Sources and References

Primary references — W. H. Kersting, Distribution System Modeling and Analysis, 3rd ed., CRC Press, 2012; T. Gonen, Electric Power Distribution System Engineering, 2nd ed., CRC Press, 2007. Supplementary texts — T. A. Short, Electric Power Distribution Handbook, 2nd ed., CRC Press, 2014; J. J. Grainger and W. D. Stevenson, Power Systems Analysis, McGraw-Hill, 1994. Online resources — IEEE Std 1547 (interconnection standards); IEEE Std C57.12 (distribution transformers); NERC reliability standards documentation.

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Chapter 1: Distribution System and Smart Grid Overview

1.1 Structure of the Electric Power System

The electric power system is conventionally divided into three tiers: generation, transmission, and distribution. Generation facilities—thermal, hydro, nuclear, and increasingly wind and solar—produce bulk electric energy at voltages ranging from roughly 11 kV to 30 kV. Step-up transformers at the generating station then raise the voltage to transmission levels (115 kV, 230 kV, 345 kV, 500 kV, or 765 kV in North American practice) to minimize resistive losses over long distances. At the load centres, step-down substations reduce the voltage to sub-transmission levels (typically 27.6 kV to 115 kV), and ultimately distribution substations further reduce it to primary distribution voltages.

Distribution systems begin at the secondary bus of the distribution substation and extend to the point of metering at the customer premises. They are characterised by:

• Radial or weakly meshed topology (unlike the looped transmission network)
• Relatively short line lengths (a few hundred metres to 30 km per feeder)
• Significant R/X ratios (resistance comparable to or larger than reactance)
• Unbalanced three-phase loading because single-phase laterals serve residential and small commercial customers unevenly

1.2 Ontario Distribution System Architecture

In Ontario, the Independent Electricity System Operator (IESO) operates the bulk electricity system. Electricity flows from generators through the high-voltage transmission grid (operated by Hydro One Transmission) to distribution substations owned by Local Distribution Companies (LDCs). The Ontario Energy Board (OEB) licenses and regulates LDCs, of which there are approximately 60 in the province.

Sub-transmission refers to the voltage tier between bulk transmission and the primary distribution bus. In Ontario, typical sub-transmission voltages are 27.6 kV, 44 kV, and 115 kV. Sub-transmission lines feed distribution substations either in radial runs or as loops to improve supply security.

Distribution substations step sub-transmission voltage down to the primary distribution voltage. Common primary voltages in Ontario are 4 kV, 8 kV, 13.8 kV, 27.6 kV, and increasingly 44 kV for large urban and suburban feeders. The substation typically contains:

• One to four power transformers (power class 5 MVA–50 MVA)
• High-side circuit switchers or circuit breakers
• Low-side bus and feeder breakers
• Protection and SCADA equipment

Primary distribution feeders extend from the substation to distribution transformers located near customers. Feeder lengths range from less than 1 km in dense urban areas to 30 km or more in rural Ontario.

Distribution transformers step primary voltage down to the customer utilisation voltage—120/240 V for single-phase residential service or 347/600 V (three-phase) for commercial and industrial customers in Canada. In Ontario, 120/240 V single-phase centre-tapped service dominates the residential sector.

1.3 Smart Grid Concepts

A Smart Grid integrates digital communication and control technology into every level of the electric power infrastructure. Key enabling technologies include:

• Advanced Metering Infrastructure (AMI): Two-way communicating meters that provide interval consumption data, enable time-of-use pricing, and support remote connect/disconnect.
• Distribution Automation (DA): Automated switches, fault indicators, and self-healing schemes that reduce outage duration without human intervention.
• Distributed Energy Resources (DERs): Rooftop photovoltaic systems, battery energy storage, electric vehicle chargers, and small combined heat-and-power units connected at the distribution level.
• Volt/VAR Optimisation (VVO): Coordinated control of voltage regulators, switched capacitor banks, and inverter-based DERs to minimise energy losses while maintaining voltage within acceptable bounds.
• Demand Response (DR): Programmes that reward customers for reducing consumption during peak periods, improving system load factor and deferring capital investment.

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Chapter 2: Load Characteristics and Load Forecasting

2.1 Load Definitions and Terminology

Understanding how loads behave in time is fundamental to distribution system planning and operation. Several key definitions establish a common vocabulary.

2.2 Individual Customer Load Characteristics

Residential customers exhibit strong daily and seasonal patterns. Summer peaks result from air conditioning loads; winter peaks from space and water heating (where electric heating is dominant). Commercial loads track business hours and ambient temperature. Industrial loads are typically steadier, with load factor often exceeding 0.85.

Load duration curves rank hourly demands over a year in descending order. The area under the load duration curve equals the total annual energy consumption. They are used to evaluate the economics of generation and storage resources.

2.3 Distribution Transformer Loading

A distribution transformer serves a cluster of customers whose individual peak demands are highly diverse. The transformer’s peak load is therefore significantly less than the sum of individual customer peaks. The ratio of transformer peak load to sum of individual peaks is the coincidence factor.

For a transformer serving \( N \) single-family residential customers, empirical relationships give the coincidence factor as a decreasing function of \( N \). Kersting (2012) provides tabulated values showing that for \( N = 1 \), \( CF = 1.0 \), while for \( N = 70 \) or more, \( CF \) approaches approximately 0.4–0.5 in typical North American climates.

2.4 Feeder Load Characteristics

The coincident peak demand of an entire feeder is built up from transformer loads, which are themselves coincident sums over customer loads. Feeder load profiles are used in voltage drop calculations, power loss estimates, and reliability studies. Planners typically work with:

• Hourly load profiles from AMI or SCADA data
• Annual load curves showing seasonal variation
• Design-hour load representing the worst-case (typically summer or winter peak) scenario for equipment sizing

2.5 Load Forecasting

Load forecasting bridges present measurements and future infrastructure needs. Distribution planners use both temporal (when total load will grow) and spatial (where new load will appear) forecasting methods.

2.5.1 Trend Extrapolation

Trend extrapolation fits historical peak demand data to a mathematical model and projects forward. Common models include:

Linear trend:

\[ P(t) = a + bt \]

Exponential (compound growth) trend:

\[ P(t) = P_0 e^{rt} \]

where \( r \) is the annual growth rate. A linearised form suitable for least-squares regression is \( \ln P = \ln P_0 + rt \).

Polynomial trend: Higher-order polynomials can capture saturation effects but risk overfitting and should be used with caution beyond a five-year horizon.

2.5.2 Spatial Load Forecasting

Spatial load forecasting divides the service territory into geographic cells (typically 0.5 km × 0.5 km to 2 km × 2 km) and forecasts load density in each cell. Inputs include:

• Land-use maps and zoning changes
• Population and employment forecasts from municipal planning
• Historical load density per land-use class (e.g., residential, commercial, industrial, institutional)
• Saturation levels for end-use appliances (HVAC, water heating, lighting)

The total feeder or substation forecast aggregates the cell-level forecasts over its service area. Spatial forecasting enables planners to identify which feeders or substations will be stressed first and to optimally site new infrastructure.

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Chapter 3: Distribution System Feeder Modelling

3.1 Per-Unit System Review

The per-unit (p.u.) system normalises all quantities by base values, allowing transformers to be eliminated from the equivalent circuit and enabling direct comparison of quantities across voltage levels.

Base selection rules: Choose a single base MVA (\( S_{base} \)) for the entire system. Choose a base voltage (\( V_{base} \)) in one zone; base voltages in other zones are set by transformer turns ratios. Base current and base impedance then follow:

\[ I_{base} = \frac{S_{base}}{\sqrt{3}\, V_{base,LL}} \]\[ Z_{base} = \frac{V_{base,LL}^2}{S_{base}} \]

where \( V_{base,LL} \) is the line-to-line base voltage. For a single-phase system substitute \( V_{base,LN} \) and use \( S_{base,1\phi} = S_{base,3\phi}/3 \).

Changing base: An impedance given as \( Z_{pu,old} \) on base \( (S_{base,old}, V_{base,old}) \) is converted to a new base by:

\[ Z_{pu,new} = Z_{pu,old} \times \frac{S_{base,new}}{S_{base,old}} \times \left(\frac{V_{base,old}}{V_{base,new}}\right)^2 \]

3.2 Phase-Frame Representation for Unbalanced Feeders

Distribution feeders are rarely balanced. Kersting’s phase-frame (also called primitive or abc) method retains all three phases (and neutral, where applicable) explicitly as separate nodes, making it the standard approach for unbalanced three-phase analysis.

For a three-phase overhead line segment of length \( \ell \), the voltage drop relationship is:

[ \begin{bmatrix} V_{a,S} \ V_{b,S} \ V_{c,S} \end{bmatrix}

\begin{bmatrix} V_{a,R} \ V_{b,R} \ V_{c,R} \end{bmatrix} + \begin{bmatrix} z_{aa} & z_{ab} & z_{ac} \ z_{ba} & z_{bb} & z_{bc} \ z_{ca} & z_{cb} & z_{cc} \end{bmatrix} \cdot \ell \cdot \begin{bmatrix} I_a \ I_b \ I_c \end{bmatrix} ]

where subscripts S and R denote the sending and receiving ends respectively. The \( 3 \times 3 \) matrix \( [z_{abc}] \) (units: \(\Omega/\text{km}\) or \(\Omega/\text{mile}\)) contains self and mutual impedance terms whose calculation is described in the next section.

3.3 Carson’s Equations for Overhead Line Impedances

Carson (1926) derived equations for the self and mutual impedances of overhead conductors with earth return, accounting for the fact that the earth carries the return current when a ground wire is absent or when the neutral is grounded.

Self-impedance of conductor \( ii \):

\[ z_{ii} = r_i + 4\omega G \times 10^{-4} + j\, 4\omega \times 10^{-4} \left(\ln \frac{1}{GMR_i} + S_{ii}\right) \quad [\Omega/\text{mile}] \]

Mutual impedance between conductors \( ij \):

\[ z_{ij} = 4\omega G \times 10^{-4} + j\, 4\omega \times 10^{-4} \left(\ln \frac{1}{D_{ij}} + S_{ij}\right) \quad [\Omega/\text{mile}] \]

where:

• \( r_i \) = AC resistance of conductor \( i \) (\(\Omega/\text{mile}\))
• \( \omega = 2\pi f \) = angular frequency (rad/s)
• \( G = 0.1609347 \times 10^{-3} \) (earth resistivity term, assuming earth resistivity \(\rho = 100\, \Omega\cdot\text{m}\))
• \( GMR_i \) = geometric mean radius of conductor \( i \) (ft or m, consistent units)
• \( D_{ij} \) = distance between conductors \( i \) and \( j \) (same units as GMR)
• \( S_{ii}, S_{ij} \) = Carson correction series terms (typically small compared to the logarithmic terms for normal power-frequency calculations)

In the simplified Carson’s equations used in most distribution analysis software, the correction terms \( S_{ii} \) and \( S_{ij} \) are approximated, and the equations reduce to:

\[ z_{ii} \approx r_i + 0.0953 + j\,0.12134 \left(\ln \frac{1}{GMR_i} + 7.93402\right) \quad [\Omega/\text{mile at 60 Hz}] \]\[ z_{ij} \approx 0.0953 + j\,0.12134 \left(\ln \frac{1}{D_{ij}} + 7.93402\right) \quad [\Omega/\text{mile at 60 Hz}] \]

3.3.1 Kron Reduction for Neutral Conductors

When the feeder includes a grounded neutral conductor (index \( n \)), the full \( 4 \times 4 \) (or larger) primitive impedance matrix \( [Z_{primitive}] \) is reduced to a \( 3 \times 3 \) phase matrix by Kron reduction:

\[ [z_{abc}] = [z_{aa}] - [z_{an}][z_{nn}]^{-1}[z_{na}] \]

where \( [z_{aa}] \) is the \( 3 \times 3 \) phase-phase submatrix, \( [z_{an}] \) is the \( 3 \times 1 \) phase-neutral coupling column, and \( z_{nn} \) is the neutral self-impedance.

3.4 Underground Cable Modelling

Underground cables are modelled similarly to overhead lines but require different geometric parameters:

• Conductor GMR: From manufacturer data (same concept as overhead).
• Phase spacing: Determined by conduit arrangement or direct-burial spacing, typically much smaller than overhead (30 cm vs. several metres). This results in higher mutual impedance magnitudes.
• Concentric neutral cables: Each phase cable is surrounded by helically wound neutral wires. A concentric neutral cable of \( k \) neutral strands with strand radius \( r_s \) has an equivalent neutral conductor whose GMR and position require special treatment.
• Tape-shielded cables: The shield is treated as a conductor at the cable radius.

For a three-phase underground cable in conduit with separate phase conductors and a common neutral duct, the same Carson’s equations apply, but using underground GMR and the actual physical spacings.

Shunt capacitance is more significant in underground cables than in overhead lines because the insulation permits a closer spacing between the conductor and the return path (shield or concentric neutral). Capacitance per unit length for a round cable:

\[ C = \frac{2\pi \varepsilon_0 \varepsilon_r}{\ln(r_o / r_i)} \quad [\text{F/m}] \]

where \( r_i \) and \( r_o \) are the inner conductor radius and inner shield radius, and \( \varepsilon_r \) is the relative permittivity of the insulation (typically 2.3–3.5 for XLPE or EPR).

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Chapter 4: Voltage Drop and Power Loss Calculations

4.1 Load Allocation Methods

Before a voltage drop or power flow calculation can be performed, loads must be assigned to specific nodes along the feeder. Three primary load allocation methods are used in distribution practice.

4.1.1 Transformer kVA Allocation

In the absence of individual metering data, loads are allocated in proportion to the nameplate kVA rating of each distribution transformer:

\[ P_j = P_{total} \times \frac{kVA_j}{\sum_k kVA_k} \]

This method assumes that all transformers are loaded to the same per-unit level, which is a rough approximation but useful for planning when detailed data are unavailable.

4.1.2 Class and Load Factor Allocation

Customers are grouped into load classes (residential, small commercial, large commercial, industrial). Each class is assigned a load profile—a set of time-varying demand factors that describe the shape of the class load curve. Allocated load at transformer \( j \) for class \( c \) at time \( t \):

\[ P_{j,c}(t) = N_{j,c} \times \bar{P}_c \times LPF_{c}(t) \]

where \( N_{j,c} \) is the number of customers of class \( c \) on transformer \( j \), \( \bar{P}_c \) is the average annual peak demand per customer, and \( LPF_{c}(t) \) is the load profile factor (dimensionless, normalised so its peak is 1.0).

4.1.3 Metered Peak Demand Allocation

When AMI data are available, the actual metered peak demand (or interval average demand) of each customer is known. Loads are allocated directly from meter reads, and the transformer or feeder load is the aggregated and time-coincident sum of customer demands. This is the most accurate method and is increasingly standard with smart meter deployment.

4.2 Voltage Drop Formulas for Radial Feeders

4.2.1 Exact Voltage Drop

For a single line section with series impedance \( Z = R + jX \) (\(\Omega\)) carrying load \( S = P + jQ \) (kVA) at receiving-end voltage \( V_R \), the exact sending-end voltage is:

\[ V_S = V_R + Z I^* \]

where \( I = (S / V_R)^* = (P - jQ)/V_R^* \). Expanding:

\[ \lvert V_S \rvert^2 = \left(\lvert V_R \rvert + \frac{PR + QX}{\lvert V_R \rvert}\right)^2 + \left(\frac{PX - QR}{\lvert V_R \rvert}\right)^2 \]

The voltage drop magnitude (approximate, ignoring the quadrature component for small angles) is:

\[ \Delta V \approx \frac{PR + QX}{V_R} \]

4.2.2 Approximate Voltage Drop Method

For a radial feeder with \( n \) load points, the total voltage drop from the source to load point \( k \) is:

\[ \Delta V_k = \sum_{j=1}^{k} \frac{R_j P_j^{total} + X_j Q_j^{total}}{V_{nom}} \]

where \( P_j^{total} \) and \( Q_j^{total} \) are the total active and reactive power flows in section \( j \) (the sum of all loads downstream of section \( j \)), and \( V_{nom} \) is the nominal voltage used as a constant denominator.

4.2.3 K Factors for Voltage Drop

The K factor method provides a quick hand calculation of voltage drop by lumping the feeder impedance characteristics. For a feeder with uniformly distributed load, the K factor is defined as the voltage drop in percent per kW per unit length at unity power factor:

\[ K = \frac{R_0}{V_{nom}^2} \times 100 \quad [\%/(kW \cdot km)] \]

where \( R_0 \) is the resistance per unit length (\(\Omega/km\)) and \( V_{nom} \) is in kV. At other power factors, a reactive component is added similarly.

The voltage drop to the end of a feeder carrying a uniformly distributed load \( P_{total} \) (kW) over length \( L \) (km) is:

\[ \Delta V_{uniform} = K \times P_{total} \times \frac{L}{2} \quad [\%] \]

The factor \( 1/2 \) arises because a uniformly distributed load behaves as if its entire weight is concentrated at the midpoint. This is the uniformly distributed load model.

4.2.4 Exact Lumped Load Model

When loads are concentrated at discrete points, the voltage drop is calculated by summing section contributions. For two load points at distances \( d_1 \) and \( d_2 \) from the source with loads \( P_1 \) and \( P_2 \):

\[ \Delta V = K \left[(P_1 + P_2) d_1 + P_2 (d_2 - d_1)\right] \]

The general formula for \( n \) lumped loads:

\[ \Delta V = K \sum_{i=1}^{n} P_i \cdot d_i \quad [\%] \]

where \( d_i \) is the distance from the source to load \( i \). This is the principle of load moments: each load is weighted by its distance.

4.2.5 Geometric Configuration Lumping

When a feeder has many loads with a complex spatial distribution, they can be combined into a single equivalent load located at the load centroid:

\[ d_{eq} = \frac{\sum_i P_i d_i}{\sum_i P_i} \]

The voltage drop to the end of the feeder is then calculated using the total load and the equivalent distance, adjusted for the end-of-feeder impedance.

4.3 Load Flow Analysis

Load flow (power flow) analysis for radial distribution feeders uses iterative methods adapted to the radial topology.

4.3.1 Backward/Forward Sweep Method

The backward/forward sweep (BFS) is the standard method for radial distribution load flow:

1. Initialise: Set all node voltages to the nominal value (flat start).
2. Backward sweep: Starting from the end nodes (leaves), calculate branch current injections using known load powers and assumed voltages. Accumulate currents toward the source.
3. Forward sweep: Starting from the source (with known source voltage), update node voltages by applying voltage drops along each branch using the currents found in the backward sweep.
4. Check convergence: If the maximum voltage change between iterations exceeds the tolerance (typically \( 10^{-4} \) p.u.), return to step 2.

BFS converges reliably for radial feeders with R/X ratios typical of distribution systems, where Newton-Raphson (designed for meshed high-voltage transmission networks) can struggle.

4.4 Power Loss Calculations

The total three-phase real power loss in a line section carrying currents \( I_a, I_b, I_c \) with resistances \( R_a, R_b, R_c \) is:

\[ P_{loss} = \lvert I_a \rvert^2 R_a + \lvert I_b \rvert^2 R_b + \lvert I_c \rvert^2 R_c \]

For a balanced three-phase feeder of total resistance \( R \) (\(\Omega\)) per phase carrying total three-phase load \( P \) (W) at voltage \( V_{LL} \) and power factor \( \cos\phi \):

\[ I = \frac{P}{\sqrt{3}\, V_{LL} \cos\phi} \]\[ P_{loss} = 3 I^2 R = \frac{P^2 R}{V_{LL}^2 \cos^2\phi} \]

For a feeder with uniformly distributed load \( P_0 \) (W/km) over length \( L \) (km) with resistance \( r \) (\(\Omega\)/km):

\[ P_{loss,total} = \int_0^L r \left(\frac{P_0 x}{V_{LL}}\right)^2 dx = \frac{r P_0^2 L^3}{3 V_{LL}^2} \]

This equals one-third of the loss that would result if the total load \( P_0 L \) were concentrated at the end of the feeder, reflecting the distributed current flow.

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Chapter 5: Distribution Substation Design

5.1 Substation Function and Voltage Levels

A distribution substation is the interface between the sub-transmission system and the primary distribution network. Its principal functions are:

• Voltage transformation from sub-transmission to primary distribution level
• Switching and isolation for protection and maintenance
• Measurement and metering
• Reactive power compensation and voltage regulation

Typical Ontario distribution substations transform 27.6 kV, 44 kV, or 115 kV to 4 kV, 8 kV, 13.8 kV, or 27.6 kV.

5.2 Substation Transformer Sizing

Transformer sizing must balance present and future load against capital cost. Key considerations:

Normal and emergency ratings: Power transformers have a nameplate rating for continuous operation and a higher emergency (overload) rating (typically 115%–140% of nameplate) that can be sustained for limited periods.

N-1 criterion: The substation should be able to supply its full load with one transformer out of service. If two equal transformers each serve half the load under normal conditions, each must be rated for at least the full substation load under emergency conditions.

Load growth margin: Transformers are typically sized 20%–30% above the present peak load to accommodate growth without replacement within the planning horizon (15–20 years).

Loss evaluation: The capitalized cost of transformer losses (no-load iron losses and load-dependent copper losses) over the transformer’s life is added to the purchase price in the economic evaluation.

5.3 Bus Configurations

The arrangement of buses and switching devices within the substation determines its reliability and flexibility. Four common configurations are used in distribution substations:

5.3.1 Single Bus (Radial)

All equipment connects to a single bus. The simplest and least expensive arrangement. Any bus fault or bus section maintenance requires complete substation outage. Suitable only for substations with very low load or high reliability on the supply side.

5.3.2 Main Bus and Transfer Bus

A transfer bus is provided in addition to the main bus. A normally open bus tie allows any feeder breaker to be transferred to the transfer bus for maintenance without interrupting supply. Improves maintainability but does not improve bus fault reliability.

5.3.3 Ring Bus

All equipment is connected around a ring, with a circuit breaker between each adjacent pair of connections. Any single breaker can be opened for maintenance without isolating any circuit. Any bus section fault is cleared by opening two breakers. The ring bus is a popular choice for medium-voltage substation buses.

5.3.4 Breaker-and-a-Half Configuration

Two main buses, with three breakers for every two circuits in a “bay.” Each circuit is connected between two breakers, sharing the middle breaker with an adjacent circuit. Provides high reliability—any single breaker can be maintained or any single fault cleared without loss of any circuit. Used for critical high-voltage substations; less common at distribution voltage due to cost.

5.4 Protection at the Substation

Primary protection of power transformers uses differential relaying (comparing currents on the high-voltage and low-voltage sides). Overcurrent relays provide backup protection. Distribution feeders are protected by feeder breakers equipped with overcurrent relays (inverse-time or instantaneous), with reclosers and sectionalising switches along the feeder providing sectional fault isolation.

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Chapter 6: Distribution System Reliability

6.1 Reliability Concepts

Reliability in the context of distribution systems refers to the ability to deliver power to customers continuously and without interruption. Two fundamental reliability events are:

• Outage: A component (line, cable, transformer, switch) fails or is taken out of service, interrupting power to some customers.
• Interruption: A customer experiences a loss of supply. An interruption is sustained if it lasts longer than 5 minutes; shorter events are momentary.

Failure rate (λ): The average number of failures per unit time per unit length (for lines) or per unit time (for equipment). Units: failures/(100 km·year) for overhead lines, failures/(component·year) for equipment.

Repair time (r): The average duration of a sustained outage for a given component, including fault location, crew dispatch, repair or switching, and restoration. Units: hours/failure.

Switching time: The time to restore customers who can be restored by switching (without repairing the fault). Typically 0.5–2 hours for manual switching, minutes for automated switching.

6.2 Standard Reliability Metrics

IEEE Std 1366 defines the standard reliability indices used by utilities and regulators worldwide. Ontario LDCs report these indices annually to the Ontario Energy Board.

6.3 Predictive Reliability Evaluation

Predictive reliability evaluation (also called analytical reliability assessment) calculates expected reliability indices from component failure rates and restoration times, before any outages occur. This enables comparison of design alternatives.

For a radial feeder with \( n \) sections, each section \( i \) having failure rate \( \lambda_i \) (failures/year) and average repair time \( r_i \) (hours):

Contribution of section \( i \) to SAIFI at a load point downstream:

\[ \lambda_{LP} = \sum_{i \in \text{upstream}} \lambda_i \]

Contribution to SAIDI:

\[ U_{LP} = \sum_{i \in \text{upstream,repair}} \lambda_i r_i + \sum_{i \in \text{upstream,switch}} \lambda_i s_i \]

where \( s_i \) is the switching time (applicable when the load point can be restored by isolation of the faulted section and closing of a normally-open tie switch).

6.4 Methods to Improve Reliability

Distribution system reliability can be improved through structural and operational changes:

• Feeder automation: Automated sectionalising switches and reclosers reduce outage duration by isolating faults and restoring healthy portions of the feeder in minutes rather than hours.
• Feeder ties and normally-open switches: Connections to adjacent feeders allow load transfers when a section is de-energised. In Ontario, most urban and suburban primary feeders have at least one normally-open tie to another feeder.
• Underground cabling: Underground cables have failure rates roughly 5–10 times lower than equivalent overhead lines per kilometre, substantially reducing SAIFI. However, repair times for cable faults are much longer (12–48 hours versus 2–4 hours for overhead), so the impact on SAIDI is less clear-cut.
• Improved protection coordination: Faster and more selective protection reduces the number of customers affected by each fault.
• Tree trimming and vegetation management: A major cause of overhead line outages in Ontario is contact with trees. Systematic trimming programmes can reduce failure rates by 20–40%.
• Distributed generation and microgrids: DERs configured as microgrids can island from the main grid during outages, maintaining supply to critical loads within the microgrid boundary.

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Chapter 7: Application of Capacitors in Distribution Systems

7.1 Purpose of Capacitor Banks

Shunt capacitor banks connected to distribution feeders provide reactive power (VAR) locally, reducing the reactive component of current flowing from the source. Benefits include:

• Voltage improvement: VAR injection raises voltage at the point of connection and downstream.
• Loss reduction: Reducing reactive current flow reduces \( I^2 R \) losses in lines and transformers.
• Capacity release: By reducing the current magnitude, capacitors free up thermal capacity in conductors, cables, and transformers that can accommodate load growth.
• Power factor correction: Capacitors correct low power factor, potentially avoiding power factor penalties imposed by the utility or reducing reactive power charges.

7.2 VAR Compensation Analysis

Consider a feeder section with impedance \( Z = R + jX \) carrying load \( P + jQ \). With a shunt capacitor of size \( Q_C \) (kVAR) at the load node, the reactive flow in the line reduces from \( Q \) to \( Q - Q_C \). The change in voltage at the load node is:

\[ \Delta V_C = \frac{Q_C \cdot X}{V_{nom}} \]

and the change in real power loss is:

\[ \Delta P_{loss} = -\frac{2Q Q_C X - Q_C^2 X}{V_{nom}^2} \cdot \frac{1}{2} + \ldots \]

For maximum loss reduction, the optimal capacitor size at a single location is:

\[ Q_{C,opt} = Q \]

That is, full reactive power compensation at the load. However, this may cause leading power factor at light load, raising voltage above acceptable limits. Practical capacitor sizing is therefore a compromise.

7.3 Optimal Capacitor Placement

When multiple capacitor banks can be placed along a feeder, the optimal placement problem minimises the combined cost of energy losses and capacitor installation. For a uniformly distributed reactive load \( q_0 \) (kVAR/km) over a feeder of length \( L \):

Two-thirds rule: For a single fixed capacitor, the optimal placement is at \( 2L/3 \) from the source, with a rating equal to \( 2/3 \) of the total feeder reactive load. This placement and rating minimises the total reactive current flowing in the feeder, and thus minimises total losses.

7.4 Fixed vs. Switched Capacitors

Capacitor banks may be:

• Fixed: Always connected. Appropriate where minimum load reactive demand is a substantial fraction of peak reactive demand (i.e., the reactive load does not vary much).
• Switched: Controlled by time clock, voltage relay, reactive power relay, or SCADA command. Appropriate where reactive demand varies significantly between light and peak load.

In Ontario practice, larger substation capacitor banks (several MVARs) are often switched, while smaller feeder banks may be fixed or time-switched. The proliferation of DERs with power factor control capability is changing the optimal mix of fixed vs. switched capacitance.

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Chapter 8: Voltage Regulation

8.1 Quality of Service and Voltage Standards

Voltage quality is a primary measure of distribution system performance. Customers depend on supply voltage staying within limits for correct and efficient operation of equipment. The key standard in Canada is CSA C235, which specifies:

• Preferred range: ±5% of nominal voltage at the point of delivery
• Acceptable range: −8.3% to +5.8% for 120 V nominal systems under normal operating conditions

The Ontario Energy Board incorporates voltage quality into its regulatory framework. Distribution companies are expected to maintain voltage within the CSA C235 preferred limits at the point of metering.

Voltage regulation of a feeder is defined as:

\[ VR = \frac{V_{no-load} - V_{full-load}}{V_{rated}} \times 100\% \]

A feeder with poor regulation has a large difference between no-load and full-load voltage at the end of the feeder, making it difficult to keep voltage within limits across all load levels.

8.2 Step-Voltage Regulators

A step-voltage regulator (SVR) is an auto-transformer with a tap changer that adjusts the turns ratio in discrete steps to maintain the output voltage within a specified band. Regulators are available as:

• Single-phase units: Three required for balanced three-phase regulation; can also be used on single-phase laterals.
• Three-phase units: A gang-operated unit in one enclosure.

8.2.1 Regulator Operation Principle

An SVR consists of a series transformer whose secondary is connected in series with the line, and an exciting transformer whose primary is connected in shunt (parallel) with the line. By changing the tap on the exciting transformer secondary, the series voltage injected into the line is varied from a maximum buck (reduce voltage) to a maximum boost (increase voltage).

Standard IEEE ratings for distribution voltage regulators:

• Regulation range: ±10% in 32 steps of 0.625% each
• Voltage ratings: 2.5 kV to 34.5 kV
• kVA ratings: 25 kVA to 833 kVA per phase (single-phase)

8.2.2 Step Calculations

The output voltage of a regulator set to tap position \( n \) relative to neutral is:

\[ V_{out} = V_{in} \times \left(1 + \frac{n \times 0.00625}{1}\right) \]

where \( n \) is positive for boost and negative for buck, ranging from −16 to +16 (for a 32-step regulator).

8.3 Line Drop Compensator (LDC)

A voltage regulator using a line drop compensator adjusts its tap position based not on the terminal voltage at the regulator, but on an estimated voltage at a remote load centre (the regulation point). The LDC is a small analog circuit inside the regulator control that models the impedance of the feeder between the regulator and the regulation point.

The LDC measures the load current \( I \) and computes an estimated voltage drop:

\[ V_{LP} = V_{source} - I (R_{set} + jX_{set}) \]

where \( R_{set} \) and \( X_{set} \) are settings in volts (on the regulator’s secondary base) that represent the scaled impedance of the feeder to the regulation point. The regulator control adjusts taps to maintain \( V_{LP} \) within the bandwidth around the set point.

8.3.1 LDC Setting Calculation

Given:

• Feeder impedance from regulator to regulation point: \( Z_{line} = R_f + jX_f \) (\(\Omega\))
• Regulator CT ratio: \( CT_R \) (e.g., 500/5 = 100)
• Regulator VT ratio: \( VT_R \) (e.g., 7200/120 = 60)

The LDC settings are:

\[ R_{set} = R_f \times \frac{CT_R}{VT_R} \quad [\text{V}] \]\[ X_{set} = X_f \times \frac{CT_R}{VT_R} \quad [\text{V}] \]

8.4 Three-Phase Voltage Regulators

Three-phase feeders can be regulated by three single-phase regulators connected in wye or open-delta, or by a three-phase regulator. Three connection schemes are common:

• Wye: One regulator per phase, each connected line-to-neutral. Each regulator independently controls its phase voltage. Used when the feeder is roughly balanced.
• Closed-delta: Three single-phase regulators in a delta configuration. More complex to analyse; used when the source is delta-connected.
• Open-delta: Two single-phase regulators regulate all three phases. Less expensive than three units; acceptable when loads are reasonably balanced.

For unbalanced feeders, independent per-phase tap control is necessary. Modern digital regulators allow independent control of each phase, whereas older electromechanical controls operated all three phases on the same tap (gang operation).

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Chapter 9: Distribution System Planning

9.1 Planning Process Overview

Distribution system planning is the systematic process of determining what infrastructure is needed, where, and when, to reliably and economically serve present and future customer loads. The planning process typically involves:

1. Load forecast: Develop spatial and temporal projections of load growth (see Chapter 2).
2. Network analysis: Model the existing system and identify deficiencies (voltage violations, thermal overloads, reliability shortfalls).
3. Alternative development: Generate candidate solutions (new feeders, substation upgrades, DERs, demand response, automation).
4. Evaluation: Assess each alternative on cost, reliability, voltage performance, and non-technical criteria (land use, environment, community impact).
5. Selection and scheduling: Choose the preferred alternative(s) and develop a capital investment plan.

9.2 Criteria and Standards for Planning

Thermal criteria: No conductor, cable, or transformer should exceed its rated current (ampacity or nameplate rating) under normal or N-1 contingency conditions. Thermal overloads accelerate insulation aging and can lead to failures.

Voltage criteria: Voltage at all customer meters must remain within the regulatory limits (CSA C235 in Canada) under all normal and N-1 contingency load conditions.

Reliability criteria: SAIDI and SAIFI targets, mandated or adopted internally, must be met. Some utilities have explicit minimum reliability standards per feeder class (urban, suburban, rural).

Economic criteria: Capital and operating costs are minimised subject to the above technical constraints. Life cycle cost analysis, including the cost of losses and the cost of reliability (value of lost load, VOLL), is used to compare alternatives.

9.3 Feeder Capacity and Reach

Two fundamental limits determine how far a feeder can extend and how much load it can serve:

• Thermal limit (ampacity): The maximum current the conductor can carry continuously without exceeding its rated temperature. For overhead ACSR conductors, ampacity ranges from a few hundred amperes for small sizes to over 1,000 A for large conductors.
• Voltage drop limit: The feeder cannot extend so far that the voltage at the end of the feeder falls below the minimum allowable level even at peak load. Voltage regulators and capacitors can extend the effective reach of a feeder.

For a given conductor size and primary voltage, the feeder reach (maximum allowable length at full load) is the shorter of the thermally-constrained length and the voltage-constrained length. Increasing the primary voltage level (e.g., from 13.8 kV to 27.6 kV) quadratically reduces the voltage drop per kW of load and increases the effective reach.

9.4 Integration of Distributed Energy Resources

The widespread adoption of DERs—rooftop solar PV, battery storage, EV charging—poses new planning challenges:

• Reverse power flow: Solar PV on residential feeders can cause power to flow from load toward the substation during midday, reversing the conventional flow direction. Equipment designed for unidirectional flow (one-way voltage regulators, protection schemes) may malfunction.
• Voltage rise: Distributed generation injects real and reactive power, potentially raising feeder voltage above maximum limits at high-generation, low-load conditions (e.g., midday in spring).
• Protection coordination: Fault current contributions from inverter-based DERs complicate fuse–recloser coordination. IEEE Std 1547-2018 specifies requirements for DER interconnection, including voltage and frequency ride-through, power quality, and protection settings.
• Hosting capacity: The maximum amount of DER generation that can be connected to a feeder without violating voltage, thermal, or protection constraints. Hosting capacity analysis is now a standard planning tool in jurisdictions with high DER penetration.

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Chapter 10: Worked Analysis Examples

10.1 Complete Voltage Drop Example with Phase-Frame Method

10.2 Reliability Index Calculation with Normally-Open Tie

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Chapter 11: Advanced Topics

11.1 Voltage Regulator Interaction and Hunting

When multiple voltage regulators are placed in series on a feeder, or when regulators operate in parallel (at a substation bus), their tap controllers can interact adversely:

• Series regulators: The downstream regulator senses the output of the upstream regulator. If both regulators are set with overlapping bandwidths relative to the same voltage reference, they can oscillate—a phenomenon called hunting—where each repeatedly changes tap to correct for the other’s action.
• Parallel regulators: Regulators in parallel must share reactive load equally. Circulating reactive currents cause one unit to boost and the other to buck. A reactive current compensation circuit in the controller prevents this.

Prevention of hunting: Use time-delay settings in the tap changer control (typically 15–60 seconds) to allow voltage transients to settle before initiating a tap change. Set the bandwidth slightly larger than the step size. Coordinate set points so downstream regulators are set to a slightly higher voltage than upstream regulators, consistent with the expected voltage drop profile.

11.2 Harmonic Considerations in Capacitor Placement

Shunt capacitors can form parallel resonant circuits with the inductive impedance of the feeder and source. If a resonant frequency coincides with a harmonic produced by nonlinear loads (adjustable speed drives, power electronics), harmonic voltage amplification can occur, damaging capacitors and connected equipment.

The parallel resonant frequency is approximately:

\[ f_r \approx f_0 \sqrt{\frac{S_{sc}}{Q_C}} \]

where \( S_{sc} \) is the short-circuit MVA at the capacitor bus and \( Q_C \) is the capacitor rating in MVAR. To avoid resonance at the dominant harmonic order \( h \) (typically 5th or 7th for six-pulse drives):

\[ h_{res} = \sqrt{\frac{S_{sc}}{Q_C}} \neq h \]

Where resonance cannot be avoided by sizing alone, harmonic filters (series inductor added to the capacitor bank to detune it from the resonant frequency) are used.

11.3 Earth Fault Protection in Unearthed and Resistance-Earthed Systems

Some distribution systems, particularly in industrial settings and some European-influenced designs, operate with unearthed (isolated) or resistance-earthed neutral. In these systems, a single phase-to-earth fault does not produce a large fault current; the system can continue to operate with one phase faulted, giving time to locate and clear the fault without causing an interruption.

The fault current in an unearthed system during a single-phase earth fault is:

\[ I_f = 3 V_{LN} j \omega C_0 \]

where \( C_0 \) is the zero-sequence capacitance per unit length of the feeder (total for all three feeders in the system). This current is capacitive and typically ranges from a few amperes to tens of amperes.

In contrast, solidly earthed systems (standard in North America, including Ontario) produce large fault currents for phase-to-earth faults, enabling fast protection operation but causing momentary service interruptions. The trade-off between fault current magnitude (affecting equipment ratings and safety) and fault detection sensitivity is a key design consideration.

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Summary Tables

Key Formulas Reference

Quantity	Formula	Units
Per-unit impedance	\( z_{pu} = Z_{actual} / Z_{base} \)	dimensionless
Base impedance	\( Z_{base} = V_{base,LL}^2 / S_{base,3\phi} \)	\(\Omega\)
Voltage drop (approx.)	\( \Delta V = (PR + QX)/V_{nom} \)	V
Uniformly dist. VD	\( \Delta V = K P L / 2 \)	%
Load factor	\( LF = E / (P_{max} \cdot T) \)	dimensionless
SAIDI	\( \sum U_i N_i / N_T \)	hr/cust/yr
SAIFI	\( \sum \lambda_i N_i / N_T \)	int/cust/yr
CAIDI	\( SAIDI / SAIFI \)	hr/int
ASAI	\( 1 - SAIDI/8760 \)	dimensionless
Regulator tap	\( V_{out} = V_{in}(1 + 0.00625\,n) \)	V
LDC R-set	\( R_{set} = R_f \cdot CT_R / VT_R \)	V
Resonant frequency	\( f_r = f_0\sqrt{S_{sc}/Q_C} \)	Hz

Standard Reliability Index Targets (Illustrative)

Customer Class	SAIDI Target (hr/yr)	SAIFI Target (int/yr)
Urban dense	< 1.0	< 0.5
Urban suburban	< 2.0	< 1.5
Rural overhead	< 6.0	< 4.0