts: RE resource availability and the conversion of RE resource to power. Uncertainties of each RE resource’s availability and the conversion of REs are represented by a probabilistic model considering random variables associated with each RE technology. In addition, the resource availability of RE is modeled in different time scales for each RE type to capture different intermittency levels of each RE technology. For example, the wind speed, irradiance and temperature, which are highly intermittent, are simulated in a sub-hourly time scale with a different hourly average value by SDEs of a constant trend at each specific hour combined with noise. The water flow rate of small hydro generation, which is likely to be more stable throughout the day, is modeled in a daily time scale with different monthly average value. Lastly, the feedstock of bio-based power generation including MSW is varied by seasons so it can be forecasted in a monthly time scale with a different quarterly average value depending on the season. Details of each RE generation model are explained as follows.

2.2.1. Wind

A power output generated from a wind power plant depends on the capacity of a wind turbine generator and wind speed. Kinetic energy in wind is converted to mechanical energy by a wind turbine. Therefore, the model of wind power generation is comprised of two main models: a wind speed model and wind operation model.

• Wind Speed Model

A wind speed model consists of two main components which are an historical average hourly wind speed ($v_{trend,t}$) and its uncertainty ($v_{noise,t}$), as shown in Equation (5).

$$v_{w,t}=v_{trend,t}+v_{noise,t}$$

(5)

By assuming that the wind speed in each hour fluctuates around its hourly mean, so that $d{v}_{trend,t}=0$. Thus, only the noise is modeled as a random variable using the Wiener process. In this paper, $v_{noise,t}$ is assumed to fluctuate with zero mean and the magnitude of the random fluctuation depends on the wind speed at that time. Hence, its model can be written in the form of SDE as shown in Equation (6). By applying Ito’s lemma [75,76] to Equation (6), a wind speed at time $t$ can be solved as shown in Equation (7). The detailed proof of the solution can be found in Appendix A.

$$d{v}_{w,t}=d{v}_{trend,t}+d{v}_{noise,t}=0+\left(0dt+\sigma v_{w,t}d{W}_t\right)={\sigma }_{v_{trend,t}}{v}_{w,t}{Z}_w\sqrt{dt}$$

(6)

$$v_{w,t}=v_{trend,t}·exp\left(-\frac{1}{2}{\sigma }_{v_{trend,t}}^{2}t+{\sigma }_{v_{trend,t}}{Z}_w\sqrt{t}\right)$$

(7)

where:

• ${\sigma }_{v_{trend,t}}$ is the variance of historical hourly wind speed at hour $t;$
• $d{W}_t$ is the Wiener process, where $d{W}_t=Z_w\sqrt{dt}$;
• $Z_w$ is a standard normally distributed random number.

2.

Wind Operation Model

A basic wind operation model with three components is considered, as shown in Figure 5. Basically, a wind generator will operate when the wind speed is high enough to turn the blades of a wind turbine. The blades of a wind turbine, which are connected to a gearbox and shaft, will drive a generator to convert mechanical power to electrical power.

The electrical power output of wind power generation depends on the wind speed. The operation mode of the wind power is classified into four levels [77] as shown in Figure 6:

• Cut-in wind speed $\left(v_{ci}\right)$: At this wind speed, a wind turbine starts moving leading to a generator gradually generating electrical power;
• Maximum rotor efficiency wind speed: This is a range of wind speed between the cut-in and a rated wind speed where power output depends on wind speed value;
• Rated or nominal output wind speed $\left(v_r\right)$: Once the wind speed reaches this level, the generator comes to its maximum capacity in generating electrical power;
• Cut-out wind speed $\left(v_{co}\right)$: It is the maximum wind speed at which the turbine is allowed to be operated. At this level, wind blades are set to be aligned with a wind flow to prevent the turbine from moving to avoid damage from excessively high wind speed.

With these wind speed levels, the output power of a wind power plant in a steady state operation can be calculated by Equation (8) [78], and then the real electric power that is injected in a power system is calculated by Equation (9).

$$P_w=\{\begin{array}{cc}0& v_w<v_{ci}, v_{co}<v_w\\ \left(\frac{v_w^3-v_{ci}^3}{v_r^3-v_{ci}^3}\right)·P_{R,w}& v_{ci}<v_w<v_r\\ P_{R,w}& v_r<v_w<v_{co}\end{array}$$

(8)

$$P_{e,w}={\eta }_w·P_w$$

(9)

where:

• $P_{R,w}$ is the rated power (W);
• ${\eta }_w$ is the efficiency of a corresponding converter.

2.2.2. Solar PV

Generally, the power output from the solar PV power plant depends on solar irradiance and ambient temperature [68]. Higher solar irradiance usually leads to higher power output whereas higher ambient temperature causes lower power output. However, solar irradiance and ambient temperature are dependent to some extent as a low temperature is not expected on a sunny day with high solar irradiance or a high temperature is not expected on a day with low solar irradiance. Their correlation should be considered in the simulation of generating solar irradiance and ambient temperature for the calculation of the power output of the solar PV power plant.

• Solar Irradiance and Ambient Temperature Models

Both solar irradiance $\left(G_t\right)$ and ambient temperature $\left(T_t^{amb}\right)$ models comprise two main components, which are the historical average and their uncertainty, as shown in Equations (10) and (11).

$$G_t=G_{trend,t}+G_{noise,t}$$

(10)

$$T_t^{amb}=T_{trend,t}^{amb}+T_{noise,t}^{amb}$$

(11)

In Equations (10) and (11), $G_{trend,t}$ and $T_{trend,t}^{amb}$ are observable hourly average values for each hour $t$. Both $G_{noise,t}$ and $T_{noise,t}^{amb}$ are random variables modeled by the SDE, as shown in (12) and (13), similar to that of the wind model. By applying Ito’s lemma, both solar irradiance and ambient temperature at time $t$ can be given as Equations (14) and (15), respectively. However, as mentioned earlier, solar irradiance and ambient temperature are dependent, whilst the $Z_G$ and $Z_T$ in (14) and (15) are not generated independently. Their correlation is taken into accounted by introducing the joint probability distribution modeled by the Copula function.

$$d{G}_{noise,t}=0dt+{\sigma }_{G_{trend,t}}{G}_{t}d{W}_t={\sigma }_{G_{trend,t}}{G}_t{Z}_G\sqrt{dt}$$

(12)

$$d{T}_t^{amb}= 0dt+{\sigma }_{T_{trend,t}^{amb}}{T}_t^{amb}d{W}_t={\sigma }_{T_{trend,t}^{amb}}{T}_t^{amb}{Z}_T\sqrt{dt}$$

(13)

$$G_t=G_{trend,t}·exp\left(-\frac{1}{2}{\sigma }_{G_{trend,t}}^{2}t+{\sigma }_{G_{trend,t}}{Z}_G\sqrt{t}\right)$$

(14)

$$T_t^{amb}=T_{trend,t}^{amb}·exp\left(-\frac{1}{2}{\sigma }_{T_{trend,t}^{amb}}^{2}t+{\sigma }_{T_{trend,t}^{amb}}{Z}_T\sqrt{t}\right)$$

(15)

where:

• ${\sigma }_{G_{trend,t}}$ is the variance of historical hourly solar irradiance at hour $t;$
• ${\sigma }_{T_{trend,t}^{amb}}$ is the variance of historical hourly ambient temperature at hour $t$;
• $Z_G$ is a standard normally distributed random number which is correlated to $Z_T;$
• $Z_T$ is a standard normally distributed random number which is correlated to $Z_G.$

2.

Copula Application for Joint Probability Distribution

The relationship between random variables of solar irradiance and ambient temperature can be described by copulas in the form of a joint cumulative distribution function (CDF) [79]. The central theory for the application of copulas is Sklar’s theorem [80], where $H$ is a joint CDF of random variables if it has a copula $C$, as stated in Equation (16).

$$H\left(Z_G,Z_T\right)=C\left(\Phi \left(Z_G\right),\Phi \left(Z_T\right)\right)$$

(16)

where:

• $\Phi \left(Z_G\right)$ is a standard normal distribution of $Z_G=\left(G_t-G_{trend,t}\right)/\left({\sigma }_{G_{trend,t}}{G}_{trend,t}\right);$
• $\Phi \left(Z_T\right)$ is a standard normal distribution of $Z_T=\left(T_t^{amb}-T_{trend,t}^{amb}\right)/\left({\sigma }_{T_{trend,t}^{amb}}{T}_{trend,t}^{amb}\right)$.

In this work, the correlation of solar irradiance and ambient temperature is classified into two forms using Gumbel and Clayton copula functions, as shown in Figure 7:

• Gumbel copula exhibits strong correlation at high values, which correspond to the nature of solar when high solar irradiance comes with a high ambient temperature. However, at the low values, Gumbel copula has weak correlation;
• Clayton copula has strong left tail dependence which captures the nature of solar when low solar irradiance comes with low ambient temperature.

In this study, copula application is applied to solar PV model since solar irradiance and ambient temperature are correlated to each other. According to Thailand’s historical data, in summer, solar irradiance and ambient temperature can be described by Gumbel copula since strong correlation at high values is found. Moreover, in winter, solar irradiance and ambient temperature can be described by Clayton copula since correlation at low values is found.

3.

Solar PV Operation Model

Since the power output of solar PV modules depends on two key factors, namely solar irradiance and ambient temperature, a basic operation model of solar PV modules can be shown in Figure 8. Based on the characteristic curves in Figure 9, the power output of solar PV modules decreases when the ambient temperature increases, while the power output increases when the solar irradiance increases.

The relationship of irradiance and ambient temperature to the power output ($P_{pv}$) can be expressed by Equations (17) and (18) [82,83] and the real electric power ($P_{e,pv}$) that is converted from direct current to alternative current and injected in a power system is calculated by Equation (19) as follows.

$$P_{pv}=P_{R,pv}·\frac{G_t}{G_{STC}}·\left(1-\gamma \left(T_{cell}-T_{STC}\right)\right)$$

(17)

$$T_{cell}=T_{amb}+\left(\frac{T_{NOC}-20°\mathrm{C}}{800}\right)G_t$$

(18)

$$P_{e,pv}={\eta }_{pv}·P_{pv}$$

(19)

where:

• $P_{R,pv}$ is the rated power of solar PV (W);
• $G_{STC}$ is the solar irradiance at the standard test condition (1000 W/m²);
• $\gamma$ is the temperature coefficient $\left({°\mathrm{C}}^{-1}\right)$ which is −0.005 to −0.003;
• $T_{cell}$ is the solar cell temperature $\left(°\mathrm{C}\right)$;
• $T_{STC}$ is the standard test condition temperature (25 $°\mathrm{C}$);
• $T_{amb}$ is the ambient temperature $\left(°\mathrm{C}\right)$;
• $T_{NOC}$ is the nominal operating cell temperature (46 $°\mathrm{C}$);
• ${\eta }_{pv}$ is efficiency of a corresponding converter.

2.2.3. Small Hydro

Small hydro power plants are classified into three types: pumped storage; regulating small reservoir; and run-of-river hydro power plants [69]. Among these three types, the run-of-river hydro power plant experiences the highest uncertainty in generating electricity since it has no water storage to harness the natural downward flow of a river. As a result, in this work, we only consider the run-of-river hydro power plant, which comprises two main following models: namely mass flow rate and generation model.

• Mass Flow Rate Model

A natural downward flow of water of the run-of-river hydro powerplant can basically be modeled by mass flow of water that hits hydro turbine blades. As it is assumed that water flow is natural, the water flow rate at time $t \left(m_t\right)$ is given by a normal distribution of mean $\left(m_{avg}\right)$ and variance $\left({\sigma }^2\right)$ or $m_t~N\left(m_{avg},{\sigma }^2\right)$. For instance, by considering the historical data of the water flow of a small hydro generation, the average water flow rate and standard deviation for the given data are 60 liters/second and 10% of the average water flow rate, respectively. The mass flow rate model of this small hydro generation can be shown as Figure 10.

2.

Small Hydro Operation Model

Basically, the power output of run-of-river hydro generation depends on three key components, namely mass flow rate, hydro turbine, and generator, as shown in Figure 11. The power output ($P_t)$ from a generator at time $t$ can be expressed as Equation (20) [84] while the generator is designed for rated power output ($P_{rated}$) corresponding to the rated water flow rate ($m_{rated}$) as expressed by Equation (21). Therefore, $P_t$ can be rewritten in Equation (22) as follows.

$$P_t=m_t·g·H_{net}·{\eta }_{system}$$

(20)

$$P_{rated}=m_{rated}·g·H_{net}·{\eta }_{system}$$

(21)

$$P_t=\frac{m_t}{m_{rated}}·P_{rated}$$

(22)

where:

• $g$ is the acceleration due to gravity at the surface of the earth (9.81 m/s²);
• $H_{net}$ is the head or falling height (m);
• ${\eta }_{system}$ is the efficiency of hydro generation.

2.2.4. Bio-Based Power and MSW

In this work, a bio-based power plant includes biomass, biogas, and municipal solid waste (MSW). Bio-based generation shown in Figure 12 is represented by two main models: the feedstock availability model and the operation model. In these models, the feedstock availability and the capability of converting feedstock to power are considered as high uncertainty factors that lead to a difference in power output. The Weibull distribution is used for modeling fuel availability due to the fact that it is harder to acquire a large amount of bio-based feedstock beyond its average level. For the fuel conversion process, normal distribution or Weibull distribution is used depending on the conversion types. Finally, the system availability is modeled by the equipment two-state model represented by the forced outage rate (FOR).

• Feedstock Availability Model

The same feedstock availability models are applied for biomass, biogas, and MSW since the material sources of these fuels come from similar sources, particularly solid and liquid waste from agricultural, municipal, and industrial activities. Biomass feedstocks are from various sources, such as wood products and agricultural residues. Their properties, i.e., availability, heating value, moisture, and organic and inorganic contents affect the design of combustion equipment and the performance of biomass in generating electricity [85]. MSW feedstocks are waste from municipal or industrial activities. Their properties and components, such as their share of combustible and non-combustible waste, also determine the technology used for generating electricity [86]. For biogas, which is generated from a decomposition of organic waste and wastewater, different feedstock materials lead to different biogas potential due to various factors of biogas production [87].

For the feedstock availability model, feedstock availability is considered as a percentage of an installed capacity and varies from season to season. We assume that the bio-based materials are likely not to be supplied at the full capacity of a power plant due to limitations in accessing and collecting those materials. In addition, it is also assumed that bio-based feedstocks are easily acquired at an average amount. However, it is unlikely to acquire the feedstocks above the average amount considering the difficulty in sourcing bio-based materials in the large amount. Thus, the feedstock availability model is expressed by a random variable with Weibull distribution, as shown in Equation (23). The example of feedstock availability with a calibrated location parameter of 30, shape parameter of 2.48, and scale parameter of 45.2 is shown in Figure 13. This example of probability distribution indicates that mostly the feedstock can be procured at approximately 70% of its full capacity.

$$f\left(x_t\right)=\frac{k}{\lambda }{\left(\frac{x_t-\theta }{\lambda }\right)}^{k-1}{e}^{-{\left(\frac{x_t-\theta }{\lambda }\right)}^k}; x_t>0$$

(23)

where:

• $x_t$ is the feedstock availability of the considered fuel type at time $t;$
• $\theta$ is the location parameter;
• $k$ is the shape parameter;
• $\lambda$ is the scale parameter.

2.

Bio-based Conversion Model

In this study, the operation models for biomass, biogas and MSW technology are similar in terms of power output dependance on feedstock availability, conversion factor, and system availability. However, for MSW technology, although its conversion system is the same as that of biomass technology, the model also considers waste sorting to sort out materials that can be converted to refuse-derived fuel (RDF). Details of the operation model for biomass, biogas, and MSW technology can be explained as follows.

• Biomass Operation Model

The biomass operation model consists of three main components: conversion technology, a turbine, and a generator, as shown in Figure 14. Conversion technologies of biomass include various technologies [88], but direct combustion, gasification, and pyrolysis are well-known technologies used around the world [15,89]. For the direct combustion technology, biomass is burnt to heat the boiler to produce high pressure steam to move the steam turbine connected to a generator. On the other hand, gasification and pyrolysis are an incomplete combustion process whereby biomass can be converted into synthesis gas containing carbon monoxide, hydrogen, carbon dioxide, and methane [90]. This synthesis gas, with a heating value of approximately 4.5–5.5 MJ/m³, can be used in a gas engine generator. Gasification is well commercialized and yields a higher portion of synthesis gas compared to pyrolysis.

B.

Biogas Operation Model

Like the biomass operation model, the biogas operation model consists of conversion technology, a turbine and a generator, as shown in Figure 15. There are various biogas technologies such as up-flow anaerobic sludge blanket (UASB), anaerobic fixed film or anaerobic covered lagoon, and continuous-flow stirred tank reactor (CSTR). Produced biogas mainly contains methane (CH₄), carbon dioxide (CO₂), and hydrogen sulfide (H₂S). A production of biogas relies on various factors, i.e., feedstock materials, process, solid concentration, etc. Therefore, it is quite difficult to predict the biogas yield.

C.

MSW Operation Model

The MSW operation model has the same components as the biomass operation model. However, waste sorting is included in the MSW operation model to produce RDF, as shown in Figure 16.

3.

System Availability

Since the power generation process of these three bio-based power generations consists of high-speed rotating machine, the availability of the conversion system is considered in this paper the system availability. The system availability is modeled by a two-state model represented by a uniform distribution with the forced outage rate (FOR), as shown in Figure 17.

With the consideration of feedstock availability, conversion model, and system availability, the power generating capability ($C_{Bio-x, t}$), and power output ($P_{e,Bio-x,t}$) of biomass or biogas generation at time $t$ can be expressed by Equations (24) and (25), respectively. For biomass, the uncertainty of the conversion factor is expressed by a normal distribution as the conversion depends on a specific technology and capacity. However, for biogas, the uncertainty of the conversion factor is given as the Weibull distribution since it is difficult to control the level of produced CH₄ in biogas; it therefore requires excessive effort to increase the CH₄ level above the average.

$$C_{Bio-x, t}=x_t·C{F}_{Bio-x}·A{V}_{Bio-x}$$

(24)

$$P_{e,Bio-x,t}=C_{Bio-x,t}·InCa{p}_{Bio-x}$$

(25)

where:

• $C{F}_{Bio-x}$ is the conversion factor of fuel to power (%);
• $A{V}_{Bio-x}$ is the availability of biogas or biomass system, which is either 0 or 1;
• $InCa{p}_{Bio-x}$ is the installed capacity of biomass or biogas generation.

Not all MSW feedstock cannot be converted into power. Only some of the waste matter ($A_t$) can be turned into electricity. This parameter is expressed by a random variable with Weibull distribution. From the waste matter that can be turned into electricity, only some of them can be converted to RDF ($RD{F}_t$). Thus, the feedstock is sorted into two groups, which are for waste that is able and unable to be converted to RDF. Therefore, the power generating capability of the RDF ($C_{MSW,1, t}$) and the remaining sorted ($C_{MSW,2, t}$) at time $t$ can be expressed by Equations (26) and (27), respectively. Power output ($P_{e,W,t}$) of the MSW generation at time $t$ relies on the power generating capability and installed capacity ($InCap$) of both parts, as shown in Equation (28).

$$C_{MSW,1,t}=\left(x_t{A}_t\right)·RD{F}_t·C{F}_{MSW,1}·A{V}_{MSW}$$

(26)

$$C_{MSW,2,t}=\left(x_t{A}_t\right)·\left(100\%-RD{F}_t\right)·C{F}_{MSW,2}·A{V}_{MSW}$$

(27)

$$P_{e,W,t}=C_{MSW,1,t}·InCa{p}_{MSW,1}+C_{MSW,2,t}·InCa{p}_{MSW,2}$$

(28)

where:

• $A_t$ is the waste matter that can be turned into electricity;
• $RD{F}_t$ is the yield of RDF from waste matter that can be turned into electricity;
• $C{F}_{MSW,1}$ is the conversion factor of the RDF;
• $C{F}_{MSW,2}$ is the conversion factor of the remaining sorted MSW;
• $A{V}_{MSW}$ is the availability of the MSW system, which is either 0 or 1.

2.3. Data and Assumptions

2.3.1. Generation System and Load

The generation system used to perform the capacity credit evaluation in this study is a part of Thailand’s Power Development Plan 2010. It is the generation system with the total installed capacity of 29,231.44 MW. An hourly load profile with the peak load of 26,355 MW is shown in Figure 18. More detail on the generation system is provided in Appendix B.

Fuels for each conventional power plants, e.g., natural gas and coal, are assumed to always be available for generating electricity through the considered timeframe, which is 40 years. The power generation and load profile used in assessing the capacity credit are sampled every 15 min. A target of LOLE for this generation system is set to be 1 day/year or 0.0027%. In the evaluation of capacity credit, the conventional generation used to find the equivalent capacity can be a thermal or combined cycle power plant since the reliability data of these two generation units are the same as details shown in Appendix B.

In this paper, the simulation for the evaluation of the capacity credit of the RE power plant was assessed for a period of 40 years. In addition, the values of capacity credit at specific periods including 2–3 PM (peak load of the day), and 10 PM–5 AM (the light load) are focused. The capacity credit is also assessed for different RE capacity sizes to investigate their impacts on the capacity credit. However, the impact of the dynamic behavior of the RE on system stability is not considered in this work.

2.3.2. RE Generation Data

This study aims to evaluate the capacity credit of six types of RE generation including wind, solar PV, small hydro, biomass, biogas, and waste. The data of RE used for assessing capacity credit are as follows:

• Wind: The wind data used in the wind generation model include the monthly average wind speed, hourly mean, and standard deviation of wind speed at each hour, as shown in Figure 19. The cut-in ($v_{ci}$ ), rated ($v_r$), and cut-out ($v_{co}$) wind speed are given as 2.8, 12.5, and 23 m/s, respectively. The efficiency of a corresponding converter (${\eta }_w$) is 98%.
• Solar PV: The solar data used for the solar PV generation model consist of the hourly mean and standard deviation of solar irradiance and ambiance temperature for summer, rainy, and winter season, as shown in Figure 20. The correlations of solar irradiance and ambient temperature during 7 AM–1 PM and 2–6 PM are given as Gumbel copula and Frank copula, respectively. The temperature coefficient ($\gamma$) is –0.0035 ${°\mathrm{C}}^{-1}$ and the nominal operating cell temperature ($T_{NOC}$) is 46 °C. The efficiency of a corresponding converter (${\eta }_s$) is 98%.
• Small hydro: Water flow rate of small hydro generation is given as a normal distribution with the average water flow rate ($m_{avg}$) of 50 liters/s and standard deviation ($\sigma$) of 25% of the mean. The rated power output ($P_{rated}$) of the generator is 37 kW with the rated water flow rate ($m_{rated}$) of 60 l/s. The probability distribution function of water flow rate is shown in Figure 21a.
• Biomass: The average feedstock availability of biomass in Thailand is assumed to be 60–70% of the installed capacity. Given that, the feedstock availability is represented by the Weibull distribution with calibrated location, shape, and scale parameters of 30, 2.48, and 45.2, respectively. The CF of the biomass generation system is expressed by the normal distribution with a mean of 95% and a standard deviation of 1% of the mean. The AV of conversion system is assumed to be equivalent to the forced outage rate (FOR) of 10%. The probability distribution function of feedstock availability and biomass conversion factor are shown in Figure 21b,c, respectively.
• Biogas: In Thailand, it is assumed that the average ratio of CH₄ in biogas production is 55% [91] and is modeled by the Weibull distribution with location, shape, and scale parameters of 0, 25, and 55, respectively. With this CH₄ ratio, the CF of biogas generation is then computed from the linear transformation that a CH₄ of 55% equals the conversion factor of 100% or CF = CH₄/0.55. The AV of the conversion system is assumed to be equivalent to the FOR of 15%. The probability distribution function of biogas production and biogas conversion factor are shown in Figure 21d,e, respectively.
• MSW: The waste matter that can be turned to electricity is expressed by the Weibull distribution with the location, shape, and scale parameters of 0.7, 12, and 90, respectively. With this distribution, the waste used for electricity generation is 90% on average. The yield of RDF is represented by the normal distribution with a mean of 20% and a standard deviation of 1% of the mean. Because RDF largely consists of combustible components, the $C{F}_{MSW,1,t}$ is given at a constant of 100%, while, for non-RDF, the $C{F}_{MSW,2,t}$ is described by the normal distribution with a mean of 95% and standard deviation of 1% of the mean. The AV of conversion system is assumed to be equivalent to the FOR of 10%. The probability distribution function of waste used for electricity generation, a yield of RDF, and non-RDF conversion factor are shown in Figure 21f,g,h, respectively.

3. Results and Discussions

The results of the capacity credit evaluation of wind, solar PV, small hydro, biomass, biogas, and MSW for different time periods and RE capacities are discussed. The capacity credits of all six of these generation types throughout the day, during the day peak period (2–3 PM), and night peak period (7–8 PM) at different installed capacities are shown in Figure 22 and their average values are shown in

Table 2. Capacity credit of RE generation.

Generation Type	Capacity Factor (%)	Capacity Credit (%)
		All Day	Day Peak (2–3 PM)	Night Peak (7–8 PM)	PDP2010r3 [47]	PDP2018 [48]
Wind	11.57	11.37	10.73	10.46	2	14
Solar PV	19.19	36.27	46.88	0.00	14	42–50
Small hydro	70.85	69.03	69.34	68.20	36	77
Biomass	58.55	60.13	60.87	61.63	36	52–80
Biogas	53.31	52.04	52.85	54.78	0	28–70
Waste	54.12	55.60	56.04	57.51	36	47–70

. Examples of power output from all RE generation models are shown in Appendix C.

It was found that the capacity credit of all RE, except the solar PV, during the day and night peak periods, are slightly different. It was also found that the capacity credit is slightly lower if the installed capacity of RE generation is higher since the effect on LOLE of an outage of large power plants is more significant than smaller units. It should be noted that the capacity credit is independent of the capacity factor of the RE. Although the values of the capacity factor and all-day capacity credit of all RE types, except the solar PV, are quite similar, their calculation methods are different in the way that the capacity credit does take the load and system reliability into account, whereas the capacity factor does not. The capacity credits calculated by the proposed method are also compared to the dependable capacity shown in Thailand’s Power Development Plan 2010 revision 3 (PDP2010r3) [47] and Thailand’s Power Development Plan 2018 (PDP2018) [52]. These dependable capacities are created by a deterministic method from the selected percentile or an average value of the hourly power output on the specific hour of several sample RE power plants. For PDP2010r3, the fifth percentile is used as a criterion to define the capacity credit, whereas for PDP2018, an average value is used. Since these dependable capacities are only used to be compared with the peak load when calculating the system’s reserve margin, the observed period is only during day peak where the peak load occurs.

The discussions of each RE generation type are below.

• Wind: Due to the continuous power output of wind generation, the capacity credits throughout the day, during the day peak period (2–3 PM), and night peak period (7–8 PM) are slightly different, which are in the range of 9.57–12.96%. This difference is caused by the difference in average hourly wind speed and noise in the proposed wind generation model used in this study since wind power generation is modeled hourly.
• Solar PV: The average capacity credits throughout the day is lower than the capacity credit during the day peak period (2–3 PM) since solar PV generation can only produce electricity during daytime (7 AM–6 PM). During the peak load period, the existence of solar PV generation can significantly enhance the LOLE. On the other hand, solar PV generation has no capacity credit during the night peak period.
• Small hydro: The average capacity credits of small hydro generation throughout the day, during the day peak period, and during the night peak period are slightly different, which are in the range of 64.45–73.05%. Since small hydro generations are modeled daily, the difference between each time of day is caused by the LOLE of the system during that period.
• Biomass: Since the power output of biomass generation is consistent, its average capacity credits throughout the day, during the day peak period, and during the night peak period are not significantly different, which are in the range of 57.57–67.38%. Due to the lack of solar PV generation during the night peak period, the LOLE is more sensitive to other types of RE generation that can supply electricity during night peak period, especially during the peak period when LOLE is quite high. Thus, the capacity credit of biomass generation during the night peak period is slightly higher.
• Biogas: Due to the nearly identical generation model, the capacity credit trend of biogas generation throughout the day, during the day peak period, and during the night peak period are almost identical to that of biomass. However, the values, which are in the range of 49.02–56.25%, are lower than those of biomass generation since the biogas generation model has a higher uncertainty from a biogas production and availability of conversion system. It was also found that the capacity credit of the biogas generation during the night peak period is slightly higher than the day peak period.
• MSW: The capacity credit trends of MSW generation for all different considered periods resemble those of biomass generation since both of them have similar generation models, except for the uncertainty of waste matter which can be turned into electricity and the ratio between RDF and non-RDF. These uncertainties lead to slightly lower capacity credits of MSW generation, which are in the range of 52.95–59.96%.

By comparing the calculated capacity credits acquired by the proposed method with the dependable capacity used in PDP2010r3 and PDP2018, shown in Table 2, it was found that the dependable capacities used in both power development plans are extremely different. It was also found that the dependable capacity of all six types used in PDP2010r3 are quite low compared to the capacity credits evaluated by the proposed method. However, the upper value of the dependable capacity used in PDP2018 is much higher than those of the capacity credits evaluated by the proposed method. This is because the determined criterion (fifth percentile) for the dependable capacity used in the PDP2010r3 might be too low, and the determined criterion (average value) for the dependable capacity used in the PDP2018 might be too high. Thus, it is obviously seen from these two plans that there is still no any rigorous criterion to determine the capacity credit of renewable energy in Thailand.

4. Conclusions

This paper proposes an evaluation of capacity credit for six RE types including wind, solar PV, small hydro, biomass, biogas, and MSW along with stochastic models to represent the resource availability of those RE generation types. The resource availability models of wind and solar PV are represented hourly by an SDE model since wind and solar resources are highly intermittent and continuously time-varying. In addition, since the proposed models are used to evaluate the capacity credit of the RE that is then used to compare with peak load when determining the reserve margin, the time of the day must be concerned. Thus, SDE models that allow the sequential Monte-Carlo simulation to be performed, which enable the consideration of time-dependent simulation results, are selected. Moreover, the copula application of solar irradiance and ambient temperature is introduced to create a joint probability distribution for representing the correlation between solar irradiance and ambient temperature. For the remaining considered RE technologies, the resource availability model of small hydro is represented daily with a normally distributed mass flow rate model since it is close to the natural process of water flow. Lastly, the resource availability model of bio-based power is represented monthly with Weibull distribution since it is assumed that it is harder to acquire a large amount of bio-based feedstock beyond its average level. In this study, the copula application is only applied to solar PV model. In other renewable energy generation types, the correlation between random variables determining their output power are not so strong. For wind and small hydro power generations, there is only one major variable, which is the wind speed and water flow rate, respectively. For bio-based power generation, although there are three main factors, which are feedstock availability, conversion factor, and system availability, these three factors are considerably independent.

To evaluate the capacity credit with the proposed resource availability models, the power outputs of RE power plants are simulated and then the LOLE of the system with RE power plants is calculated using the Monte Carlo simulation technique. Those RE power plants are then removed and several conventional power plants are added into the system instead until the new system’s LOLE is equal to the LOLE of the system with the RE power plants calculated earlier. Finally, the installed capacity of additional conventional power plants is compared with the installed capacity of RE power plants to evaluate the capacity credit. According to the result, it was found that the capacity credit acquired by the proposed method is quite different from the dependable capacity of RE generation used in Thailand’s earlier power development plans. However, since the capacity credit from the proposed method was evaluated based on a probabilistic model derived from the historical data of RE resources’ availability and the availability of the generation system rather than the deterministic method—based on a system discretion planner as used in Thailand’s power development plans—the proposed method can provide a reasonable and more accurate capacity credit of RE generation. The findings from this study may provide crucial information about how to determine the capacity credit for renewable energy generation in other countries.