7.2.1.1 Geothermal Models

The internal heat of the earth can be used to produce electricity through the use of steam turbines. The temperature $T$ at a depth $D$ is given by $T=T_{\mathrm{surface}}+D\Gamma$, where $T_{\mathrm{surface}}$ is the average temperature at the surface, and $\Gamma$ is the temperature gradient. The temperature gradient is related to the heat flow Q and rock conductivity K by $Q=K\Gamma$. The values for the temperature gradient and heat flow vary depending on the age of the geological formation, but are typically 20°C/km. Electricity production from geothermal sources are liquid based, where hot, compressed water fills the fractured and porous rocks of a reservoir. As the fluid moves upward toward the surface, the pressure decreases until the saturation pressure is reached corresponding to the geofluid temperature. The fluid begins to boil creating a two-phase liquid–vapor mixture that will flow through the upper section of the well. A separator will divert steam to turbine generators for electricity production and brine–water for either direct heating uses or will be returned to the reservoir through injection wells. After the steam has been used for electricity production, it passes through a condenser, is cooled, and is returned to the geothermal reservoir as shown in Fig. 7.1 [4].

The potential power output of a geothermal plant depends primarily on two factors: The temperature of the geofluid and the fluid flow rate. Figs. 7.2 and 7.3 show these dependencies [5].

7.2.1.2 Biomass Models

Organic plant matter, known as biomass, can be converted to energy by burning, gasification, fermentation, or other processes. Wood products are the most common form of biomass power, but agricultural waste, yard-clippings, and municipal solid waste can also be used. To generate power, biomass is typically burned in a boiler, which produces high-pressure steam that is used to drive a turbine that generates electricity.

The power output potential for biomass depends on the heat content of the biomass fuels which is highly dependent on the plant species. Some combustion technologies can accept a wide range of biomass feedstock, while others require a much more homogenous feedstock. The heat content can also vary depending on the moisture content of the biomass fuel and the conversion technologies.

7.2.2.1 Wind Unit Models

The power generated by a wind turbine is calculated by a simple application of the kinetic energy equation $\mathit{KE}=1/2m{v}^2$ through a cross sectional area A. The available power in Watts can be shown to be

$P=\frac{1}{2}\rho v^{3}A=\frac{1}{2}\rho v^3\frac{\pi d^2}{4}$ (7.1)

(7.1)

where $v$ is the wind speed in meter per second (m/s), $\rho$ is the density of air in kg/m³ (typically 1.225 kg/m³ at 15.55°C and 101,325 Pa), $\mathrm{and}d$ is the rotor diameter in m.

Clearly, as the wind speed increases, the power output will increase with the cube of the wind speed. This model must be adapted to account for cut-in speed, rated speed, and cut-out speed.

Low-wind speeds do not provide sufficient torque to the turbine blades to make them rotate. Power generation typically begins at wind speeds between 3 m/s and 4 m/s (6.71 and 8.95 mi/h). The wind speed at which power generation begins is called the cut-in speed.

As the wind speed exceeds the cut-in speed, power generation will rise steeply as a function of the cube of the wind speed. However, when the speed reaches a threshold, the wind turbine will reach its rated power output and will no longer increase its power as the velocity increases. This typically occurs between 12 m/s and 17 m/s (26.84 and 38.03 m/h). With large turbines, the wind power is usually maintained at a constant level by adjusting the angle of the blades.

As the speed increases above the rated speed, there is a danger that excessive forces will cause damage to the turbine. A braking system will engage and bring the rotor to a standstill. This occurs at the cut-out speed which is typically around 25 m/s (Fig. 7.4).

Energy production, measured in kW h, is $E=\int P\left(t\right)\mathit{dt}$. Since the power output of a wind turbine depends on the wind speed, predicting energy production requires the use of PDF to account for the variances in wind speed in time. There are two commonly used PDFs that are used to describe wind velocity variation: The normal (Gaussian) distribution and the Weibull distribution. The Weibull distribution is often a better fit for modeling wind speeds due to the asymmetry of measured distributions.

The Weibull PDF applied to wind speeds is defined as

$f\left(v\mathrm{;}c,k\right)=\frac{k}{c}{(\frac{v}{c})}^{k-1}{e}^{-{(v/c)}^k}$ (7.2)

(7.2)

where $k$ is the shape factor, $c$ is the scale factor, and $v$ is a vector of the measured wind speeds.

The value of the shape factor $k$ determines the width of the wind speed distribution around the average. Smaller values indicate a relatively wider wind speed distribution, while a larger indicates the opposite. The scale factor $c$ defines where the majority of the distribution lies and how stretched out the distribution is.

Fig. 7.5 shows a Weibull PDF for wind speed data collected during a 1-year period in Gilbert, IA and includes the shape and scale factors. MATLAB was used to analyze the data.

Fig. 7.6 shows the fitted Weibull PDF with c=5.0298, k=1.9579 compared to the empirical PDF plotted using historical hourly wind speed for the Ames site. The Weibull curve fitting is used to obtain the scale and shape factors using historical wind speed data. As shown, the fitted Weibull PDF is a suitable fit and an appropriate probabilistic model for the wind speed data.

7.2.2.2 PV Unit Models

Solar irradiance is the amount of power that reaches the surface of the earth per unit area. For a given location, this amount will vary on a daily and yearly basis due to the motion of the earth relative to the sun. The amount of solar radiation also depends on the geographical location (latitude and longitude) and the climatic conditions. Cloud cover is the main factor affecting the amount of solar radiation that reaches the surface of the earth. Fig. 7.7 shows the seasonal and hourly variability in solar irradiance measured at Ames, Iowa for the year 2015.

There are a variety of models that are used to estimate the solar irradiance and PV power generation for example, the Meneil, Hottel, and Iqbal models for irradiance. Two models will be examined: The ASHRAE model [6,7] and a model developed by Tina et al. [8].

The ASHRAE model was developed by the American Society of Heating, Refrigerating, and Air Conditioning engineers to estimate the beam portion of the irradiation reaching the earth’s surface for a clear sky [6,7]. It is based on original empirical work and has the advantage that it does not depend on any atmospheric data; yet, it has been widely used in many solar applications. The direct beam radiation at sea level, $I_o$, is determined by the following equation:

$I_o=A*\exp \left(-\frac{B}{\cos {(\theta }_Z)}\right)*\cos {(\theta }_Z)[\mathrm{W}/{\mathrm{m}}^2]$ (7.3)

(7.3)

where $A$ represents the apparent extraterrestrial flux, $A$=1140 + 75 $\sin \left(n\right)$, $B$=0.132 + 0.023 $\cos \left(n\right),{\theta }_Z$ is the solar zenith angle (depends on declination, latitude, and hour for a given location), and n is the day number.

Fig. 7.8 shows the maximum direct beam radiation as a function of the day of the year for latitudes ranging from 0° to 60°.

To account for the differences between the maximum solar radiation available and the actual amount reaching earth’s surface, an hourly clearness index, $k_t$, has been defined as the ratio of the irradiance on a horizontal plane, $I_t$ [kW/m²] to the extraterrestrial maximum solar irradiance $I_o$ [kW/m²]:

$k_t=\frac{I_t}{I_o}$ (7.4)

(7.4)

Since most PV units are installed at an angle relative to the horizontal plane, the irradiance can be modeled by an equation that accounts for this angle and the reflectivity of the ground.

$I_{\beta }=\left[R_b+\left(\frac{1+\cos \beta }{2}-R_b\right)*k+\rho *\frac{1-\cos \beta }{2}\right]*I_t[\mathrm{kW}/{\mathrm{m}}^2]$ (7.5)

(7.5)

where $R_b$ is the ratio of beam radiation on the tilted surface to that on a horizontal surface, $k$ is the fraction of the hourly radiation on the horizontal plane which is diffused, and $\rho$ is the reflectance of the ground.

The power output of a PV can then be modeled using the following equation:

$P_{\mathit{pv}}=A_C*\eta *I_{\beta }$ (7.6)

(7.6)

where $A_C$ is the surface area of the PV array (m²), $\eta$ is the efficiency of the PV in realistic reporting conditions (RRC), and $I_{\beta }$ is irradiance on a surface with an angle to the horizontal plane (kW/m²).

The efficiency of a PV system will depend on a number of factors such as the temperature of the solar cell, intensity, and spectrum of the incident sunlight. The open circuit voltage is highly sensitive to changes in temperature, which could be due to variation in ambient temperature, irradiance, or wind speed at the panel site [9]. Like all electronics, PV systems have lower efficiency in the heat than the cold.

Because $I_{\beta }$ will vary throughout the year based on latitude, the day of the year, and the hour angle, sunset hour angle, and reflectance of the ground, the parameters T and T′ will be used to account for these variations. Thus, the expressions for the power output generated by a PV array is given by the following equation:

$P_{\mathit{pv}}=A_C*\eta *(T*k_t-T^{\prime }*k_t^2)$ (7.7)

(7.7)

where

$T=\left[\left({\mathrm{R}}_{\mathrm{b}}+\mathrm{\rho }\frac{1-\cos \mathrm{\beta }}{2}\right)+\left(\frac{1+\cos \beta }{2}-R_b\right)\rho \right]*r_d*\frac{H_o}{3600}$

$T^{\prime }=\left(\frac{1+\cos \beta }{2}-R_b\right)*q*r_d*\frac{H_o}{3600}$

in which $R_b$ is a geometric factor, the ratio of beam radiation on the tilted surface to that of a horizontal surface at any time, $\rho$ is the reflectance of the ground, $\beta$ is the tilt angle measured in degrees, $r_d$ is the ratio of the diffuse radiation in an hour to the diffuse radiation in a day, $H_o$ is the extraterrestrial radiation on a horizontal surface ($\mathrm{mJ}/{\mathrm{m}}^2$), and $q$ is the parameter in the functional form for $k$ ($k_t$).

The hourly clearness index,$k_t$, can be considered a random variable due to weather conditions such as cloud cover and visibility factors such as mist and haze. From sampled values of $k_t$over a period of time, it is possible to obtain a probability density function ${f_p}_{\mathit{pv}}\left(\mathit{Ppv}\right)$ for $P\mathit{pv}$. There are four PDFs typically used for solar radiation. In all cases, the value of $x$ represents the hourly clearness index.

Normal Distribution

$f\left(x\mathrm{;}\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-{(x-\mu )}^2/2{\sigma }^2}$ (7.8)

(7.8)

where $\mu$ is the mean of the distribution, and $\sigma$ is the standard deviation of the distribution.

Weibull Distribution

$f\left(x\mathrm{;}a,b\right)={b*a}^{-b}{x}^{b-1}{e}^{-{(x/a)}^2}$ (7.9)

(7.9)

where $a$ is the shape parameter of the distribution, and $b$ is the scale parameter of the distribution.

Extreme Value Distribution

$f\left(x\mathrm{;}\mu ,\sigma \right)={\sigma }^{-1}*e^{\left[(x-\mu )/\sigma \right]}*e^{-e\left[(x-\mu )/\sigma \right]}$ (7.10)

(7.10)

where $\mu$ is the location parameter of the distribution, and $\sigma$ is the scale parameter of the distribution.

Beta Distribution

$f\left(x\mathrm{;}\alpha ,\beta \right)=\frac{\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{(1-x)}^{\beta -1}*x^{\alpha -1}$ (7.11)

(7.11)

where $\alpha$ is the first shape parameter of the distribution, and $\beta$ is the second shape parameter of the distribution.

It has been observed that the choice of an adequate PDF model may depend on latitude [10].

7.3.3.1 An Introduction on Importance of Reliability Studies in Distribution Systems

Over the past 100 years, the role of electricity has evolved. In today’s information age, reliable electricity is no longer a luxury; it is now a basic requirement of many customers. It is critical to all aspects of safely operating our cities, businesses, and homes. However, the electric grids have not kept pace with surging demand. Even with substantial improvements in energy-efficient buildings, electricity demand has increased from 1500 billion kW h in 1970 to over 3700 billion kW h in 2004 and is projected to reach 5600 billion kW h by 2030 [17]. With the growing demand and increasing dependence on electricity energy, the necessity to achieve an acceptable level of reliability becomes a crucial issue for decision makers of power systems [18]. Therefore, the utilities in electricity industry have to improve systems continuously to meet customers’ reliability requirements.

Distribution systems, as the final level in delivery of electricity to the end users, have an undeniable role in providing reliable electricity to the costumers. However, the technical studies show that more than 90% of the customer service interruptions occur due to failures in different equipment of these systems [19]. This represents the importance of this sector in reliability level of customers and emphasizes on the reliability improvement of distribution systems.

Reliability is a measure of the system’s ability to meet the electricity needs of customers, which can be used as a means of assessing the past performance of a system or predicting its future. For distribution systems, at the present time, reliability evaluations are more widely used in assessing the system performance. Assessment of system performance is a valuable procedure for three important reasons.

Reliability assessment necessitates evaluation techniques and indices to quantify current system status and compare the merits of various reinforcement schemes.

7.3.3.2 Reliability Evaluation Methods

Power system reliability evaluation methods can be divided into two categories: (1) Analytical methods, and (2) simulation methods. In addition, the reliability evaluation may be a qualitative study, in which the main factors that impact system reliability can be determined and prioritized, or a quantitative study, where the reliability is assessed through different parameters and indices defined and calculated for the system or load points [20].

Analytical techniques represent the system by a mathematical model and evaluate the reliability indices from this model using direct numerical solutions. They generally provide expectation indices in a relatively short computing time. There are several methods to analytically evaluate reliability, including fault tree analysis, failure mode, effect, and criticality analysis, Markov processes, minimum cut-set method, and network reduction method [21]. The most common evaluation techniques using a set of approximate equations are failure mode analysis and minimum cut set analysis. Analytical approaches are based on assumptions concerning the statistical distributions of failure rate and repair times. On the contrary, assumptions are frequently required in order to simplify the problem and produce an analytical model of the system. This is particularly the case when complex systems and complex operating procedures have to be modeled. The resulting analysis can therefore lose some or much of its significance. The use of simulation techniques is very important in the reliability evaluation of such situations. Simulation methods estimate the reliability indices by simulating the actual process and random behavior of the system. Monte Carlo technique is the widely used simulation method which has two different categories [21]: (1) Sequential, and (2) nonsequential.

In a nonsequential Monte Carlo simulation (MCS), the samples are taken without considering the time dependency of the states or sequence of the events in the system. Therefore, using this method, a nonchronological state of the system is determined. On the other hand, a sequential MCS can address the sequential operating conditions of the system, and may be used to include time correlated events and states such as output generation of renewable-based generating units, demand profile, and customer decisions, which is more applicable for power system reliability studies.

The MCS methods are classified based on the methods used for sampling. Three common sampling approaches in MCS are: (1) State sampling approach; (2) system state transition sampling approach; and (3) state duration sampling approach [21].

7.3.3.3 Reliability Indices

Reliability indices are used by system planners and operators as a comparative tool to improve the quality level of service to customers. Planners use these parameters to determine the requirements for capacity additions, while operators use them to ensure that the system is robust enough to withstand possible failures without catastrophic consequences.

The vast majority of reliability indices reflect faults and failures in the distribution system. These metrics are based upon the customers’ outage data and the unserved load. They describe how often electrical service are interrupted, how many customers are involved with each outage, how long the outages lasts, and how much load goes unsupplied.

Distribution reliability indices falls into two major categories, costumer-oriented and load- and energy-oriented, which both provide detailed information about the system [18].

1. Customer-oriented indices

a. System average interruption frequency index (SAIFI)
SAIFI indicates how often an average customer is subjected to sustained interruption over a predefined time interval:

$\mathrm{SAIFI}=\frac{\mathrm{Total\; Number\; of\; Customer\; Interruptions}}{\mathrm{Total\; Number\; of\; Customers\; Served}}=\frac{\sum {\lambda }_i{N}_i}{\sum N_i}$ (7.19)

(7.19)

where ${\lambda }_i$ is the failure rate, and $N_i$ is the number of customers of load point i.

b. System average interruption duration index (SAIDI)
It indicates the total duration of interruption an average customer is subjected for a predefined time interval:

$\mathrm{SAIDI}=\frac{\mathrm{Sum\; of\; Customer\; Interruption\; Durations}}{\mathrm{Total\; Number\; of\; Customers}}=\frac{\sum U_i{N}_i}{\sum N_i}$ (7.20)

(7.20)

where $U_i$ is the annual outage time.

c. Customer average interruption duration index (CAIDI)
This index indicates the average time required to restore the service:

$\mathrm{CAIDI}=\frac{\mathrm{Sum\; of\; Customer\; Interruption\; Durations}}{\mathrm{Total\; Number\; of\; Customer\; Interruptions}}=\frac{\sum U_i{N}_i}{\sum {\lambda }_i{N}_i}$ (7.21)

(7.21)

d. Average service availability/unavailability index (ASAI/ASUI)
ASAI specifies the fraction of time that a customer has received the power during the predefined interval of time and is vice versa for ASUI:

$\mathrm{ASAI}=\frac{\mathrm{Customer\; Hours\; of\; Available\; Service}}{\mathrm{Customer\; Hours\; Demanded}}=1-\frac{\sum U_i{N}_i}{8760\times \sum N_i}$ (7.22.a)

(7.22.a)

$\mathrm{ASUI}=1-\mathrm{ASAI}=\frac{\mathrm{Customer\; Hours\; of\; Unavailable\; Service}}{\mathrm{Customer\; Hours\; Demanded}}=\frac{\sum U_i{N}_i}{8760\times \sum N_i}$ (7.22.b)

(7.22.b)

where 8760 is the number of hours in a calendar year.

2. Load- and energy-oriented indices

a. Energy not supplied (ENS)
ENS specifies the average energy the customer has not received in the predefined time:

$\mathrm{ENS}=\mathrm{Total\; Energy\; Not\; Supplied\; by\; the\; System}=\sum L_{\alpha \left(i\right)}{U}_i$ (7.23)

(7.23)

where $L_{\alpha \left(i\right)}$ is the average load connected to load point i.

b. Average energy not supplied (AENS)

$\mathrm{AENS}=\frac{\mathrm{Total\; Enery\; Not\; Supplied}}{\mathrm{Total\; Number\; of\; Customers\; Served}}=\frac{\sum L_{\alpha \left(i\right)}{U}_i}{\sum N_i}$ (7.24)

(7.24)

c. Average customer curtailment index (ACCI)

$\mathrm{ACCI}=\frac{\mathrm{Enery\; Not\; Supplied}}{\mathrm{Total\; Number\; of\; Customers\; Affected}}$ (7.25)

(7.25)

This index differs from AENS in the same way that CAIFI differs from SAIFI. It is therefore a useful index for monitoring the changes of average energy not supplied between one calendar year and another.

7.3.3.4 DGs and Electric Distribution System Reliability

Today, changing consumer needs, combined with advances in DG technologies making them more efficient and less costly, have opened new opportunities for consumers to use DGs to meet their own energy needs, as well as electric utilities to explore possibilities to meet electric system needs with distributed generation.

Implementation of DGs offers potential benefits to distribution systems. They provide an enhanced power quality, reduce peak loads, and provide ancillary services such as reactive power and voltage support. Using DGs to meet these local system needs can add up to improvements in overall electric system reliability. Therefore, DGs can be used by electric system planners and operators to improve reliability in both direct and indirect ways.

7.3.3.5 Reliability Models of Renewable Generators

Having a reliable distribution network requires the reliability assessments of renewable generators with stochastic nature such as wind turbines, solar arrays, electric vehicles, and others. Several reliability models have been developed for wind turbines and solar generators such as Markov-based analytical models, MCS, universal generating function, homogenous Poison process, and others. A Markov model is a set of states defined by random variables where a system can transit between states. MCS technique can generate multiple scenarios and calculate the outage frequency of a system. A two-state model (ON–OFF model) is used for a system including a wind turbine/solar generator. However, for a distribution level wind/solar farm, a multistate model is used for more accurate calculations [28,29].

7.3.5.1 First Voltage Stability Index

Fast voltage stability index (FVSI) [32,33] determines the stable operating region of the load using the reactive power transfer. In order to obtain FVSI, we use the two-bus model (Fig. 7.10) and the following definitions:

$I=\frac{V_s-V_r}{Z}=\frac{V_s-V_r}{R+\mathit{jX}}=\frac{\left|V_s\right|\measuredangle {\delta }_s-\left|V_r\right|\measuredangle {\delta }_r}{R+\mathit{jX}}$ (7.32)

(7.32)

$Q_r=\mathit{Im}\left(V_r{I}^{*}\right)=[\frac{X}{R^2+X^2}\left(\left|V_s\right|\left|V_r\right|\cos \left({\delta }_s-{\delta }_r\right)-{\left|V_r\right|}^2\right)+\frac{R}{R^2+X^2}\left(\left|V_s\right|\left|V_r\right|\sin \left({\delta }_s-{\delta }_r\right)\right)]$ (7.33)

(7.33)

Solving Eq. (7.33) for |$V_r$| gives:

${\left|V_r\right|}^2-\left[\frac{R}{X}\sin \left({\delta }_s-{\delta }_r\right)+\cos ({\delta }_s-{\delta }_r)\right]\left|V_s\right|\left|V_r\right|+\frac{R^2+X^2}{X}{Q}_r=0$ (7.34)

(7.34)

The following condition needs to be satisfied for |$V_r$| to have a real solution

$\frac{4(R^2+X^2)Q_r}{{\left|V_s\right|}^2{\left[R\sin \left({\delta }_s-{\delta }_r\right)+X\cos ({\delta }_s-{\delta }_r)\right]}^2}\le 1$ (7.35)

(7.35)

Assuming ${\delta }_s-{\delta }_r$≈0, the FVSI for line sr can be defined using (7.35) as follows:

${\mathrm{FVSI}}_{\mathit{sr}}=\frac{4(R^2+X^2)Q_r}{{X\left|V_s\right|}^2}\le 1$ (7.36)

(7.36)

Based on (7.36), the system remains stable as long as FVSI is less than or equal to unity. This equation can be used to determine the maximum reactive power loading capability of a node where FVSI equals unity.

7.3.5.2 Line Stability Factor (LSF)

A similar concept applies to LSF as well [32,33]. Both active and reactive power transfer equations are used to obtain LSF. Using the two-bus model, the active and reactive powers at the receiving bus can be calculated by

$P_r=\mathit{Re}\left(V_r{I}^{*}\right)=[\frac{R}{R^2+X^2}\left(\left|V_s\right|\left|V_r\right|\cos \left({\delta }_s-{\delta }_r\right)-{\left|V_r\right|}^2\right)+\frac{X}{R^2+X^2}\left(\left|V_s\right|\left|V_r\right|\sin \left({\delta }_s-{\delta }_r\right)\right)]$ (7.37.a)

(7.37.a)

$Q_r=\mathit{Im}\left(V_r{I}^{*}\right)=[\frac{X}{R^2+X^2}\left(\left|V_s\right|\left|V_r\right|\cos \left({\delta }_s-{\delta }_r\right)-{\left|V_r\right|}^2\right)+\frac{R}{R^2+X^2}\left(\left|V_s\right|\left|V_r\right|\sin \left({\delta }_s-{\delta }_r\right)\right)]$ (7.37.b)

(7.37.b)

when $R/X\ll 1$, the line losses could be neglected. Then, the active and reactive powers at the receiving end of a lossless line can be written as

$P_r=\frac{1}{X}\left|V_s\right|\left|V_r\right|\sin \left({\delta }_s-{\delta }_r\right)$ (7.38.a)

(7.38.a)

$Q_r=\frac{1}{X}(\left|V_s\right|\left|V_r\right|\cos \left({\delta }_s-{\delta }_r\right)-{\left|V_r\right|}^2)$ (7.38.b)

(7.38.b)

Since ${\sin }^2\left({\delta }_s-{\delta }_r\right)+{\cos }^2\left({\delta }_s-{\delta }_r\right)=1$, replacing $\sin ({\delta }_s-{\delta }_r)$ and $\cos ({\delta }_s-{\delta }_r)$ using (7.37) gives

${{(\frac{X{P}_r}{\left|V_s\right|\left|V_r\right|})}^2+(\frac{{\left|V_s\right|}^2+X{Q}_r}{\left|V_s\right|\left|V_r\right|})}^2=1$ (7.39)

(7.39)

Eq. (7.38) could be written as a quadratic equation of $\left|V_r\right|$

${\left|V_r\right|}^4+\left({-\left|V_s\right|}^2+2X{Q}_r\right){\left|V_r\right|}^2+X^2{P}_r^2+X^2{Q}_r^2=1$ (7.40)

(7.40)

The following condition needs to be satisfied for ${\left|V_r\right|}^2$ to have a real solution:

${\left({-\left|V_s\right|}^2+2X{Q}_r\right)}^2-4(X^2{P}_r^2+X^2{Q}_r^2)\ge 1$ (7.41)

(7.41)

As the line is assumed lossless, the sending and receiving active powers are equal ($P_S=P_r$). Therefore, Eq. (7.41) can be written as

$\frac{4X}{{\left|V_s\right|}^2}(Q_r-\frac{X{P}_r}{{\left|V_s\right|}^2})\le 1$ (7.42)

(7.42)

The line stability index (LSI) for line sr is then defined by

${\mathit{LSI}}_{\mathit{sr}}=\frac{4X}{{\left|V_s\right|}^2}(Q_r-\frac{X{P}_r}{{\left|V_s\right|}^2})\le 1$ (7.43)

(7.43)

Both indices under normal and contingency conditions can be used to recognize the weakest nodes in distribution networks where there might be voltage collapse issues as the load increases. The weakest nodes in a distribution system could be the candidates for installing new DG units.

7.3.6.1 Fault Current Contribution by Upstream Grid and the DG Interface

A fundamental requirement for the connection of DG resources to the network is the total fault level which is contributed by the short circuit of the upstream grid and the DG. In medium voltage (MV) and low voltage (LV) radial networks, the fault current contribution of the upstream grid is determined by the short-circuit impedance of the HV/MV or MV/LV transformers. On the other hand, the contribution of DG sources is smaller.

DG systems have maximum acceptable fault current. The fault current level should not exceed this maximum value at every point of the distribution system. Different DG sources and their interconnection have different fault current level contributions. Three types of DG connection interfaces are synchronous generator, induction generator, and power-electronic-interfaced DG units. With DGs, the fault current levels passing through the protective device change. The amount of change in fault current level depends on the type of interface [34].

7.3.6.2 Limit Fault Current

Installing DG units increases the fault current level. Due to the increasing fault current level, protection devices such as breakers and relays that are used for normal distribution may not work for DGs. To minimize the contribution of fault current level of DG units, there are a few methods suggested such as replacing breakers and protection devices. However, these methods are costly.

One approach is to use fault current limiters (FCLs). These devices can be categorized in passive and active types. Passive FCLs are inserted permanently to the distribution system to limit fault current levels. Adding passive FCLs can cause voltage drop, lead to power loss in normal operation, and decrease the power stability of the circuit when satisfying rapidly changing demand. To avoid short coming of passive FCLs, active FCLs have low impedance during normal operation. This nonlinear impedance increases significantly during a fault condition to limit the fault current. FCLs are installed in series with synchronous-based DGs.

7.3.6.3 Calculating Fault Current

There are number of factors of a DG that contribute to the fault current level: The type of DG, the distance of DG from the fault, whether or not a transformer is placed between the fault locations, the configuration of the network between the DG and the fault, and whether we have directly connected DGs or DGs are connected via power electronics.

The fault current total impedance value $Z_{\mathit{Tk}}$ is given by

$Z_{\mathit{Tk}}=\sqrt{(R_{\mathit{Tk}}^2+X_{\mathit{Tk}}^2})$ (7.44)

(7.44)

The 3-phase fault current is given by

$I_{3\mathrm{\varnothing }}=\frac{c{U}_n}{\sqrt{3}*Z_{\mathit{Tk}}}$ (7.45)

(7.45)

where c is the voltage factor, and $U_n$ is the nominal voltage of the network.

7.3.6.4 Contribution of Upstream Grid

The contribution of the upstream grid is calculated by

$I_k^{"}=\frac{c_{\max }{U}_n}{\sqrt{3}(Z_Q+Z_{\mathit{Tk}})}$ (7.46)

(7.46)

where $Z_Q$ is the impedance of the network feeder (upstream grid) at the connection point Q, and $Z_T$ is the impedance of the transformer. $K_T$ is a correction factor used for the impedance of the transformer.

7.3.6.5 Contribution of DG Unit

The contribution of the DG unit is calculated by

$I_k^{"}=\frac{c_{\max }{U}_n}{\sqrt{3}(Z_G+Z_T+Z_L+Z_R+Z_F)}$ (7.47)

(7.47)

where the impedance of the generator ($Z_G$), the transformer ($Z_T$) (if used), the interconnection line ($Z_L$) to the substation, fault impedance ($Z_F$) and the reactor ($Z_R$) (if used) are included, all referred to the voltage at the fault location F.

7.3.7.1 Harmonic Power Flow Analysis

When modeling harmonic power flow calculations, the system can be considered as a combination of passive elements and harmonic current sources [36]. An admittance matrix is generated according to the harmonic frequency such that linear loads utilize resistance in parallel with reactance. Nonlinear loads are modeled as current sources that inject harmonic currents into the system.

The fundamental current of the nonlinear components installed at any bus with real power P and reactive power Q is modeled as

$I^{\left(1\right)}={[\frac{P_i+j{Q}_i}{V^{\left(1\right)}}]}^{*}$ (7.48)

(7.48)

The hth harmonic current is then modeled as

$I^{(h)}=C\left(h\right)I^{\left(1\right)}$ (7.49)

(7.49)

where $C\left(h\right)$ is the ratio of the hth harmonic current to its fundamental current, $I^{\left(1\right)}$.

The branch current resulting from harmonic currents occurring in the system can be calculated by

$B_{\mathit{ij}}^{(h)}=\left[A_{\mathit{ij}}^{\left(h\right)}\right][I^{\left(h\right)}]$ (7.50)

(7.50)

where $\left[I^{\left(h\right)}\right]$ are the system harmonic currents, and $\left[A_{\mathit{ij}}^{\left(h\right)}\right]$ is the coefficient vector of harmonic currents for the branch between bus i and j at the hth harmonic order [37].

The coefficient vector, $\left[A_{\mathit{ij}}^{\left(h\right)}\right]$, is defined to record the existence of system harmonic currents that flow through the branch. The backward current sweep technique [37] is used to build the coefficient vectors. If a bus harmonic flows through the branch, then the coefficient vector will have a value of 1 in its corresponding position. The coefficient vector will only have binary values of 0 or 1. Then, the branch voltage drop resulting from the system harmonic vector can be expressed as

$\Delta V_{\mathit{ij}}^{(h)}=Z_{\mathit{ij}}^{(h)}{B}_{\mathit{ij}}^{(h)}$ (7.51)