*3.1. Reliability Parameters at Load Point 'p'*

The reliability indices are the function of reliability parameters mentioned in Equations (36)–(38). The reliability parameters have been calculated at load point 'p' as follows.

$$\text{Failure rate (average)}; \lambda\_p = \sum\_{k \in n} num\_k \times F\_k \quad \text{failure per year} \tag{36}$$

$$\text{Outage duration (annual):} \, lI\_p = \sum\_{k=n} F\_k D\_{pk} \quad \text{hour per year} \tag{37}$$

$$\text{Outage duration (average):}\newline D\_p = \frac{\text{II}\_p}{\lambda\_p} \quad \text{hour} \tag{38}$$

where *Fk* = failure rate (average) of the *kth* element, *n* = number of elements in the EDS, *numk* = number of *kth* elements in the EDS, *Dpk* = duration of failure at *pth* load point due to *kth* failed element. The calculation equations of *λ<sup>p</sup>* and *Up* are given in Appendix A.2.

### *3.2. System-Based Indices*

These indices are further categorized in load-oriented indices and customer oriented indices as given in Equations (39)–(40) and Equations (41)–(46c), respectively.

#### 3.2.1. Load-Oriented Indices

Load-Oriented Indices have been calculated at load point 'p', as mentioned in Equations (39)–(40).

$$EENS\_p = P\_p l I\_p \quad \text{megawatt hour per year} \tag{39}$$

$$AENS\_p = \frac{\sum\_{p=1}^{n\_p} EENS\_p}{\sum\_{p=1}^{n\_p} N\_p} \quad \text{megawatt hour per customer per year} \tag{40}$$

where *Pp* = demand/load (average) of the *pth* load point, *EENSp* = expected ENS at *Pth* load or customer point. where *np* = total load points, *Np* = number of customers at *Pth* load point.

#### 3.2.2. Customer Oriented Indices

These indices have allowed to enhance the EDS's reliability related to the improvement of customer or load services. The two of the indices namely ECOST and IEAR are related to the cost reliability and thus, termed as reliability worth of the system.

$$ECOST\_p(=LOEE\_p) = P\_p \sum\_{k=n} f(D\_{pk}) F\_k \quad \text{k}\\$ \text{ per year} \tag{41}$$

$$IEAR\_p = \frac{ECOST\_p}{EENS\_p} \quad \text{\(\text{\(\)}\)}\tag{42}$$

where *ECOSTp* = expected interrupted cost at *Pth* load point, *IEARp* = interrupted energy assessment rate at *Pth* load point, *LOEEp* = loss of expected energy, *f*(*Dpk*) = system composite customer damage function (\$ per kilowatt) as provided in Table A3 of Appendix A.

$$SAIFI = \frac{\sum\_{p=1}^{n\_p} \lambda\_p N\_p}{\sum\_{p=1}^{n\_p} N\_p} \quad \text{failure per customer per year} \tag{43}$$

$$SAIDI = \frac{\sum\_{p=1}^{n\_p} \, ^{\mathcal{U}\_p} \mathcal{N}\_p \mathcal{N}\_p}{\sum\_{p=1}^{n\_p} \, ^{\mathcal{N}} \mathcal{N}\_p} \quad \text{hour per customer per year} \tag{44}$$

$$\text{CAIDI} (= \frac{SAIDI}{SAIFI}) = \frac{\sum\_{p=1}^{n\_p} IL\_PN\_p}{\sum\_{p=1}^{n\_p} \lambda\_p N\_p} \quad \text{hour per customer per interruption} \tag{45}$$

where CAIDI = Customer Average Interruption Duration Index.

$$ASAI = \frac{8760 \sum\_{p=1}^{n\_p} N\_p - \sum\_{p=1}^{n\_p} \mathcal{U}\_p N\_p}{8760 \sum\_{p=1}^{n\_p} N\_p} \quad \text{per unit} \tag{46a}$$

Also, *ASAI* can be derived as follows.

$$ASAI = 1 - \frac{SAIDI}{8760} \tag{46b}$$

$$ASIII = 1 - ASAI \quad \text{per unit} \tag{46c}$$

where *ASUI* = Average Service Unavailability Index.

#### **4. Modeling of WTG, SPV, and BSS**

The reliability assessment of the IEEE 33 bus EDS is accomplished, considering the optimal siting(s) and sizing(s) of SPV, WTG, and BSS. In this regard, a brief modeling and specifications of these RESs are illustrated in Sections 4.1–4.3.

#### *4.1. Wind Turbine Generator*

The V162-5.6MW(IECS based on IEC IIB), a WTG, manufactured by General Electric Company is considered for its output power rating. The specifications of the WTG considered in this study are provided in Table 4. The mechanical power of WTG (*Pmech*) is a function of generator rotor speed and wind speed as formulated in Equations (47)–(51) [70].

$$P\_{mech}(v\_{wind, \prime}, \omega\_{rotor}) = \frac{1}{2} \times \rho \times v\_{wind}^3 \times \mathbb{C}\_{\mathbb{P}}(\lambda, \theta) \tag{47}$$

$$
\lambda = \frac{\omega\_{rotor} \times GR \times R\_{rotor}}{v\_{wind}} \tag{48}
$$

$$\mathbb{C}\_{p}(v\_{wind}, \omega\_{rotor}, \theta) = \mathbb{C}\_{1}(\mathbb{C}\_{2}\frac{1}{\mathfrak{a}} - \mathbb{C}\_{3}\theta - \mathbb{C}\_{4}\theta^{x} - \mathbb{C}\_{5}) \times \exp(\frac{-\mathbb{C}\_{6}}{\mathfrak{a}}) \tag{49}$$

$$\frac{1}{\alpha} = \frac{1}{(\lambda + 0.08\theta)} - \frac{0.035}{1 + \theta^3} \tag{50}$$

where *ρ* = air density, *As* = area swept by the turbine rotor blades, *Vwind* = speed of the wind, *Cp* is the non-linear function of the tip speed ratio (*λ*) and pitch angle (*θ*), *ωrotor* = generator rotor speed, GR = gear ratio, *Rrotor* = rotor radius at the turbine blades, *C*<sup>1</sup> to *C*<sup>6</sup> and *x* are constants and computed in [70].

$$P\_{WTG} = \begin{cases} 0; & 0 \le V \le V\_{cin} \text{ or } V \ge V\_{cout} \\ P\_{WTG,rated} \times \left(\frac{V - V\_{cin}}{V\_{rated} - V\_{cin}}\right); & V\_{cin} \le V \le V\_{rated} \\ P\_{WTG,rated} & V\_{rated} \le V \le V\_{cout} \end{cases} \tag{51}$$

where *PWTG* = output WTG power, *PWTG*,*rated* = rated output WTG power, *Vrated* = rated wind speed.


**Table 4.** Wind Turbine (V162-5.6 MW) specifications.

#### *4.2. Solar Photovoltaic*

The SPR-P5-545-UPP, a Solar PV Module, manufactured by Sunpower Company is considered for its output power rating. The specifications of the SPV module considered in this study are provided in Table 5. The SPV module is developed by implementing several cells. The power output for the SPV module can be derived as described by Equations (52)–(65) [71–73].

$$P\_{SPV(AC)}(t) = P\_{SPV(out)}(t) \times \eta\_{invert \tau} \tag{52}$$

$$P\_{SPV(out)}(t) = FF\_A(t) \times I\_{short}(t) \times V\_{open}(t) \tag{53}$$

$$I\_{short}(t) = \frac{SR}{1000} [I\_{short,STC} + C\_I(T\_{SM}(t) - 25)] \tag{54}$$

$$V\_{open}(t) = V\_{open,STC} + C\_V \left[ T\_{SM}(t) - 25 \right] \tag{55}$$

$$FF\_A(t) = FF\_i(t) \times \left[1 - R\_s(t)\right] \tag{56}$$

$$FF\_{\bar{1}}(t) = \frac{V\_{open,0}(t) - \ln[V\_{open,0}(t) + 0.72]}{V\_{open,0}(t) + 1} \tag{57}$$

$$V\_{open,0}(t) = V\_{open}(t) \times \frac{Q}{N \times K[T\_{SM}(t) + 273.15]} \tag{58}$$

$$R\_s(t) = R\_s \times \frac{I\_{short}(t)}{V\_{open}(t)}\tag{59}$$

$$T\_{SM}(t) = T\_{amb}(t) + SR \frac{(T\_{nom} - 20)}{0.8} \tag{60}$$

where *t* = time instant, *PSPV*(*AC*)(*t*) = AC output power, *PSPV*(*out*)(*t*) = maximum output power, *ηinverter* = inverter efficiency, *FFA*(*t*) = fill factor actual, *Ishort*(*t*) = short circuit current under operating conditions, *Vopen*(*t*) = open circuit voltage under operating conditions, SR = solar radiation intensity (W/m2), STC = standard test conditions, *CI* = temperature coefficient for current (A/◦C), *TSM*(*t*) = solar module temperature (◦C), *CV* = temperature coefficient for voltage (V/◦C), *FFi*(*t*) = fill factor ideal, *Rs*(*t*) = normalized series resistance of solar module, *Vopen*,0(*t*) = normalized open circuit voltage, *Q* = an electron charge, *N*(≈1) = ideality factor, *K* = Boltzmann's constant, *Rs* = series resistance of SPV module, *Tamb*(*t*) = ambient temperature of SPV module, *Tnom* = nominal operating cell temperature.

The *Rs* is being evaluated as follows [74].

$$R\_s = R\_{s,STC} = r\_{s,STC} \times \frac{V\_{open,STC}}{I\_{short,STC}} \tag{61}$$

$$R\_{s,STC} = 1 - \frac{FF\_{A,STC}}{FF\_{i,STC}}\tag{62}$$

$$FF\_{A,STC} = \frac{V\_{mpp,STC} \times I\_{mpp,STC}}{V\_{open,STC}} \times I\_{short,STC} \tag{63}$$

$$FF\_{i,STC} = \frac{V\_{open,0,STC} - \ln[V\_{open,0,STC} + 0.72]}{V\_{open,0,STC}} + 1\tag{64}$$

$$V\_{open,0,STC} = V\_{open,STC} \times \frac{Q}{N \times K [T\_{SM,STC}]} (t) + 273.15 \,\text{J} \tag{65}$$

where *rs*,*STC* = normalized series resistance under STC, and all other parameters in Equations (61)–(65) are evaluated at STC.


**Table 5.** Bifacial Solar Panel (SPR-P5-545-UPP) specifications.

#### *4.3. Battery Storage System*

The BESS 3000, a Lithium-Ion Battery System, manufactured by Freqcon Company is considered for its output power rating. The specifications of the BSS considered in this study are provided in Table 6. The flowchart of BSS dispatch modeling is illustrated in Figure 6. Mathematical modeling is further defined. The BSS dispatch strategy starts functioning by monitoring the peak load hours. If the peak load is greater/less than the capacity of WTG and SPV, the BSS discharges/charges to support the distribution system; otherwise the BSS operates as a neutral device. Also, the BSS SOC will decide the charging or neutral operation during the off-peak hours. The battery model is derived with the help of Equations (66)–(68) assuming that no battery is self-discharging [75].

**Figure 6.** General BSS dispatch strategy opted for battery operation.

$$P\_{Battery(DC)}(t) = \frac{E\_{Battery}(t) - E\_{Battery}(t - \Delta t)}{\Delta t} \tag{66}$$

$$P\_{\text{Rattery}(A\text{C})} \{ t \} = \begin{cases} \frac{P\_{\text{Rattery}(D\text{C})} \{ t \}}{\eta\_{\text{Rattery}}} ; P\_{\text{Rattery}(D\text{C})} \{ t \} > 0 \\ P\_{\text{Rattery}(D\text{C})} \{ t \} \times \eta\_{\text{Rattery}} \times P\_{\text{F}\_{\text{Invert}}} \text{Otherwise} \end{cases} \tag{67}$$

$$\text{SOC(t)} = \text{SOC(t-\Delta t)} + \eta\_{\text{Charging}} \times \frac{P\_{\text{Battery(DC)}}(t)}{\mathbb{C}\_{\text{Battery}} \times V} \times \Delta t \tag{68}$$

where *PBattery*(*DC*)(*t*) = DC charging/discharging power of the battery in Δ*t* interval (W), *EBattery*(*t*) = energy of battery (Wh), *PBattery*(*AC*)(*t*) = AC power discharged/charged state of battery, *ηBattery* = efficiency of the battery, *PFInverter* = inverter power factor, *SOC*(*t*) = state of charge of the battery, *ηCharging* = charging efficiency of the battery, *CBattery* = battery capacity (Ah), *V* = nominal voltage of the battery (V).


**Table 6.** Battery Storage System (BESS 3000) specifications.
