Generating capacity adequacy evaluation of large-scale, grid-connected photovoltaic systems

Abstract

Large-scale, grid-connected photovoltaic systems have become an essential part of modern electric power distribution systems. In this paper, a novel approach based on the Markov method has been proposed to investigate the effects of large-scale, grid-connected photovoltaic systems on the reliability of bulk power systems. The proposed method serves as an applicable tool to estimate performance (e.g., energy yield and capacity) as well as reliability indices. The Markov method framework has been incorporated with the multi-state models to develop energy states of the photovoltaic systems in order to quantify the effects of the photovoltaic systems on the power system adequacy. Such analysis assists planners to make adequate decisions based on the economical expectations as well as to ensure the recovery of the investment costs over time. The failure states of the components of photovoltaic systems have been considered to evaluate the sensitivity analysis and the adequacy indices including loss of load expectation, and expected energy not supplied. Moreover, the impacts of transitions between failures on the reliability calculations as well as on the long- term operation of the photovoltaic systems have been illustrated. Simulation results on the Roy Billinton test system has been shown to illustrate the procedure of the proposed frame work and evaluate the reliability benefits of using large-scale, grid-connected photovoltaic system on the bulk electric power systems. The proposed method can be easily extended to estimate the operating and maintenance costs for the financial planning of the photovoltaic system projects.

Keywords

adequacy assessment / Markov method / large-scale grid-connected photovoltaic(PV) systems / long-term operation

Introduction

There has been a rapid growth in the development of renewable energies in recent years [ 1]. Of renewable energy sources, solar energy is the most plentiful and pollution free energy source that has been harnessed so far [ 2]. An enormous amount of research has been conducted and vast improvements have been made in the performance of photovoltaic (PV) systems [ 3].

Utilization of solar energy by photovoltaic equipment requires no mechanical medium, thus, efficiency is higher [ 4]. The robustness and simplicity in design, as well as little maintenance required for photovoltaic devices, make them much easier to use and the most cost efficient generator of electricity compared to other renewable energies [ 5]. Because of the multiple applications of solar energy, requests for photovoltaic devices are increasing each year [ 6].

The manufacturers mainly assume that a PV system is quite reliable with a high efficiency that can easily past 25 years in operation [ 7]. However, the reliability of various components in the PV system affects the power output of the PV system. The reliability estimation of the PV systems has been one of the most crucial issues in the last few decades.

In the literature, there are several probabilistic reliability evaluation methods including fault tree analysis (FTA), Monte Carlo, and the Markov method. Additionally, reliability evaluation of the large-scale, grid-connected PV systems has drawn significant research attention based on the various methods. For instance, the performance degradation of PV systems based on the Markov method has been investigated [ 8]. The Markov method has also been used [ 9] for the development of an energy storage model for the power supply reliability assessment of a PV generation system. An analytical approach has been proposed [ 10] in order to evaluate the reliability of large-scale, grid-connected PV systems.

The fault tree method with an exponential probability distribution function has been used to analyze the components of large-scale PV systems. On the other hand, in the technical literature, there are several proposed methods to estimate the reliability of different PV components (e.g., inverter [ 11], capacitor [ 12], and IGBT [ 13]), while others focus on the simulating PV modules and arrays. For instance, a technique based on the simplified bird mating optimizer is proposed to estimate the electrical equivalent circuit parameters of photovoltaic arrays [ 14]. The performance of mono-crystalline silicon type PV modules at different tilt angles and orientations has been discussed [ 15]. Another study has determined the electrical characteristics and temperature equations of the PV arrays installed in the tropics based on Sandia National Laboratory model [ 16]. The operation strategy of the grid-connected PV systems integrated with the battery energy storage has been studied [ 17]. A method based on the evaluation of new current and voltage indicators for automatic fault detection in grid connected PV systems has been proposed [ 18].

This paper presents a novel method using an exponential distribution for estimating the reliability impact on the bulk electric power systems incorporating large-scale, grid-connected PV systems. An analytical reliability evaluation framework based on the Markov method is used to quantify the adequacy studies of the bulk power systems considering PV systems. The Markov method serves as a useful tool to explain system states and behaviors and the transition between these states as well as the generation capacity of each state for the PV systems.

The Markov chain is well suited to evaluate the availability of systems based on a continuous stochastic process which follows two assumptions. The first assumption is that the system has neither the long memory nor the short memory. In other words, the future probabilities are only a function of the existing condition of the current state. The second assumption is that the transition states are constant and do not vary with the time.

The failure rate of the component of PV systems is considered in the proposed model to assess a comprehensive reliability framework. System adequacy is estimated using the conventional loss of load expectation (LOLE) and expected energy not supplied (EENS) indices.

PV systems are mainly less reliable than the conventional generating units and this effect is illustrated in the case studies. Furthermore, the implementations are taken into account by considering replacing conventional generating units with the PV systems.

Brief description of large-scale, grid-connected PV systems

The large-scale, grid-connected PV system consists of series components such as PV modules, string protection, DC switch, inverter, AC circuit breaker, grid protection, AC switch, differential circuit breaker, connector (couple), battery system, and charge controller, which are shown in Fig. 1. It is worth noting that some components can be omitted in real installations. The PV module and inverter characteristics are demonstrated as

{{\displaystyle V}}_{\max }=\mathrm{1000} V,{{\displaystyle I}}_{DC,\max }=\mathrm{235} A, Inverter(\mathrm{100}kW),{{\displaystyle V}}_{mpp,\max }=\mathrm{450} V,{{\displaystyle V}}_{mpp,\max }=\mathrm{820} V,

{{\displaystyle \mu }}_I=\mathrm{3.3}mA/°C,{{\displaystyle V}}_{oc}=\mathrm{37.2}V,{{\displaystyle I}}_{mpp}=\mathrm{7.60}A,{{\displaystyle \mu }}_v=-\mathrm{120}mV/°C. PV\mathrm{mod}ule(\mathrm{230}W),{{\displaystyle I}}_{SC}=\mathrm{8.24} A,{{\displaystyle V}}_{mpp}=\mathrm{30.2}V,

The PV system operation is assumed to operate 8.5 h per day. Thus, the failure rates for all components are failures/hour. Table 1 lists the failure rates of the component of PV systems [ 10]. During the calculation and analysis in this paper, the following assumptions are made:

1) The PV system can operate between 0% and 100% of its rated capacity and the failure rate of each component is constant at any load operations. Other electrical and electronic components are assumed to be 100% reliable. So, the failure rate of cables and the transformer are not considered. In other words, the implementations are assumed to have a flawless system installation.

2) The inverter is protected by the surge protection device (SPD) from lightning. It is assumed that the SPD is flawless with no failure rate.

Figure 2 depicts the adequacy evaluation model for a composite power system and the basic system structure including large-scale, grid-connected PV systems.

Reliability analysis methodology

General model

The reliability describes quantitative terms of the security of the system, the probability of a component performing its function sufficiently, the probability of uninterrupted power supply and outage avoidance. The term adequacy is defined as having adequate electricity generating capacity to supply the load by defined customer requirements. In a series system with n components, the failure in each single component leads to an overall system failure. The time to failure or the lifetime of a component is defined by a random parameter T. The cumulative distribution function (CDF) represents the probability of failure in the interval [0,t], which can be obtained as

(1) F(t)=P(T⟨t),\mathrm{0}⟨t.

If there is no failure in the interval [0,t], the reliability function is

(2) R(t)=P(T⟩t)=\mathrm{1}-F(t).

The system reliability can also be expressed by the failure rate of each component in the system as

(3) R(T)=e^{-({\lambda }_{\mathrm{1}}+{\lambda }_{\mathrm{2}}+\mathrm{...}+{\lambda }_n)\times T}.

If the failure rate of all components are equal, there is

(4) R(T)=e^{-n\times \lambda \times T}.

Moreover, it is not difficult to show that

(5) \lambda ={\displaystyle \frac{-\ln (R(T))}{n\times T}}.

A detailed discussion of the basic reliability concepts and mathematics can be found in Refs. [ 19, 20].

Markov method

The Markov method is an approach based on the state probabilities in the stochastic process. The Markov approach is a powerful method based on system states and transition between these states. Markov techniques are well suited to a variety of modeling problems and can be applied to several areas related to reliability analysis [ 21– 24]. In the Markov method, future states are only based on the present state and the previous states do not affect the process. A system can consist of n components and each component at any given time might be operating successfully or not. The system is considered in only two states (i.e., operating state or fail state). In fact, the Markov method considers the failure of individual components in a system and gives a clear representation of all the states and the transition between these states. Consider a continuous-time stochastic process based on the assumption of X = {X(t), t≥0}. If X satisfies the Markov property, it can be concluded that X is a continuous-time Markov chain which follows the following equation of probability [ 25]

(6) \mathrm{Pr}\{X({{\displaystyle t}}_n)=iX({{\displaystyle t}}_{n-\mathrm{1}})={{\displaystyle j}}_{n-\mathrm{1}},\mathrm{...},X({{\displaystyle t}}_{\mathrm{1}})={{\displaystyle j}}_{\mathrm{1}}\}=\mathrm{Pr}\{X({{\displaystyle t}}_n)=iX({{\displaystyle t}}_{n-\mathrm{1}})={{\displaystyle j}}_{n-\mathrm{1}}\},

where i, j₁, ..., j_{n- 1} are a countable set of S. In other words, S takes its values in a finite set of 0, 1, 2,..., n. Also, 0, 1, 2,..., n-1 denotes system configuration resulting in component failures. If X has homogeneous transition probabilities, then

(7) \mathrm{Pr}\{X(t)=iX(s)=j\}=\mathrm{Pr}\{X(t-s)=iX(\mathrm{0})=j\},\forall i,j\in S,\mathrm{0}⟨s⟨t.

If the chain X is irreducible, then

(8) \mathrm{Pr}\{X(t)=iX(\mathrm{0})=j\}⟩\mathrm{0.}

Suppose f_i(t) is the probability of state i at time t, and t≥0, it follows that f(t) = [f₀(t), f₁(t),..., f_n(t)]. Thus, based on the Chapman-Kolmogorov equations, there is

(9) \phi (t)=\phi (t)\Lambda .

Using f_n(0) = 1, f_i(0) = 0, and i = 0, 1, ..., n- 1, L is the failure matrix which can be obtained by component failure and repair rates. In addition, L is not invertible because the sum of the rows is zero. Let e denote a row of ones, thus

(10) \phi \Lambda =\mathrm{0},\phi e^{\Lambda }=\mathrm{1.}

Moreover, the failure rate (l) is defined as the transition probability from state up (which is the fully operational state of the system) to state down (where the system is in the degraded operational state), and the repair rate (µ) is called the transition probability from state down to state up. A Markov model consisting of the three states is described schematically in Fig. 3. The steady-state occupational probabilities of different states of such a system are [ 26]

(11) {\displaystyle \sum _{c=\mathrm{1}}^{C}P{b}_c}=\mathrm{1},

(12) [T]{[P{b}_{\mathrm{1}}\mathrm{...}P{b}_C]}^T=\mathrm{0},

where Pb_C is the probabilities of the state of C, and c = 1, 2,..., C. The squared matrix T consists of the failure and repair rates between the states. The transition rate with the location of (c, c) in T is equal to the negative sum of probabilities of all off diagonal array in the cth column. Also, the transition probability in the diagonal terms for a state c is called the rate of departure for state c. By solving Eqs. (11) and (12), the steady-state probabilities for the C states can be obtained.

State probabilities based on PV system model

The transitions between the up and down states are defined by the failure and repair rates. The failure rate of a PV system can then be obtained as [ 27]

(13) \lambda ={\displaystyle \frac{N_f}{h_s}}\times \mathrm{8760},

where Nf is the number of transitions due to the forced outages, and h_s is the duration of the PV system service. The repair time (r) of a PV system is determined by maintenance strategies and climate conditions. The repair rate of the system can be calculated as \mu =\mathrm{1}/r . The number of states that result from the Markov chain with N PV systems is 2N . The Markov model for a plant including different PV systems schematically is displayed in Fig. 4. Furthermore, the behavior of a PV system can be modeled as a stochastic process in the Markov method as seen in Fig. 5. The number of states varies based on the accuracy of the model. The transition rates from state j–1 to state j are identified by the transition rate l_(j–1)j. To deduce an adequate Markov chain, the transition rates between different states should be stationary and constant during the whole study period. The transition rate between any two states of such a system can be calculated by

(14) {\lambda }_{ij}={\displaystyle \frac{N_{ij}}{D_{ij}}},

where N_ij is the number of transitions from state i to state j, and D_ij is the duration of state i before going to state j. If N is the total number of states, the corresponding state probability is given by

(15) P_i={\displaystyle \frac{{\displaystyle \sum _{j=\mathrm{1}}^N{D}_{ij}}}{{\displaystyle \sum _{k=\mathrm{1}}^N{\displaystyle \sum _{j=\mathrm{1}}^N{D}_{kj}}}}}.

Besides, the transition rates between states can be expressed as

(16) {\lambda }_{ij}={\displaystyle \frac{F_{ij}}{P_i}},

where F_ij is the frequency of transitions between state i and state j. Moreover, the Markov method associated with the combination of the PV systems and the PV system models can be developed and represented in a state-space diagram [ 28].

From Fig. 4 (b), a transition matrix can be created with certain dimensions, depending on the number of components in the system. For a system with n components [ 29], the dimension of the transition matrix is n × n. The parameters in the transition matrix are defined by either the failure or repair rate between states [ 30]. For instance, in a transition between the state i and j (with i\ne j ), the transition rate enters into the ith row and jth column of the transition matrix. The diagonal parameters of the matrix must be equal to 1 minus the sum of the other parameters on the row. The transition matrix (T) of Fig. 5 becomes

(17) T=\left[\begin{array}{llll}\mathrm{1}-({\lambda }_1+{\lambda }_2)& {\mu }_1& {\lambda }_2& 0\\ {\mu }_{\mathrm{1}}& 1-({\lambda }_{\mathrm{2}}+{\mu }_{\mathrm{1}})& \mathrm{0}& {\lambda }_{\mathrm{2}}\\ {\mu }_{\mathrm{2}}& \mathrm{0}& 1-({\lambda }_{\mathrm{1}}+{\mu }_{\mathrm{2}})& {\lambda }_{\mathrm{1}}\\ \mathrm{0}& {\mu }_{\mathrm{2}}& {\mu }_{\mathrm{1}}& 1-({\mu }_{\mathrm{1}}+{\mu }_{\mathrm{2}})\end{array}\right],

where T is the transition matrix. Based on the Markov method, the limiting state probability cannot be changed in the further transition procedure. Thus, this statement can be expressed as [ 31]

(18) PT=P

or

(19) P(T-I)=\mathrm{0},

where P represents the limiting state probability vector and I is the identity matrix. Then, based on Eq. (19), the following results can be obtained.

(20) \left[\begin{array}{llll}{P}_{\mathrm{1}}& P_{\mathrm{2}}& P_{\mathrm{3}}& P_{\mathrm{4}}\end{array}\right]\left[\begin{array}{llll}-({\lambda }_{\mathrm{1}}+{\lambda }_{\mathrm{2}})& {\lambda }_{\mathrm{1}}& {\lambda }_{\mathrm{2}}& \mathrm{0}\\ {\mu }_{\mathrm{1}}& -({\lambda }_{\mathrm{2}}+{\mu }_{\mathrm{1}})& \mathrm{0}& {\lambda }_{\mathrm{2}}\\ {\mu }_{\mathrm{2}}& \mathrm{0}& -({\lambda }_{\mathrm{1}}+{\mu }_{\mathrm{2}})& {\lambda }_{\mathrm{1}}\\ \mathrm{0}& {\begin{array}{l}\mu \end{array}}_{\mathrm{2}}& {\mu }_{\mathrm{1}}& -({\mu }_{\mathrm{1}}+{\mu }_{\mathrm{2}})\end{array}\right]=\left[\begin{array}{llll}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right],

(21) \left[\begin{array}{llll}-({\lambda }_{\mathrm{1}}+{\lambda }_{\mathrm{2}})& {\mu }_{\mathrm{1}}& {\mu }_{\mathrm{2}}& \mathrm{0}\\ {\lambda }_{\mathrm{1}}& -({\lambda }_{\mathrm{2}}+{\mu }_{\mathrm{1}})& \mathrm{0}& {\mu }_{\mathrm{2}}\\ {\lambda }_{\mathrm{2}}& \mathrm{0}& -({\lambda }_{\mathrm{1}}+{\mu }_{\mathrm{2}})& {\mu }_{\mathrm{1}}\\ \mathrm{0}& {\lambda }_{\mathrm{2}}& {\lambda }_{\mathrm{1}}& -({\mu }_{\mathrm{1}}+{\mu }_{\mathrm{2}})\end{array}\right]\left[\begin{array}{l}{P}_{\mathrm{1}}\\ P_{\mathrm{2}}\\ {\begin{array}{l}P\end{array}}_{\mathrm{3}}\\ P_{\mathrm{4}}\end{array}\right]=\left[\begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right],

where P₁, P₂, P₃, and P₄ are the probability of states 1, 2, 3, and 4 in Fig. 5, respectively. In addition, the individual probabilities can be calculated as

(22) {{\displaystyle P}}_{\mathrm{1}}+{{\displaystyle P}}_{\mathrm{2}}+{{\displaystyle P}}_{\mathrm{3}}+{{\displaystyle P}}_{\mathrm{4}}=\mathrm{1.}

This condition should be able to solve the above equation based on the n–1 independent equations in which there are four state variables involved. Thus, any row within the above equation is

(23) \left[\begin{array}{llll}\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}\\ {\lambda }_{\mathrm{1}}& -({\lambda }_{\mathrm{2}}+{\mu }_{\mathrm{1}})& \mathrm{0}& {\mu }_{\mathrm{2}}\\ {\lambda }_{\mathrm{2}}& \mathrm{0}& -({\lambda }_{\mathrm{1}}+{\mu }_{\mathrm{2}})& {\mu }_{\mathrm{1}}\\ \mathrm{0}& {\lambda }_{\mathrm{2}}& {\lambda }_{\mathrm{1}}& -({\mu }_{\mathrm{1}}+{\mu }_{\mathrm{2}})\end{array}\right]\left[\begin{array}{l}{P}_{\mathrm{1}}\\ P_{\mathrm{2}}\\ P_{\mathrm{3}}\\ P_{\mathrm{4}}\end{array}\right]=\left[\begin{array}{l}\mathrm{1}\\ \mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right]\mathrm{.}

Finally, based on Eq. (23), the following results can be obtained.

(24) \begin{array}{l}{P}_{\mathrm{1}}={\displaystyle \frac{{\mu }_{\mathrm{1}}{\mu }_{\mathrm{2}}}{({\mu }_{\mathrm{1}}+{\lambda }_{\mathrm{1}})({\mu }_{\mathrm{2}}+{\lambda }_{\mathrm{2}})}},\\ P_{\mathrm{2}}={\displaystyle \frac{{\lambda }_{\mathrm{1}}{\mu }_{\mathrm{2}}}{({\mu }_{\mathrm{1}}+{\lambda }_{\mathrm{1}})({\mu }_{\mathrm{2}}+{\lambda }_{\mathrm{2}})}},\\ P_{\mathrm{3}}={\displaystyle \frac{{\mu }_{\mathrm{1}}{\lambda }_{\mathrm{2}}}{({\mu }_{\mathrm{1}}+{\lambda }_{\mathrm{1}})({\mu }_{\mathrm{2}}+{\lambda }_{\mathrm{2}})}},\\ P_{\mathrm{4}}={\displaystyle \frac{{\lambda }_{\mathrm{1}}{\lambda }_{\mathrm{2}}}{({\mu }_{\mathrm{1}}+{\lambda }_{\mathrm{1}})({\mu }_{\mathrm{2}}+{\lambda }_{\mathrm{2}})}}.\end{array}

Reliability indices in adequacy assessment of composite system

There are several quantitative indices that can be used to estimate the adequacy of a composite power system. In this section, the two most significant and basic adequacy indices, LOLE and EENS, are introduced which can be evaluated by the loss of load methods [ 32]. These indices can determine the system capability and can be calculated for each load point of the system. Additionally, these indices can be used in the prediction of the reinforcements and plans for the future expansions including reliability/worth analysis. The LOLE (h/a) and EENS

(MWh/a)

indices can be obtained as

(25) \mathrm{LOLE}={\displaystyle \sum _{i=\mathrm{1}}^n{P}_i\times t_i}.

When the cumulative probability is taken into account, the LOLE is given as

(26) \mathrm{LOLE}={\displaystyle \sum _{i=\mathrm{1}}^n{P}_i\times (t_i-}{t}_{i-\mathrm{1}}),

(27) \mathrm{EENS}={\displaystyle \sum _{i=\mathrm{1}}^n{C}_i\times F_i\times D_i={\displaystyle \sum _{i=\mathrm{1}}^{n}t\times C_i\times }{P}_i}.

In all equations, P_i is the probability of system state i, t_i is the duration of loss of power supply in days, L is the annual system peak load in MW, C_i is the load curtailment of system state i, n is the set of all system states associated with load curtailments, t is the period of study. F_i and Di are the frequency and the duration of system state i, respectively.

Study results

The Roy Billinton test system (RBTS) is a composite power system which has been developed for research and educational purposes at the University of Saskatchewan, Saskatchewan, Canada. The RBTS is used to study the feasibility and effectiveness of the proposed method. The single line diagram of the RBTS is exhibited in Fig. 6. This system has 6 buses, 5 load buses, 9 transmission lines, and 11 generators in buses 1 and 2 which are ranged from 5 MW to 40 MW. The total installed generating capacity is 240 MW and the peak load of the system is 185 MW. The system voltage level is 230 kV. The generating unit ratings and reliability data (e.g., force outage rate (FOR)) for the RBTS are given in Table 2.

It is worth noting that the parameters such as N_ij, and D_ij in Eqs. (14) should be available in order to calculate the lij of the components. However, the data of the manufactory are used, as reported in Tables 1 and 2. Other basic data from the RBTS such as bus data, line data, weekly peak load, daily peak load, and hourly peak load can be found in Ref. [ 33]. The probability values less than 1×10-8 have not been considered in Ref. [ 34]. However, for getting more accurate results, all the probabilities are considered. The results and indices can be used as an invaluable reference for comparing with other methods. The comprehensive capacity outage probability table (COPT) will show a large number of states. To quantify all states, a Matlab program is developed for the reliability evaluation. The study period and the system load in the RBTS are considered by the daily peak load variation curve which is also modeled as a straight line from 100% to 40% of the peak load. Figure 7 depicts the system load model which is assumed to be linear, although such a linear representation is not likely to occur in practice. The study period is based on hour and therefore 100% on the abscissa declares to 8760 h. Considering the proposed methodology in the previous sections, two well-known adequacy indices (i.e., the LOLE and the EENS) as well as the results are obtained in the following case studies.

Adequacy assessment of bulk power systems integrated with large-scale PV systems

In this section, reliability analysis in the presence of large-scale, grid-connected PV systems is performed. The improvement in the system reliability is obtained regarding the reduction in the LOLE and EENS. The results are shown in Figs. 8, and 9, from which the following observations can be made. First, the EENS and the LOLE are calculated when the grid is powered with its 11 default generators. Then, these indices are evaluated again in the presence of four different PV systems with the nominal output power of 5 MW, 10 MW, 15 MW and 20 MW, which are connected to the grid. Each five test states are conducted at the load peak of the RBTS which varies from 180 MW to 235 MW in steps of 5 MW. It is assumed that the FOR of PV systems and their nominal output power are directly proportional. The reliability of all PV systems is evaluated assuming one year of operation and then 20 years of operation. It can be seen that a larger PV system, even with larger FOR, results in less EENS and LOLE of the overall system.

For instance, at the peak load of 235 MW in one year of operation, with the help of a 5 MW PV system, the EENS and the LOLE of the system have declined from 4350 MWh/a and 265 h/a to 3170 MWh/a and 212 h/a, respectively. In other words, adding a 5 MW PV system deduces a 27% improvement in the EENS and 20% in the LOLE. As seen in Figs. 8 and 9, adding a larger PV generation system to the RBTS, further mitigates LOLE and EENS. Moreover, in the sample of results shown in Figs. 8, and 9, higher peak loads result in bigger LOLE and EENS. It can be seen from the inductive results that at a peak load of 235 MW, the system LOLE is 265 h/a before adding PV system. The system LOLE then decreases by approximately 20%, 37%, 54%, and 63% by adding 5 MW, 10 MW, 15 MW, and 20 MW of the PV system, respectively. Moreover, using PV systems with higher capacity further improves the system LOLE and the EENS. However, a smaller increase in improvement is observed each time. It is worth noting that, adding a PV generation grants smaller reliability compared to the conventional form of generation with the same capacity added. In other words, due to the intermittent nature of almost all of the renewable energy sources, the power generated from these sources are less reliable compared to that of conventional power generators. To get further results of the proposed implementations, the improvements of adding the PV systems as well as the conventional power generators, both with the capacity of 20 MW are compared. As shown in Fig. 9, in one year of operation, conventional power generators improved EENS index by 2680 MWh/a which is larger than 2010 MWh/a for the PV system. Sensitivity analyses show that the reliability indices are dramatically improved by adding more PV systems with high generation capacity. However, it is important to note that the implementations of this study are only case sensitive and the optimal decisions can be deduced when the economic issues are considered based on the reliability/worth analysis.

Degradation effects of the PV systems

To explore the impact of PV systems on the adequacy of the bulk power system, the failure effects of the PV systems are also investigated. It can be seen that the system indices increases significantly as the FOR of the PV system increases. Tables 3 and 4 demonstrate the variation of the LOLE, and the EENS indices versus the FOR of the PV system at the peak loads of 190 MW, 205 MW, 220MW, and 235MW. As it can be seen, an increase in the FOR results in less compensation by the PV systems and consequently more EENS and LOLE compared to those with less FOR. These results are given in Tables 3 and 4. The first rows of Tables 3 and 4 associate with the default RBTS with no PV system compensation. The rows that follow are the results of adding PV systems with incremental order of FORs. The important conclusion drawn from these results is that the existence of a compensator improves system status and increase in its FOR has only a minor influence on this improvement. For instance, at the peak load of 190 MW, the LOLE and the EENS are 10.437 h/a and 119.885 MWh/a, which after adding a PV system with FOR of 8%, improves to 2.795 h/a and 25.248 MWh/a, respectively. However, with the increase of FOR to 11%, the indices are increased to 3.044 h/a and 28.334 MWh/a. Further conclusions can be drawn by considering the relative deterioration caused by the FOR increment in different peak loads. As it can be noticed, reliability improvements brought by the PV systems are decreased as the FOR of the PV system is increased from 8% to 11%. As an example, at the peak load of 190 MW, increasing FOR from 8% to 11% changes the LOLE from 2.795 h/a to 3.044 h/a, which is approximately an 8.9% of deterioration. On the other hand, at the peak load of 235 MW, increasing the FOR results in only a 5% of deterioration. From the aforementioned results, it also can be found that the results for the EENS, which are not discussed here, also reveal the same phenomenon as discussed for the LOLE which prove the accuracy of the proposed method. Improving the FOR of the PV systems is a very controversial issue. An APV system with an optimal FOR is more reliable and more expensive. Thus, it is important to trade-off between reliability and cost.

Conclusions

There is an ever-increasing demand for large-scale, grid-connected PV systems in bulk power systems. To fully exploit the benefits of a PV system, a few new modifications as well as new concepts in distribution system planning are required. Thus, PV systems ought to be quantified and modeled properly. This paper proposed a novel approach for modeling the reliability impact of the PV system on the reliability of the bulk power system. Moreover, it discussed the degradation caused by long-term operation of the PV system. The results confirm the merits of the proposed method. For instance, at the peak load of 235 MW in one year of operation, with the help of a 5 MW PV system, the EENS and the LOLE of the system were improved from 4350 MWh/a and 265 h/a to 3170 MWh/a and 212h/a, respectively, which yielded an improvement of up to 27% with just a 5 MW PV system. Since the power output of PV systems is influenced by different intermittent conditions, probabilistic modeling for the PV systems should be used. Furthermore, it proposed an analytical methodology based on the Markov method. The implementations reveal that the proposed method can be useful for the planners even though the data showing precise reliability of the systems is not available. Sensitivity studies were conducted to confirm the assumption that the generation of the PV system can be modeled by the Markov method considering the multi-state process. The tools developed in this paper are useful to quantify the probabilistic modeling as well as reliability studies of the large-scale, grid-connected PV systems. The main advantage of the proposed technique is that it relies on a simple approach to estimate the generation states with their probabilities to conduct power system adequacy and sensitivity analysis. The implementations were performed on the RBTS to highlight and validate the effectiveness of the proposed approach. It is conceivable that the solar radiation affects the reliability of the system. However, this effect could be very dependent on the system conditions. Although the solar radiation is not considered in this research paper, it can be easily taken into consideration by the proposed framework. Instead of using the proposed method, any technique for bulk power system adequacy, such as the Monte-Carlo method, can be embedded into the proposed adequacy approach without changing the model of the PV system. The methodologies, models, and case study results presented in this paper can be easily implemented to the real power systems and can be employed to assist power system planners and operators to quantitatively assess the capacity benefits of PV systems and provide useful input to the managerial decision process (i.e., upgrading, evaluate, size, and optimize the power system). The proposed models of the PV systems will be applied for a practical case study in future research.

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