21

Engineering Principles of Electricity Pricing

Lawrence J. Vogt

Mississippi Power Company

21.1    Electricity Pricing Overview

21.2    Electric Cost-of-Service Study

21.3    Cost-of-Service Study Framework

21.4    Minimum Distribution System Analysis

21.5    Analysis of Load Diversity

21.6    Analysis of Demand and Energy Losses

Methodology for Evaluating Losses

21.7    Electric Rate Design

21.8    Cost Curve Development

Coincidence Factor–Load Factor Relationship • Allocation of Unit Demand Cost Components

21.9    Rate Design Methodology

Reference

Electricity is produced by both utilities and nonutility entities, and it is sold in retail transactions with end-use customers and in wholesale transactions with other utilities for ultimate resale. Pricing of these transactions is market based and/or cost based. For example, in some jurisdictions electricity producers are permitted to compete in retail markets in which customers are allowed to choose an energy service provider and pay market-based prices for electricity. However, the electricity must then be transported from the producers to the customers by means of transmission and distribution (T&D) facilities that are owned and operated by the local utility. T&D power delivery service is typically regulated by a government agency under cost-based pricing. In many other cases, local utilities are fully integrated and have exclusive jurisdictional service rights; thus, their production and T&D functions are fully regulated under cost-based pricing.

Electricity pricing is a multidisciplinary function involving accounting, economics, engineering, finance, and regulatory law. The costs of building, operating, and maintaining a power system arise from meeting the capacity, energy, reliability, and power quality needs of a diverse base of electricity consumers. Engineering principles play a particularly key role in developing effective cost-based prices for electricity by taking into account these technical requirements of providing electric services [1].

21.1  Electricity Pricing Overview

To adequately capture the cost-related attributes of electric service through pricing, while preserving equity among customers, three fundamental cost/pricing components are recognized:

•  Energy component: variable costs associated with a customer’s requirements for a volume of kWh, for example, generation fuels

•  Demand component: fixed costs associated with a customer’s maximum kW or kVA load requirements, for example, T&D line and transformer capacity

•  Customer component: fixed costs that are independent of a customer’s energy or capacity requirements, for example, electric service meter

These components are represented as fundamental unit prices or rates, and they are combined and often modified in a variety of fashions to form the basic billing mechanism or rate structure for the various rate classes.

The form of rate structure is varied for different customer segments in order to reflect the distinct load characteristics and cost-of-service differences of each of the groups. Typically, simple rate structures based solely on kWh consumption are utilized to recover energy-related and demand-related costs of service for small watt-hour metered residential and commercial customer rate classes. The composite energy/demand charges may be flat (same price per kWh for all kWh) or blocked, as exemplified in the following.

Residential/small commercial rate example:

Monthly Rate for Secondary Service

Declining block rate structure version	Inverted block rate structure version
Customer charge: $20.00 per month	Customer charge: $20.00 per month
Energy/demand charges:	Energy/demand charges:
First 150 kWh @ 5.6¢/kWh	First 300 kWh @ 3.0¢/kWh
Next 350 kWh @ 4.8¢/kWh	Next 450 kWh @ 4.5¢/kWh
Next 500 kWh @ 4.1¢/kWh	Next 750 kWh @ 6.5¢/kWh
Excess kWh @ 3.7¢/kWh	Excess kWh @ 7.5¢/kWh

In contrast, large demand metered commercial and industrial customer rate structures are much more complex because of wide variations in the cost to serve and the load characteristics between individual customers within a given rate class. In particular, maximum demands and load factors (LFs) often vary significantly between such customers due to unique end-use energy requirements and daily operational cycles. Typically, rate structures for demand metered commercial and industrial customer rate classes include an explicit demand charge along with either a flat or blocked energy charge configuration. When the LFs of customers within a given rate class vary widely, an hours use of demand (kWh per kW) energy charge structure is often utilized, as exemplified in the following.

Large commercial/industrial rate example:

Monthly Rate for Secondary Service

Customer charge		$250.00 per month
Demand charges
First 30 kW	@	$5.25 per kW
Over 30 kW	@	$4.95 per kW
Energy charges:
First 200 kWh per kW
First 6,000 kWh	@	4.0¢ per kWh
Over 6,000 kWh	@	3.0¢ per kWh
Next 250 kWh per kW
First 10,000 kWh	@	2.0¢ per kWh
Over 10,000 kWh	@	1.0¢ per kWh
Over 450 kWh per kW
All kWh	@	0.5¢ per kWh

Establishment of cost-based rate structures and their unit prices is accomplished by first defining homogenous groups (classes) of customers, including residential, lighting, and various subgroups of commercial and industrial, based on load characteristics (primarily demand requirements and LF) and electric service methodology (including delivery voltage level and single-phase service vs. three-phase service). Next the total costs of providing electric service to customers are assigned to these groups and subgroups using cost causation principles. The total costs assigned to a given group are then apportioned to the customers within that group on the basis of monthly customer LFs or kWh per kW usage. This intra-group cost assignment represents an extension of the basic class cost-of-service study, and its results, along with other information, provide guidance for setting practical rate structures and associated prices for each of the customer rate groups or classes. The following sections focus on some of the essential engineering principles that are utilized in the production of an electric cost-of-service study and the subsequent rate design for retail electric service.

21.2  Electric Cost-of-Service Study

Cost-of-service studies are data intensive, and the numerous input requirements, as shown in Table 21.1, call for information from across many of the operational areas of the utility. The cost-of-service study requires both monetary and technical operating data and information. Cost and revenue figures for the study are acquired from corporate financial operating reports, which summarize data entries from various accounting systems (i.e., mass property accounting, revenue accounting, etc.). Having cost information booked in a configuration, such as the Federal Energy Regulatory Commission’s (FERC) Uniform System of Accounts, facilitates the organization of the data elements within the actual cost-of-service study. Technical data are acquired from the utility’s customer information system (CIS) and various engineering sources.

TABLE 21.1 Major Cost-of-Service Study Input Data Requirements

Rate base (investment) items •  Electric plant in service •  Intangible plant •  General plant •  Common plant •  Construction work in progress •  Plant held for future use •  Fuel stock •  Materials and supplies •  Working capital •  Prepayments •  Accumulated depreciationa •  Property insurance and other operating reservesa •  Accumulated deferred income taxesa •  Investment tax creditsa •  Customer advancesa •  Customer depositsa Expenses •  Fuel burned •  Purchased power •  Operations expenses •  Maintenance expenses •  Customer accounts, assistance, and sales expenses •  Salaries and wages •  Depreciation expense •  Administrative and general •  Taxes other than income taxes •  Miscellaneous fees •  Interest on customer deposits •  Amortization of nonrecurring expenses

Income taxes •  Federal •  State •  Tax rates •  Deductible interest Financial information •  Cost of capital •  Capital structure Revenues •  Revenues from sales •  Late charges and forfeited discounts •  Miscellaneous service fees •  Leases and rentals •  Nonterritorial sales credits •  Revenue credits Customer information •  Class designation •  Service voltage level •  Billing determinants Class information •  Hourly load shapes •  Unbilled sales System information •  Generation output •  Territorial purchases •  T&D System loss characteristics •  Substation diagrams •  Distribution circuit characteristics •  Current distribution •  Equipment costs

a Rate base deductions.

21.3  Cost-of-Service Study Framework

The cost-of-service study is fundamental for establishing cost-based rates for electric service. Stated from a financial point of view, the cost-of-service study is a methodology for measuring the earnings position, or profitability, of the various classes of electric service. A key result of the study is the quantification of an annual test period rate of return yield for each class under the existing rates for electric service. A rudimentary framework of the cost-of-service study process is presented in Figure 21.1. The process consists of four principal cost analysis steps:

1.  Functionalization of costs: Basic cost elements are organized in accordance with major operating functions of the power system and its supporting business functions. These power system expenditures are assigned to function based on the voltage level at which the costs are incurred.

2.  Classification of costs: The levelized functional costs are then categorized as being energy related, demand related, or customer related. Since plant costs are associated with specific equipment and facilities, some distribution system cost elements are related to both the customer and demand cost components and must be separated accordingly.

FIGURE 21.1  Principal steps of the cost-of-service analysis process. Functionalization and classification of rate base and expenses are executed for total company costs. The cost assignment process then apportions the total company costs to each of the classes where rates of return and other metrics can be determined on a class basis.

3.  Assignment of costs: The levelized functional cost components are then proportionally assigned to each customer class using (a) class kWh usage to allocate the energy cost components, (b) class load responsibilities to allocate the demand cost components, and (c) the numbers of customers in each class to allocate the customer-related cost components. Cost elements that are incurred exclusively for a specific class of electric service (or even a single customer) are directly assigned to that class (or to the class in which the specific customer is included).

4.  Quantification of results: Allocated expenses and taxes are deducted from class revenues in order to determine class net income amounts, which are then divided by the allocated rate base to determine class rates of return. Other results, such as customer, demand, and energy cost component summaries, can be produced for rate design guidance and support purposes.

As noted in Figure 21.1, the functionalization and classification steps represent analyses of cost data for the utility as a whole. The accounting data and the engineering inputs interact in the cost assignment step in accordance with the cost causation criterion. The completed cost-of-service study indicates results on both a class and a total company basis.

The cost functionalization step is reasonably straightforward as the Uniform System of Accounts defines a utility’s operational cost structure in terms of six principal functions: production, transmission, distribution, customer accounts, customer service and information, and sales. A series of accounts under each major function provide further information, and utilities typically create subaccounts that yield additional detail for cost of service. For example, FERC Account 368—Line Transformers contains the book costs and counts not only for distribution line transformers but also for distribution capacitors, voltage regulators, and cutouts. Levelization, which is a subfunctionalization step, arranges costs according to system voltage levels. For example, the FERC prescribes a 5-level system for cost-of-service studies as follows:

Level 1: production, including generator step-up (GSU) transformers

Level 2: transmission, including subtransmission and associated substations

Level 3: distribution substations, defined by level 2 to distribution voltage conversion

Level 4: primary distribution lines, including primary line equipment and facilities

Level 5: distribution line transformers, including secondary distribution lines and facilities

In the cost classification step, the functionalized costs are classified as fixed or variable in terms of the fundamental cost components: customer, demand, and energy. Customer-related costs are fixed as they are independent of energy or load. Demand-related costs are fixed as they relate to peak load conditions, and a fixed amount of capacity is installed to meet maximum demands even though a customer’s actual demand can vary considerably from hour to hour or even minute to minute. Energy-related costs are variable as they relate to the production and delivery of kWh as energy is consumed. In general, the production, transmission, and distribution substation plant functions (levels 1, 2, and 3) are demand related. The distribution system (levels 4 and 5) is partially demand related and partially customer related. A key analysis that separates distribution level costs between the demand and customer cost components is referred to as the minimum distribution system. Level 1 production fuels and some other O&M expenses are energy related. Classification further aligns the functionalized costs in preparation for assignment to jurisdiction or classes using customer, demand, and energy allocation factors.

The cost assignment step distributes the functionalized and classified total company costs to the specified customer or rate classes. Much of the system functions are jointly used by different classes; thus, an allocation of costs is required at each voltage level. Allocation factors are based on cost causation criteria, which are driven by various technical characteristics of the power system. For instance, all customers do not peak at the same time due to load diversity throughout the system. Thus, coincidence of load is a major issue, which must be considered in the development of rational demand allocation factors. In addition, demand and energy losses are key issues to be considered in the formulation of the ultimate demand and energy allocators to ensure cost allocation equity among the customer groups, which are served at various voltage levels.

TABLE 21.2 Example Cost-of-Service Study Results for Major Customer Classes

Once all investment items (i.e., the rate base), O&M expenses, and income taxes are allocated to the classes, various class financial metrics and other cost-of-service study results can be quantified. An example summary of cost-of-service study results is shown in Table 21.2. Financial performance measures include the return on investment (ROI) and the return on equity (ROE) under present rates. The ROI for the total utility or for a given customer group j is determined by the following:

$RO{I}_j=\frac{RE{V}_{PRE{S}_j}-EX{P}_j-I{T}_j}{R{B}_j}$

(21.1)

where

ROI_j is the return on investment for group j in percent

REV_PRESj is the group j annual revenue under present rates

EXP_j are O&M expenses assigned to group j

IT_j are income taxes assigned to group j

RBj is the rate base (investment) assigned to group j

The numerator in Equation 21.1 is referred to as the net operating income (NOI), while the result of (REV_PRESj – EXP_j) is referred to as the NOI before income tax (NOI_BIT).

For an investor-owned business, common equity (common stock) is a component of the overall cost of capital, which also includes debt (bonds) and preferred stock. Like ROI, the ROE can also be measured on both a total and a customer or rate class basis using the results of the cost-of-service study. The ROE for the total utility or for a given customer group j is determined by the following:

$RO{E}_j=\frac{NO{I}_j-\left[R{B}_j\times \left(D+PS\right)\right]}{R{B}_j\times E{Q}_{CAP}}\times 100$

(21.2)

where

ROE_j is the return on equity for group j in percent

NOI_j is the net operating income for group j

RB_j is the rate base for group j

D is the cost rate for long-term debt (bonds) in percent

PS is the cost rate for preferred stock in percent

EQ_CAP is the common equity capitalization in percent

The common equity capitalization ratio represents the amount of the utility’s common equity relative to its total amount of capital.

As noted in Table 21.2, the ROI and ROE values for the commercial customer class are significantly higher than the total company values, whereas the same values for the residential class are significantly lower than the total company values. ROI and ROE values of both the industrial class and the lighting class are fairly close to the total company values. Such information is useful when allocating rate revenue increases or decreases to the customer or rate classes of electric service, particularly when the objective is to move the class rate of return values closer to the total company values.

The cost-of-service study also provides information that is vital for establishing rate structures and unit prices for each of the rate schedules for electric service. In particular, the basic customer, demand, and energy cost components derived from the cost-of-service study allocations can be cast as component revenue requirements, which include O&M expenses, income taxes, and an ROI. These data serve as inputs to the rate design process by providing a starting point for establishing rates that are intended to recover costs on a cost causation basis.

The following three sections address key cost-of-service analyses that specifically rely on engineering principles and methodologies. These sections include the minimum distribution system analysis, analysis of load diversity, and analysis of demand and energy losses. These analyses ensure a cost causation approach to cost classification and cost assignment.

21.4  Minimum Distribution System Analysis

The concept of a minimum distribution system recognizes that the costs of the primary and secondary distribution system have both customer-related and demand-related attributes. As discussed previously, the customer cost component is associated with no-load conditions, whereas the demand cost component is associated with load conditions, that is, equipment and facilities have the capacity to meet peak loads. Some devices can be categorized as being either customer related or demand related, while some individual devices are related to both components.

A direct approach to identifying and quantifying the customer and demand cost components is to evaluate each major distribution item in terms of its mission. For example, a distribution system protection scheme consists of a mix of circuit breakers, reclosers, and fused cutouts that are coordinated to minimize customer outages in the event of faults on the system. A simple feeder schematic including protective devices is shown in Figure 21.2.

FIGURE 21.2  A schematic of a primary feeder system. A fault occurs on the primary tap line, which connects customer #26 to the main feeder. Under a temporary fault condition, customers downstream from the recloser will experience a momentary outage, but service is restored to all customers. Under a permanent fault condition, the same momentary outage will occur to all customers, but only customer #26 will lose service as fuse F-1 blows to clear the fault from the main feeder. A switched capacitor located on the line between fuses F-3 and F-4 provides feeder voltage support under heavy loading conditions.

The protective devices work together to preserve service to as many customers as possible under various fault conditions. In essence, the protection scheme safeguards the “voltage path,” which connects each customer on the circuit to the source, that is, the substation. Consider a case where all customer load is removed from the circuit, and a fault occurs as before; the protection scheme will operate exactly as it did under load in an effort to maintain the voltage path to as many customers as possible. The protection scheme’s mission is independent of load or demand and thus is a customer-related function. Furthermore, other facilities that provide a connection of a customer to the source have a customer-related attribute.

Consider again the circuit in Figure 21.2 during peak demand periods. Under light loading conditions, the capacitor is switched off. However, once high load currents create an unacceptable voltage drop, the capacitor connects to the circuit to raise the voltage along the feeder profile, thereby releasing capacity. The capacitor’s mission is to support voltage under load and is thus dependent on load. Absent customer load, the capacitor is not required at all, even as the circuit is energized to create a voltage path. The capacitor is independent of the voltage path; thus, it is a demand-related component.

The missions of a number of other distribution system devices have both customer-related and demand-related attributes. Primary and secondary conductors clearly create a path interconnecting customers to a source, but they also have capacity to carry load. Line transformers are integral to the path as they connect the secondary conductor system to the primary conductor system, but they also provide capacity to serve customers’ loads.

When a single device has both customer-related and demand-related attributes, its total cost must be allocated. The minimum-intercept or zero-intercept methodology provides a rational basis for separating the cost of a device between its customer and demand components. The zero-intercept methodology is a weighted linear regression of the unit costs of standard ratings or sizes of a specific device, such as a single-phase overhead line transformer, plotted as a function of its capacity-related characteristic, which would be kVA for a line transformer. The objective of the regression analysis is to determine the y-intercept. The y-intercept represents that portion of a device’s total cost that is associated with zero capacity and thus the customer-related component. The unit costs must be weighted by the numbers of devices because of the uneven distribution of the various ratings or sizes of the devices in service. The slope and the y-intercept of the weighted linear regression equation are given by the following:

$m=\frac{\left[N\times {\displaystyle {\sum }_{i=1}^{\tau }{n}_i{X}_i{Y}_i}\right]-\left[{\displaystyle {\sum }_{i=1}^{\tau }{n}_i{X}_i\times {\displaystyle {\sum }_{i=1}^{\tau }{n}_i{Y}_i}}\right]}{\left[N\times {\displaystyle {\sum }_{i=1}^{\tau }{n}_i{X}_i^2}\right]-\left[{\displaystyle {\sum }_{i=1}^{\tau }{n}_i{X}_i}\right]}$

(21.3a)

$b=\frac{{\displaystyle {\sum }_{i=1}^{\tau }{n}_i{Y}_i}}{N}-m\times \left[\frac{{\displaystyle {\sum }_{i=1}^{\tau }{n}_i{X}_i}}{N}\right]$

(21.3b)

where

m is the slope

b is the y-intercept

X_i is the unit size

Y_i is the unit cost

n_i is the number of units of a particular size or rating

N is the total number of all units (i.e., Σn_i)

τ is the total number of sizes or ratings

It is not critical for the regression analysis to have unit cost data points for all transformers in service; however, a range of the smaller size units in each category is important to ensure data linearity and reasonable y-intercept results. In addition, the data points of a particular regression should represent the same input and output voltages (e.g., 15 kV, 208Y/120 volt pad mount transformers). Given the wide range of characteristics (voltages, one vs. two bushings, etc.) within a major transformer category, the regression analysis can be conducted using the most common or representative transformer type in each major category with the results then applied to the population of transformers in each respective category.

A problem arises often when embedded (book) costs are used in the regression analysis because of wide variations in equipment vintage. Over time, equipment ratings have generally trended upward due to higher distribution voltages and increased customer loads. However, the book records may reflect a significant number of small size devices in service that have relatively low average unit costs due to the older age of their installations. Concurrently, larger size devices have much higher average unit costs as they were more recently installed. This temporal disparity in unit costs can distort the regression results and even produce a negative y-intercept. An alternative approach overcomes such incongruities as it is based on an estimate of the cost required to “rebuild” all of the devices in service all at once using current unit construction costs applied to the full inventory of devices. The y-intercept from a regression using current unit costs in lieu of embedded unit costs can then be applied to determine the customer component proportion of the total “rebuild” cost. This ratio is then applied to the book cost in order to estimate the customer component on an embedded cost basis.

An example of zero-intercept regression results for single-phase and three-phase overhead (pole mount) and underground (pad mount) line transformers is shown in Figure 21.3. The y-intercept (where kVA = 0) represents the no-load customer cost component for each of the four transformer types.

Once the y-intercept unit cost is calculated for a given category of devices, the total cost of all such devices (units) in that category is apportioned between the customer cost component and the demand cost component by

FIGURE 21.3  Zero-intercept regression analysis for four categories of distribution line transformers.

TABLE 21.3 Example Results of a Level 5 Line Transformer Classification Analysis

$\text{Customer}\text{cost}=y\text{-intercept}\times \text{Total}\text{number}\text{of}\text{units}$

(21.4a)

$\text{Demand}\text{cost}=\text{Total}\text{category}\text{cost}-\text{Customer}\text{cost}$

(21.4b)

As shown in Table 21.3, the y-intercepts from the regression examples shown in Figure 21.3 are multiplied by the total number of units of line transformers, by major category, to compute the respective customer-related costs. The demand-related amounts are then calculated as the difference between the total cost of all transformers and the customer cost.

Conductors and poles also fall into the category of distribution facilities that must be classified by both customer and demand components. Separate regressions are needed for overhead and underground and primary and secondary distribution conductors. Separate regressions are needed for wood, concrete, and steel distribution poles. The regression analyses are similar to line transformers, but the independent variables that represent capacity are different. For example, conductor capacity is typically addressed by using conductor ampacity, although conductor size (MCM) also serves as a very good proxy for capacity.

21.5  Analysis of Load Diversity

The assignment of costs depends on customer loads being quantified at each level of the system. Due to load diversity, a customer’s load ratio share of a line transformer is vastly different than the customer’s load ratio share of generation or transmission. Load diversity both between individual customers and between customer groups varies considerably throughout the power system. This effect occurs as a result of the interaction of the customer’s load with other customer loads at that particular service level and at the higher levels of the system. Thus, the cost causation effect of an individual customer’s load varies when viewed at the point of power delivery and at all other voltage higher within the system.

Monthly load shapes for an example commercial customer, an entire commercial class of customers, and a total system are illustrated in Figure 21.4. As shown in the upper chart of Figure 21.4, the system peak demand is established on the third Friday of the month at a magnitude of about 1,600 MW. In the center chart of Figure 21.4, the commercial class peak for the month is established on Friday of the following week at a magnitude of about 230 kW. Because of the time difference between the occurrence of the system peak and the commercial class peak, the value of the class maximum load is referred to as the class non-coincident peak (NCP). The magnitude of the commercial class demand at the time of the system peak is about 200 kW, and this value is referred to as the class coincident peak (CP). As shown in the lower chart of Figure 21.4, an individual customer within the commercial class is found to establish a monthly maximum demand of approximately 37 kW during the third Tuesday of the month. However, the customer’s demand at the time of the commercial class peak is found to be less than 5 kW. Furthermore, the customer’s demand at the time of the system peak is also less than 5 kW as its operation is observed to have shut down to a minimum load level just prior to the system peak hour. A comparison of the three load shapes illustrates the nature of load diversity throughout the system. The example customer’s peak demand is significantly diverse with respect to both the peak load of the commercial class and the peak load of the system.

Demand-related cost drivers are functions of load diversity. The cost to serve a customer at the local level is based on sizing the capacity of local facilities to meet the customer’s peak demand, which could occur at any hour of the day and on any day of the year. In this example, line transformer capacity that can properly handle the kVA load associated with the customer’s 37 kW peak demand would be required. As observed in Figure 21.4, the customer’s load at the time of either the commercial class peak or the system peak would not provide an accurate representation of the customer’s service transformer capacity requirements.

FIGURE 21.4  An example of load diversity during the course of a month. A single customer’s peak demand (lower chart) does not occur at the same time as either the class peak (center chart) or the system peak (upper chart). Load diversity characteristics are key to providing an adequate amount of capacity at each functional level of the power system.

In contrast, a customer’s peak demand does not provide a true representation of that customer’s production level capacity requirements. The example commercial customer actually requires less than 5 kW of production capacity, which is a small fraction of its peak demand of 37 kW. While some customers may establish their peak demands simultaneously with the system peak, many other customers’ peaks occur in other hours. Production capacity is sized to meet the diversified loads of all customers as a whole, that is, where load diversity is greatest. Since costs are typically allocated to rate or customer classes, class CP load values are highly indicative of production cost responsibilities assignable to the classes.

Building production capacity to serve an extreme load circumstance that may happen only occasionally would not be the most economical decision when other operational options exist for providing capacity. For example, supplemental capacity might be acquired from tie-line interchange with neighboring utility systems, and/or interruptible service contracts with large customers might be invoked during system critical loading conditions. An understanding of the typical system load diversity characteristics is a key factor in not only providing adequate capacity to serve load but also determining the capacity-related cost responsibilities of the customer or rate classes.

Figure 21.4 provides just a single month’s view of load diversity. During another month, the example customer may indeed peak at the same time as the class peak, system peak, or both as the class peak might also occur at the time of system peak in any given month. A more comprehensive view of load diversity at the upper functional levels of the system is provided by viewing the interactions of load over multiple months or a whole year. Compared to a single month CP, an average of multiple monthly class peaks relative to the average of multiple monthly system peaks is more appropriate for determining the typical amount of load diversity at the production and transmission levels of the system. A “12-CP” methodology is utilized frequently for the allocation of Level 1 demand–related production plant costs as it yields a very high load diversity characteristic to the demand allocation factors. The 12-CP allocation factor for each customer or rate class is determined by the following equation:

$DA{F}_{C{P}_j}=\frac{{\displaystyle {\sum }_{i=1}^{12}C{P}_{ij}}}{{\displaystyle {\sum }_{i=1}^{12}S{P}_i}}\times 100$

(21.5)

where

DAF_CPj is the 12-CP allocation factor for the jth class, as a percent

CP_ij is the coincident load of the jth class in the ith month, in MW

SP_i is the system monthly peak load in the ith month, in MW

The result of Equation 21.5 is a class allocation factor representing an average month of a test year. Averaging across the 12 monthly system and class peaks captures the diversity in weather-sensitive end-use load devices in utility systems that experience appreciable seasonal peak load cycles.

Coincidence factor (CF) values, which correspond to the major functional levels of the power system, are plotted for example residential, commercial, and industrial customer classes in Figure 21.5. A substantial difference in LF exists among the three classes. The correlation between LF and load diversity is evidenced by the difference in shapes of the class CF profiles. Higher LF class loads are more coincident with system peaks than are lower LF class loads. With the exception of a 100% LF customer, this relationship holds for groups (classes) of customers as opposed to individual customers since even a solitary low LF customer could peak simultaneously with the total system load.

As discussed previously, a 12-CP allocation factor is utilized often for allocation of level 1 production demand costs. At lower system levels, other allocation factors based on load diversities that are characteristic of each level are more reflective of cost causation principles. For example, the average of monthly peak season class CP values may be more typical of transmission and/or distribution substation load diversities. Load diversity at the primary distribution system level is well represented by class NCP loads since the times at which feeders peak vary widely based on each feeder’s particular load mix. Secondary distribution customer NCP values, that is, individual customer maximum demands, are more appropriate for representing load diversity at the line transformer and secondary voltage level. The customer NCP allocation factor is based on the maximum annual demands of customers for each secondary voltage customer or rate class and is determined by the following:

FIGURE 21.5  A plot showing the effect of LF on the coincidence of load at various power system levels. The number of observations (customer demands), upon which the calculations of the class CFs are based, is a key aspect of the level of load diversity realized. The CFs for the residential, commercial, and industrial classes decrease at different rates when moving from the service transformer to the generator due to differences in class LFs. The decrease is more pronounced with lower LFs; however, at 100% LF, a load is unconditionally coincident with the peak at every level of the system regardless of the number of observations considered.

$DA{F}_{MA{X}_j}=\frac{{\displaystyle {\sum }_{k=1}^n{D}_{jk}}}{{\displaystyle {\sum }_{j=1}^N{\displaystyle {\sum }_{k=1}^n{D}_{jk}}}}\times 100$

(21.6)

where

DAF_MAXj is the maximum load allocation factor for the jth class, as a percent

D_jk is the maximum annual demand of the kth secondary voltage customer of the jth customer or rate class, in MW

n is the total number of secondary voltage customers in each class

N is the total number of secondary voltage customer or rate classes

An example of load diversified demand allocation factors used to assign costs to customer classes is shown in Table 21.4. A given allocator is indicated by a series of percentage-based factors, which represent the cost causation share of the total cost element for each applicable class.

TABLE 21.4 Example Customer Class Demand Cost Allocation Factors

21.6  Analysis of Demand and Energy Losses

Electrical losses throughout the T&D system represent a major factor in the development of energy and demand cost allocators. Energy sales are recorded by revenue meters located at each customer’s point of service. Since customers receive service at different voltage levels, the adjustment of the metered kWh units to incorporate total system line and transformation losses ensures that allocations of the fuel and variable O&M costs incurred at the production level of the system are accomplished in a fair and equitable manner. In other words, prior to allocation, class sales at the meters are transformed to their equivalent share of energy production at level 1 generator buses. For cost allocation purposes, energy-related losses are assessed on an annual basis.

Demand-related losses are assessed on a system peak load basis, since they are related to system capacity requirements. Class demands at the service points are adjusted for losses up through the system to the level or levels at which they will be applied as demand-related cost allocators. For example, a CP type of allocator used for assigning production plant costs would require that the associated class demands be adjusted for total system line and transformation losses in order to reflect their equivalent share of production capacity at level 1. In contrast, secondary voltage class NCP demands would first be adjusted for secondary line and line transformer losses to make them equivalent with primary voltage class NCP demands. The loss-adjusted secondary voltage class demands would then be further adjusted, along with the primary voltage class demands, to incorporate primary feeder losses for development of a level 4 NCP set of allocation factors. Core, or no-load, losses are constant in time as they represent a steady-state condition caused simply by energizing the transformers. Core losses are independent of a transformer’s loading conditions. On the other hand, load losses are a function of load current, that is, i²R, and are present in both the windings of transformers and line conductors.

Figure 21.6 illustrates the i²R aspect of losses. The load-related losses in a given hour are proportional to the square of the load in that hour. As a result, the losses at peak loading conditions are proportionately higher relative to the system load than during off-peak loading conditions. At 5:00 am, when system load is at its minimum for the day, the associated load losses are slightly more than 4% of the load in that hour. However, at the 5:00 pm system peak hour, losses represent nearly 10% of the system load. The magnitude of the no-load losses is constant in every hour.

FIGURE 21.6  Peak day profile of system load and the associated T&D losses consisting of load- and no-load-related losses.

A plot comparing a unitized system load with its associated load-related losses, on an annual load duration basis, is shown in Figure 21.7. Compared to a system LF of 55% (i.e., the unitized average load), the i²R nature of losses results in a 29% LF for the losses. The LF of the losses is a key component for modeling line conductor and transformer load–related energy and demand losses. The LF of the load-related losses is a function of the square of the load, as determined by the following equation:

$L{F}_{Loss}=\frac{1}{T}{{\displaystyle \sum _{i=1}^T\left(\frac{L_i}{L_{Max}}\right)}}^2$

(21.7)

where

LF_Loss is the annual LF of the load losses, in percent

L_i is the load in the ith hour, in MW

L_Max is the maximum hourly load occurring during the year, in MW

T is the number of hours in the year

FIGURE 21.7  Annual unitized load duration curves of system load and the associated load-related losses. The LF of the losses is much less than the LF of the load itself as the average losses relative to the peak system losses are much lower. The LFs are equal to the unitized average values.

For the cost-of-service study, system losses are modeled for each voltage level. A simplified schematic diagram for an entire power system is used as a framework for modeling peak demand and annual energy flows from the territorial inputs (generators and intersystem tie lines) to the customer delivery points. Transformer and line losses are accounted for at each voltage level. Each voltage level is analyzed as a node with power flowing in from various sources, including the higher voltage subsystems, and with power flowing out in terms of level sales, level losses, and outputs to lower voltage subsystems. The sum of the input and output power flows for each node is equal to zero.

A detailed schematic diagram of a 34.5 kV subsystem is shown in Figure 21.8. Power flows into this 34.5 kV node, which is indicated by the encircling dashed line, from the 138 and 69 kV subsystems at points “A” and “B.” In addition, one or more generators are connected to the 34.5 kV system by means of 13–34.5 kV GSU transformers. Note that the GSU and the input substations (SUB-1 and SUB-2) represent the total of all such voltage transformations. For instance, SUB-2 may represent the composite of just a few or even dozens of 69 to 34.5 kV substations that are scattered throughout the power system territory. The input substations are metered at the low voltage buses of the transformers (M-1 and M-2), while the generators are metered at their output terminals (M-G).

FIGURE 21.8  A nodal model of a 34.5 kV subsystem. Power inputs to the node are shown at “A,” “B,” and the generator “G.” Power outputs are shown at “C,” “D,” and the 34.5 kV customers. Implicit outputs to be determined include transformer losses and line losses occurring along the 34.5 kV lines, which are indicated by the line segment “X” to “Y.”

Sales to customers taking service at 34.5 kV are metered for billing purposes (M-R). In reality, customer delivery points are distributed along the lines; however, for modeling losses for the cost-of-service study, sales can be treated as a single nodal output from the composite 34.5 kV line represented by “X” to “Y.” The 34.5 kV level outputs to the two lower voltage subsystems are indicated as “C” and “D;” however, these outputs are metered at the low voltage buses of SUB-3 and SUB-4 (M-3 and M-4), which are located a voltage transformation beyond the edge of the 34.5 kV node.

As indicated in Figure 21.8, metered load data are essential to determining the power flows into and out of the node. Differences in power flows are key to quantifying the energy and demand losses. Basic watt-hour metering adequately captures the annual energy flows, but interval metering is necessary to determine the demands during the time of system peak. While large commercial and industrial customers typically are metered with interval recorders for billing purposes, peak demands for residential and smaller commercial and industrial customers might have to be estimated from load research sample data.

Transformer data are also required for the loss analysis, including factory results of full-load-loss and no-load-loss tests. Individual test results are best for substation power transformers, but typical results may be sufficient for line transformers. Installed transformer MVA capacities are needed as well, including substation OA/FA/FOA ratings. Transformer demand and energy losses are determined from the factory test data. Since the no-load-related transformer loss data are constant under any transformer loading condition, the associated demand and energy no-load losses for a given transformer are determined simply by the following:

$D_{TNL}=NLL$

(21.8a)

$E_{TNL}=NLL\times T$

(21.8b)

where

D_TNL are peak hour demand-related transformer no-load losses in MW

NLL is the no-load loss rating of the transformer in MW

E_TNL are annual energy-related transformer no-load losses in MWh

T is the number of hours in the year

The no-load-related demand and energy losses of individual transformers are summed to get the total for each of the different voltage transformations. For the 34.5 kV subsystem shown in Figure 21.8, total no-load-loss values are calculated for each of the four substation transformers and for the GSU transformers.

Load-related transformer demand and energy losses are functions of transformer loading conditions. The demand load losses of a transformer at the time of system peak are determined by the following:

$D_{TLL}=FLL\times {\left(\frac{L_{@Peak}}{T_{CAP}\times PF}\right)}^2$

(21.9a)

where

D_TLL are peak hour demand-related transformer load losses in MW

FLL is the full-load loss rating of the transformer in MW

T_CAP is the transformer capacity in MVA

PF is the power factor of the transformer load

L_@Peak is the transformer load coincident with the system peak in MW

The quantity contained within the brackets of Equation 21.9a represents the percent loading of the transformer at the time of system peak.

A transformer’s energy load loss across the year is a function of its demand load loss as determined by the following:

$E_{TLL}=\frac{D_{TLL}}{{\left(L_{@Peak}/L_{Max}\right)}^2}\times T\times L{F}_{Loss}$

(21.9b)

where

E_TLL are annual energy-related transformer load losses in MWh

D_TLL are peak hour demand-related transformer load losses in MW

L_@Peak is the transformer load coincident with the system peak in MW

L_MAX is the transformer maximum annual load

T is the number of hours in the year

LF_Loss is the annual LF of the losses in percent

The denominator of Equation 21.9b adjusts the transformer’s demand load losses at the time of system peak to the transformer’s demand losses at the time of its peak loading condition. In other words, this adjustment takes into account load diversity between the system peak and the transformer peak. Without this adjustment, the annual energy load losses would be understated.

21.6.1  Methodology for Evaluating Losses

The loss characteristics of the power system are evaluated by modeling the power flows from the generators to the loads. Generally, the analysis begins at the highest voltage levels and ends with the lowest voltage levels. However, each voltage level (node) can be analyzed as a subsystem and then consolidated with the other voltage subsystems. The 34.5 kV subsystem schematic diagram in Figure 21.8, along with the input data provided in the following, will serve as the basis for this example. Note that even at three decimal places, some rounding difference will be observed.

Transformer Data	Capacity (MVA)	NLL (MW)	FLL (MW)
GSU	19.412	0.008	0.038
SUB-1	70.185	0.100	0.301
SUB-2	10.910	1.015	0.042
SUB-3	20.605	0.029	0.091
SUB-4	7.035	0.002	0.008

^a Load factor, loss factor, and coincidence factors determined from hourly loads.

Other data

Annual hours = 8760

Power factor @ peak = 98%

> Calculation of 34.5 kV subsystem inflow @ “A”	> MW	> MWh
SUB-1 metered demand and energy (M-1)	29.772	251,170
SUB-1 NL demand losses (Equation 21.8a)	0.100
SUB-1 NL energy losses (Equation 21.8b)		879
SUB-1 demand load losses (Equation 21.9a)	0.056
SUB-1 energy load losses (Equation 21.9b)		467
Total inflow from 138 kV subsystem	29.929	252,515

In the same manner, the demand and energy inflows from the 69 kV subsystem to SUB-2 @ “B,” the demand and energy outflows to the 4.16 kV subsystem (SUB-3) @ “C,” and the demand and energy outflows to the 14.4 kV subsystem (SUB-4) @ “D” are determined to be the following:

	MW	MWh
34.5 kV subsystem inflow @ “B”	4.522	38,153
34.5 kV subsystem outflow @ “C”	6.672	27,472
34.5 kV subsystem outflow @ “D”	19.027	89,886

Note that the sum of the demands and energies for “C” and “D” represents the 34.5 kV subsystem outflows at “Y,” which are 25.699 MW and 117,358 MWh.

Generation is also connected to the 34.5 kV subsystem. The output of the GSU is determined by subtracting the calculated transformer no-load and load-related demand and energy losses from the generator terminal demand and energy meter readings (M-G), which are determined to be 18.218 MW and 24,781 MWh. The total input to the 34.5 kV lines @ “X” can now be determined:

	MW	MWh
SUB-1 metered demand and energy (M-1)	29.772	251,170
SUB-2 metered demand and energy (M-2)	4.498	37,950
GSU output:	18.218	24,781
Total input to the 34.5 kV lines @ “X”	52.488	313,901

While line losses are distributed along the lengths of the conductors and demand and energy sales are also distributed based on customer delivery points, both the load-related losses and sales can be represented as single nodal outputs. The line losses between “X” and “Y” can be determined, as residual values, given the known inputs and outputs:

	MW	MWh
Total input to the 34.5 kV lines @ “X”	52.488	313,901
Less: total output of the 34.5 kV lines @ “Y”	25.699	117,358
Less: metered demand and energy sales (M-R)	25.710	192,000
Total line losses	1.079	4,543

The results of the calculated losses and power flows through the example 34.5 kV subsystem are organized in Table 21.5. While the initial analysis of system losses proceeded by evaluating power flows from the generators and the upper level voltage subsystems to the loads and to the lower voltage subsystems (a downward direction), the loss analysis model developed for the cost-of-service study works in reverse (an upward direction).

TABLE 21.5 Total Power Flow through an Example 34.5 kV Subsystem

To complete the model framework, (a) load-related loss factors for lines and transformers and (b) power flow branching factors are determined. For example, the 34.5 kV line losses were calculated to be 1.079 MW at peak load and 4,542 MWh across the year. Thus, the respective line loss factors as a percentage of the subsystem demand and energy outflows are 2.100% (1.079 MW/51.409 MW = 0.020988) and 1.468% (4,542 MWh/309,358 MWh = 0.014683). Similarly, transformer load–related loss factors are determined by the ratio of transformer loss to transformer outflow. For example, the load-related loss factor for the 138–34.5 kV substation transformers is 0.189% (0.056 MW/29.772 MW = 0.001894).

Branching factors indicate how the system power flows split between different voltage levels. For example, the total power inflow to the 34.5 kV subsystem from 34.5 kV generation and the higher voltage subsystems is 52.712 MW at peak load. The inflow from the 138 kV subsystem alone is 29.992 MW or 56.7789% of the total subsystem inflow. Due to no-load and load-related losses, the power flow out of the 138 kV transformers to the 34.5 kV lines is 29.722 MW. Since the total 34.5 kV subsystem transformer outflow to the 34.5 kV lines is 52.488 MW at peak load, the 138–34.5 kV transformer output represents 56.7219% of the total transformer outflow.

All information needed to model power flows and calculate losses for each of the customer or rate classes has now been determined. The model integrates all of the voltage subsystems together into a single, uniform analysis. Given the determination of the various system loss and branching factors, the only input required to determine the results for a given customer or rate class is that class’s annual energy sales and load at the system peak.

21.7  Electric Rate Design

Rate design is the process of establishing a rate structure for a given class of electric service, along with its constituent price components, that is capable of achieving a specified revenue requirement at a commensurate level of risk. The cost-of-service study is a key resource for the rate design function, as it provides electric service costs on a customer, demand, and energy component basis for the major power system functions of production, transmission, distribution (primary and secondary), customer accounts, customer assistance, and sales. Such information, along with the associated customer, demand, and energy billing units of the rate classes, provides a basis from which to develop elementary rates for electric service.

21.8  Cost Curve Development

A cost curve bridges the gap between the cost-of-service study and rate design. Basic unit costs are translated into prices within a potentially complex rate structure. Cost curve development can be thought of as an intra-rate cost allocation methodology that further encompasses the characteristics of load diversity relative to the costs of electric service across all levels of energy and demand usage for a specified rate class. Like the cost-of-service study, the more intricate part of the cost curve allocation process revolves around the demand cost components. The cost curve serves as an essential model for designing and evaluating alternative demand rate structures, particularly when displayed in graphical form.

A flowchart describing the input data and analytical processes needed to develop a cost curve for a given demand rate class is shown in Figure 21.9. Input data include the cost-of-service study component revenue requirements for the given rate. Another input is an hours use of demand bill frequency, which is based on the subject rate’s kWh usage and actual kW maximum demands. The frequency is a distribution of customer monthly maximum demands as a function of LF in the form of kWh per kW. Since customer maximum demands represent customer NCP values (kW_MAX), they must be diversified in order to determine the equivalent class kW_NCP and kW_CP values. The relationship that exists between CFs and LFs is used to diversify the customer maximum demands. The unit costs of service for an example rate class consisting of 12,000 commercial and industrial secondary three-phase customers having an annual energy use of 2,100,000 MWh are calculated as follows:

Unit customer cost = $6,885,417/(12,000 customers × 12 months)

    = $47.8154 per bill

Unit demand costs:

•  Production/transmission = $67,560,485/4,099,734 kW_CP

    = $16.4792 per kW_CP

•  Substation/primary lines = $9,451,570/5,273,832 kW_NCP

    = $1.7922 per kW_NCP

•  Transformers/sec. lines = $4,372,917/6,888,228 kW_MAX

   = $0.6348 per kW_MAX

FIGURE 21.9  Cost curve development process. Three major analytical steps are implemented to produce a cost curve for a given rate class. The cost curve is of a form similar to an hours use of demand rate structure, and it can be easily plotted and compared to the associated rate structure that is currently used for billing.

Unit energy costs:

•  Nonfuel = $19,637,444/2,100,000,000 kWh

  = $0.009351 per kWh

•  Fuel/purchased power = $51,292,168/2,100,000,000 kWh

   = $0.024425 per kWh

21.8.1  Coincidence Factor–Load Factor Relationship

The relationship between the coincidence of load, measured in terms of a CF, and customer monthly LF is a key cost causation driver and thus a central model for development of a rate class cost curve. In essence, the relationship shows that CF has a nonlinear relationship to LF, as illustrated by the CF–LF curve plotted in Figure 21.10. Specifically, the relationship is in the form of a third-order polynomial.

The CF–LF curve in Figure 21.10 is based on the conventional calculation of the CF, which is the ratio of the maximum demand of a group of individual loads to the sum of the maximum demands of the individual loads. In this context, the individual loads are customers having a common value of LF, and several LF groups are required to accurately plot the CF–LF curve across the entire LF spectrum. When considering an LF group’s position with respect to the total system load, the resulting maximum demand of the customer group represents an NCP type of factor. In other words, a group peak may or may not occur simultaneously with the system peak demand. The cost-of-service study regards an NCP type of factor as being highly indicative of the cost causation of the demand-related component of the primary distribution system. Thus, the conventional CF–LF relationship, designated here as CF_NCP, can be utilized as an intra-rate cost allocator for the primary distribution demand costs assigned to a particular rate class.

The conventional CF–LF relationship is not a plausible indicator of cost causation for production and transmission system demand-related costs since load diversity is much greater at these levels of the system than at the distribution system level. A strong indicator of the cost causation driver of production and transmission is a CP type of demand cost allocation factor. Consequently, an adapted CF is required for the development of a modified CF–LF relationship in order to properly allocate these costs in an intra-rate cost curve development. The conventional NCP and the modified CP CFs are given by the following:

FIGURE 21.10  The nonlinear relationship between CF and LF. The x-axis represents monthly LFs of individual customers. The y-axis represents the CFs of groups of customers having the same monthly LFs. The CF–LF relationship was first identified by Constantine W. Bary; thus, it is often referred to as the Bary Curve.

$C{F}_{NCP}=\frac{D_{MAX}}{{\displaystyle {\sum }_{i=1}^n{d}_i}}C{F}_{CP}=\frac{D_{SYS}}{{\displaystyle {\sum }_{i=1}^n{d}_i}}$

(21.10)

where

CF_NCP is an LF group’s NCP type of CF

CF_CP is an LF group’s CP type of CF

D_MAX is an LF group’s maximum demand

D_SYS is an LF group’s demand at the time of system peak

d_i is the maximum demand of the ith customer in an LF group

n is the number of customers in the LF group

The significant demand-related cost causation driver of local facilities is customer maximum demands. In essence, a single customer’s maximum demand is fully coincident with itself under all conditions (i.e., 1 kW/1 kW = 1 kW), since diversity of a single load is undefined. Thus, the CF based on customer maximum demands is represented as 100% at all levels of LF, and it can be designated as CF_MAX. A chart comparing CF_CP, CF_NCP, and CF_MAX curves is shown in Figure 21.11. As shown, a kWh per kW scale has been added to the x-axis of the CF–LF chart. A cost curve for a rate class is in a form similar to a multi-step hours use of demand rate structure; thus, it is more practical to work with kWh per kW units than with LF percentages. The average month is 730 h; thus, 730 kWh per kW is equivalent to a monthly LF of 100%.

21.8.2  Allocation of Unit Demand Cost Components

The unit demand costs developed previously for the example rate class consist of the following functional components: $16.4792 per kW_CP for production and transmission, $1.7922 per kW_NCP for distribution substations and primary lines, and $0.6348 per kW_MAX for line transformers and secondary lines. The CF–LF curves are used to prorate the unit demand costs for production and transmission and for distribution substations and primary distribution lines, as shown in Table 21.6. The prorated unit demand costs are keyed to various hours use of demand levels. The prorated unit costs are applicable to a customer’s peak demand, based on that customer’s hours use of demand, since the unit costs themselves account for the characteristics of load diversity between a customer’s peak demand and its effective contribution to the system peak and the LF group peak loads. The three unit demand cost components are summed to obtain a total unit cost assessment as a function of hours use of demand. Equation 21.11 is used to develop the average demand cost on a ¢ per kWh basis between any 2 kWh per kW points of usage:

FIGURE 21.11  Comparison of the CP and NCP CF–LF curves. The CP curve lies below the NCP curve due to greater customer load diversity at the time of the system peak as opposed to the time of an LF group’s peak. The MAX curve represents a customer’s demand at the local facilities level where diversity is nonexistent for all practical purposes at maximum loading conditions. Also shown is a second x-axis scale having units of kWh per kW, which facilitates the chart’s use for the development of cost curves.

TABLE 21.6 Allocation of Unit Demand Costs

$E_{AVG}=\frac{\Delta D_{TOT}}{\Delta HUD}\times 100$

(21.11)

where

E_AVG is the average demand cost in ¢ per kWh

ΔD_TOT is the difference in unit demand cost in $ per kW

ΔHUD is the difference in hours use of demand in kWh per kW

Generally, increments of 50 kWh per kW are sufficient enough to graph the average cost curve, particularly on a hyperbolic scale of hours use of demand. For example, the average cost between 100 and 150 kWh per kW, based on Table 21.6 data, is calculated as follows:

$\left[\left(\$8.2978\text{/kW}-\$5.8731\text{/kW}\right)/\left(150\text{kWh/kW}-100\text{kWh/kW}\right)\right]\times 100=4.8494\text{¢}\text{per}\text{kWh}$

All or a portion of fuel-related costs may be embedded in the basic rate structure. Assuming for this example that all fuel-related costs are to be recovered through a separate rate mechanism, only the nonfuel energy cost component, that is, $0.009351 per kWh, needs to be added to the average demand costs to obtain the total average demand and energy cost as a function of kWh per kW. The total cost structure is thus defined by the following:

Customer cost: $47.8154 per bill

Demand/energy cost:

kWh/kW	¢/kWh	kWh/kW	¢/kWh
0	0.0000	400	2.2597
20	9.2493	450	1.9790
50	6.2103	500	1.8492
100	6.1905	550	1.8704
150	5.7845	600	2.0424
200	4.5308	650	2.3655
250	4.0074	700	2.8394
300	3.2739	≥730	3.3232
350	2.6913

The hours use–based cost structure is plotted in Figure 21.12. The combined demand and energy cost structure is plotted both with and without the customer cost component. Since the customer cost component is truly fixed, that is, not a function of either kW or kWh, plotting of the cost curve with the customer cost component requires an assumption of demand so that its average cost per kWh can be calculated as a function of kWh per kW. A 50 kW value was assumed for this plot as it is approximately equal to the average monthly kW per customer for this example rate class. The y-intercept at x = 20 kWh per kW is approximately 14¢ per kWh. A cost curve based on a demand higher than 50 kW would have a comparable intercept that is somewhat less than 14¢ per kWh because of the relatively higher amount of kWh that exists, even at low hours use. The opposite would occur with demands that are less than 50 kW. While this cost curve could serve as a billing rate, it is rather complicated. Rate design utilizes the cost curve as a guide for developing more practical rate structures.

FIGURE 21.12  Demand rate cost curves. A cost curve in ¢ per kWh is plotted as a function of the hours use of demand using a hyperbolic scale for the x-axis. The demand and energy cost structure is plotted both with and without the customer cost component. A demand of 50 kW is assumed for plotting the curve with the customer component. The selection of different demand assumptions would result in a family of curves when the customer cost component is included.

21.9  Rate Design Methodology

The design of a practical rate structure, which tracks the cost curve in a reasonable manner, could be approached in different ways. For this example, a three-step rate design is illustrated for the hours use ranges of 0–200 kWh per kW, 200–400 kWh per kW, and over 400 kWh per kW. To simplify the rate design process, the design initially addresses only the kWh charges (the finalization of the customer charge is considered during the fine tuning process). Thus, the combined demand and nonfuel energy revenue target is $101,022,417 (the total base rate revenue target with the customer-related cost is $107,907,834). The demand and energy-related cost curve, without the customer cost component, is shown in Figure 21.13.

The 0–200 kWh per kW price step is considered first. The average cost is found to be 6.808¢ per kWh at 100 kWh per kW and 5.850¢ per kWh at 200 kWh per kW. By inspection of the cost curve, it is reasonable to set the initial unit price of the first 200 kWh per kW step in the middle of these values, that is, 6.33¢ per kWh. Without inclusion of the customer charge, the first hours use of demand price of the rate structure will display as a flat line when graphed, as observed in Figure 21.13.

Now consider the tail step (i.e., 400 kWh per kW) of the concept rate structure. By inspection of Figure 21.13, a straight-line extrapolation of the cost curve based on the range of 400–730 kWh per kW indicates an average rate at infinity of approximately 2.3¢ per kWh. This rate at infinity would also represent the value of the tail step price. Thus, initial values for the first and last steps of the rate structure have been selected. The revenue produced by each of these 2 h use steps can then be determined by applying the two prices to their associated kWh quantities, which results in $82,131,266.

The remainder of the combined demand and energy revenue amount is to be collected by the middle 200–400 kWh per kW step. The unit price for this middle step is determined by dividing its revenue responsibility by its associated kWh, which yields a result of 2.84¢ per kWh.

FIGURE 21.13  Hours use of demand rate design. A three-step hours use of demand rate designed to track the cost curve closely between 100 kWh per kW and 730 kWh per kW. The plots exclude the customer charge. The hours use of demand rate structure was originally developed by Arthur Wright, a British engineer.

Due to rounding, when the initial kWh prices are applied back to the billing determinants, a revenue target overcollection of $12,622 is calculated. The customer-related cost component is now incorporated for fine tuning to the total base rate revenue target. The kWh prices for the final rate design are shown in Figure 21.13, and the rate curve is plotted in comparison to the cost curve.

As can be observed in Figure 21.13, the rate curve does not track the cost curve below 100 kWh per kW. If low LF customers are served under this rate, then another rate design could be considered by adding an explicit demand charge to the hours use of demand structure. The addition of such a fixed charge will increase the average rate per kWh in the lower LF range. Reducing the first 200 kWh per kW price by ½¢ would free up revenue, in the amount of $6,096,856, for the explicit demand charge. The demand charge is then determined by dividing this revenue amount by the sum of the associated customer maximum demands, which yields a result of $0.8851/kW. After fine tuning of the charges to achieve the target revenue, the modified hours use of demand rate structure is plotted in comparison to the cost curve in Figure 21.14. Although the rate structure has been made somewhat more complex, it tracks the cost curve very closely across the whole spectrum of usage.

Alternatively, a simple rate structure with an explicit demand charge and a flat energy charge (as well as the customer charge) can be developed. The CF–LF relationship was utilized previously to develop a cost curve based on diversified demands. A cost curve, which is not based on diversified demands, can also be developed from the cost-of-service study and demand unit data for the example general service rate. In this case, the total demand-related revenue requirements for production, transmission, distribution substations, primary distribution lines, line transformers, and secondary distribution lines are summed and then divided by the total of the customer maximum demands. Thus, the nondiversified cost curve is represented by $11.8151 per kW_MAX and $0.009351 per kWh. As illustrated in Figure 21.15, the nondiversified cost structure has been designated as the “fundamental Hopkinson” rate design as its demand charge consists of only demand-related costs while its energy charge consists of only the nonfuel energy–related costs. By shifting a portion of demand cost recovery responsibility from the demand charge to the energy charge, that is, by applying the principle of rate tilt, the average rate per kWh is observed to decrease on the left side of the chart and to increase on the right side of the chart. The point of pivot is located at 305 kWh per kW, which is also the hours use of demand at which the fundamental Hopkinson rate and the tilted Hopkinson rate intersect. Furthermore, 305 kWh per kW represents the average hours use of demand for the example rate class. Any degree of applied rate tilt will cause the average rate curve to pivot at this point. With full rate tilt, the average rate would be plotted as a straight line across the entire chart, that is, a uniform rate per kWh; thus, the Hopkinson rate structure would be transformed into a nondemand, flat energy rate structure.

FIGURE 21.14  Modified hours use of demand rate design. By adding an explicit demand charge to a basic hours use of demand rate structure, the average rate is elevated at low hours use of demand since the demand charge is a fixed charge and thus provides the same declining average rate effect as caused by a customer charge. This design is referred to as a Wright-Hopkinson rate structure. Like Wright, Dr. John Hopkinson was a British engineer.

FIGURE 21.15  Hopkinson rate design. The nondiversified cost curve is designated as the “fundamental Hopkinson” rate design. By tilting the fundamental Hopkinson rate, a portion of the demand cost recovery is transferred to the kWh price. In so doing, the average rate per kWh of the “tilted Hopkinson” rate structure rate shifts downward on the left and upward on the right. The pivot point is 305 h use of demand, that is, the average hours use of the rate class.

Note in Figure 21.15 how the tilted Hopkinson rate tracks the HUD diversified cost curve generally over the range of 175–500 kWh per kW. Further rate tilt could be applied so as to make the Hopkinson rate coincide with the HUD diversified cost curve at 100 kWh per kW. Then the Hopkinson rate would track the HUD diversified cost curve generally over the range of 100–275 kWh per kW. However, at hours use above about 275 kWh per kW, the average rate would rise well above the HUD diversified cost curve.

Selection of a particular rate structure design is guided to a great extent by the load characteristics of the customers served under each particular rate class of service. Rate classes of high LF customers are best served under the fundamental Hopkinson rate structure. A rate class of medium to high LF customers would be well served under a basic Wright rate structure. If a rate class serves customers with a wide distribution of LFs from low to high, the Wright–Hopkinson design would be most practical.

Reference

1.  Vogt, L. J. 2009. Electricity Pricing: Engineering Principles and Methodologies. Boca Raton, FL: Taylor & Francis.