Understanding Why Electricity Markets Over-Price Transmission Losses

Examining the universal factor-of-2 over-recovery in electricity market loss allocation systems and exploring Average Loss Factor alternatives

By Shankar Karki | Published: September 2025 |

This analysis examines the mathematical foundations of marginal loss factor systems used in electricity markets globally, with particular focus on AEMO's implementation in the National Electricity Market. We demonstrate that all marginal loss factor systems universally exhibit factor-of-2 over-recovery due to the quadratic nature of transmission losses combined with marginal cost pricing principles.

An alternative Average Loss Factor (ALF) framework is presented that addresses this mathematical over-recovery while maintaining locational efficiency signals. The analysis includes detailed mathematical proofs, computational comparisons, and implementation scenarios using real AEMO operational data.

Section 1: Marginal Loss Factor Fundamentals

1.1 The Physics of Transmission Losses

When electricity flows through transmission lines, energy is inevitably lost due to the physical properties of conductors. These losses occur because electrical current flowing through resistance creates heat, following fundamental laws of physics.

The Fundamental Loss Equation

For any transmission line carrying electrical current, the power loss is governed by:

P_loss = I² × R

Where:

P_loss = power lost as heat (MW)
I = current flowing through the line (amperes)
R = electrical resistance of the conductor (ohms)

In power system analysis, we typically work with power flows rather than currents. Using the relationship between current and power:

I = P / (V cos φ)

Where P is power flow, V is voltage level, and cos φ is power factor (typically close to 1.0 for transmission systems).

The Quadratic Relationship

Substituting this relationship into the loss equation gives us the fundamental quadratic loss relationship that drives all marginal loss factor mathematics:

AEMO's Visual Explanation of the Quadratic Relationship

AEMO's own documentation provides an excellent illustration of why marginal loss factors differ from average losses. Consider the parabolic loss curve below:

Key Insight from AEMO's Example:

At 100 MW power transfer, total losses = 3 MW
Average loss rate = 3% (3 MW ÷ 100 MW)
Marginal loss rate = 6% (slope of tangent at that point)
Factor of 2 relationship: 6% = 2 × 3%

As AEMO explains: "when the load increases by 1MW, the generator will need to increase output by 1.06MW" because marginal losses are actually 6 per cent, even though total losses are only 3 per cent.

This mathematical relationship exists in all power systems globally and creates systematic differences between marginal and average loss allocation methods.

1.2 Economic Rationale for Loss Allocation

In electricity markets, generators produce power and consumers use it, but some energy is lost in transmission. This creates a fundamental accounting question: Who should pay for these losses?

The Market Challenge

The total losses in a power system are:

L_total = Σ_lines k_line × P_line²

But losses on any individual line depend on power flows from multiple generators. This creates an allocation problem: how do we fairly assign loss responsibility to individual market participants?

Economic Efficiency Principles

Economic theory suggests that prices should reflect marginal costs—the cost of producing one additional unit. In transmission losses, this means:

Marginal loss cost = additional losses caused by a small increase in generation
Average loss cost = total losses divided by total generation

The marginal approach aims to:

Send correct price signals for economic dispatch
Incentivize efficient generator location in low-loss areas
Minimize total system costs through optimal resource allocation

The Mathematical Challenge

However, the quadratic nature of losses creates a mathematical complication:

Marginal Loss Rate = ∂L_total/∂P = 2kP
Average Loss Rate = L_total/P = kP

This means: Marginal = 2 × Average

This mathematical relationship exists in all power systems globally and creates systematic differences between marginal and average loss allocation methods.

1.3 Global Approaches to Loss Factor Implementation

Electricity markets worldwide have developed different approaches to handle transmission loss allocation, but all face the same fundamental mathematical relationships.

Static vs Dynamic Loss Factor Approaches

This conceptual comparison illustrates the relative differences between static and dynamic loss factor approaches used globally. The values shown are qualitative assessments for comparison purposes, not empirically measured data.

Static approaches (like AEMO's) calculate factors annually and apply them consistently, offering high predictability but lower real-time accuracy. Dynamic approaches (like US ISOs) recalculate loss effects every 5 minutes, providing high real-time accuracy but greater complexity and volatility.

Despite these operational differences, both approaches use the same quadratic loss functions in their optimization objectives, leading to identical factor-of-2 over-recovery relationships.

Static Loss Factor Approach

Used by: Australia (AEMO), New Zealand, some European markets

Methodology:

Calculate loss factors annually based on projected operation
Apply fixed factors throughout the year for each generator/load
Update factors once per year with forward-looking analysis

Mathematical Process:

Model expected system operation for entire year (8,760 or 17,520 intervals)
Calculate marginal loss sensitivity for each connection point
Compute generation-weighted average to create annual factor
Apply static factor in real-time market operations

Dynamic Loss Factor Approach

Used by: United States ISOs (PJM, ERCOT, CAISO, etc.), many European markets

Methodology:

Calculate loss effects real-time during market dispatch
Incorporate loss impacts directly into Locational Marginal Prices (LMP)
Update every 5 minutes based on current system conditions

Mathematical Process:

Solve Security-Constrained Economic Dispatch every 5 minutes
Include transmission loss costs directly in optimization objective
Extract loss component from resulting shadow prices
Publish LMPs with embedded loss signals

Mathematical Equivalence Across Approaches

Despite different implementation methods, all marginal loss factor systems share the same fundamental mathematical structure:

Common Mathematical Foundation

Both static and dynamic approaches calculate:

Loss Factor = 1 - ∂L_total/∂P_generator

Using chain rule expansion:

∂L_total/∂P_generator = Σ_lines 2k_line × P_line × PTDF_{line,generator}

The factor of 2 appears in both approaches from the quadratic loss function derivative.

Universal Result

Regardless of implementation approach (static annual vs. dynamic real-time), all marginal loss factor systems produce:

Total Marginal Allocation = 2 × Total Actual Losses

This mathematical relationship is universal across all power system configurations and operating conditions.

Section 2: Static MLF Mathematics - AEMO Approach

2.1 AEMO's Two-Stage MLF Framework

The Australian Energy Market Operator (AEMO) calculates Marginal Loss Factors through a two-stage process that separates MLF calculation from operational dispatch:

Stage 1: Annual MLF Calculation

AEMO runs comprehensive AC power flow analysis modeling the entire National Electricity Market (NEM) operation for the upcoming financial year
This stage uses detailed AC network models to capture voltage effects, reactive power flows, and precise loss calculations
Results in static MLF values published annually for each connection point

Stage 2: Operational Economic Dispatch

AEMO uses the pre-calculated static MLF values in real-time economic dispatch
Operational dispatch typically uses simplified DC optimal power flow for computational speed
The static MLFs are applied as fixed coefficients rather than being recalculated in real-time

This two-stage approach represents one of the most sophisticated implementations of static loss factor methodology globally.

Stage 1: MLF Calculation Scope and Scale

AEMO's annual MLF calculation (Stage 1) covers:

AEMO MLF Calculation Scale

Time Horizon

365 Days

Full financial year (July 1 - June 30)

Time Resolution

30 Min

Trading intervals

Total Intervals

17,520

Half-hour periods per year

Regional Scope

5 Regions

QLD, NSW, VIC, SA, TAS

2.2 Mathematical Formulation of AEMO's MLF Calculation

Decision Variables

For each trading interval t ∈ {1, 2, ..., 17520} and the AC network model:

P_g,t = active power generation from generator g in interval t (MW)
Q_g,t = reactive power generation from generator g in interval t (MVAr)
P_ℓ,t = active power flow on transmission line ℓ in interval t (MW)
V_i,t = voltage magnitude at bus i in interval t (per unit)
θ_i,t = voltage angle at bus i in interval t (radians)

Note: This comprehensive AC formulation is used for MLF calculation only. Operational dispatch uses the resulting static MLF values in simplified DC models.

Objective Function for MLF Calculation

AEMO's annual MLF calculation minimizes the total expected cost of serving load across all intervals using AC power flow constraints:

min Σ_t=1¹⁷⁵²⁰ [Σ_g∈𝒢 C_g(P_g,t) + Σ_ℓ∈ℒ ψ · k_ℓ P_ℓ,t²]

Where:

C_g(P_g,t) = generation cost function for generator g
k_ℓ = loss coefficient for transmission line ℓ
ψ = loss penalty weight (typically ~$100/MWh)

Critical Mathematical Element: The quadratic loss terms k_ℓ P_ℓ,t² in this MLF calculation stage create the marginal loss factors through the optimization's first-order conditions.

Constraint Set

Power Balance Constraints

For each bus i and interval t:

Σ_{g∈𝒢_i} P_g,t - D_i,t = Σ_{ℓ∈δ⁺(i)} P_ℓ,t - Σ_{ℓ∈δ^-(i)} P_ℓ,t + L_i,t

DC Power Flow Constraints

For each transmission line ℓ = (m,n) and interval t:

P_ℓ,t = B_ℓ (θ_m,t - θ_n,t)

Generator and Line Limits

P_g^min ≤ P_g,t ≤ P_g^max
|P_ℓ,t| ≤ P_ℓ^max

2.3 Shadow Price Analysis and MLF Derivation

The annual published MLF reports from AEMO considers generation costs, transmission losses, and network constraints to determine optimal dispatch patterns.

The shadow prices from this optimization contain the marginal loss sensitivities that become the published MLF values. Notice how generators closer to load centers receive MLFs closer to 1.0, while remote generators receive lower MLFs reflecting transmission distance.

Lagrangian Formulation

The Lagrangian for AEMO's optimization problem incorporates all constraints with dual variables (shadow prices):

ℒ = Σ_t=1¹⁷⁵²⁰ [Σ_g C_g(P_g,t) + Σ_ℓ ψ k_ℓ P_ℓ,t²]
+ Σ_t,i λ_i,t [power balance constraints]
+ Σ_t,ℓ μ_ℓ,t [flow constraints] + ...

First-Order Optimality Conditions

Generator Output Stationarity

For generator g at bus i in interval t:

∂ℒ/∂P_g,t = ∂C_g(P_g,t)/∂P_g,t + λ_i,t = 0

This gives: λ_i,t = -∂C_g(P_g,t)/∂P_g,t

Economic Interpretation: λ_i,t is the shadow price representing the marginal cost of serving additional load at bus i in interval t.

Line Flow Stationarity

For transmission line ℓ = (m,n) in interval t:

∂ℒ/∂P_ℓ,t = 2ψ k_ℓ P_ℓ,t - λ_m,t + λ_n,t + μ_ℓ,t = 0

Critical Observation: The factor 2 appears explicitly from differentiating the quadratic loss term ψ k_ℓ P_ℓ,t².

MLF Calculation Methodology

Step 1: Marginal Loss Sensitivity

For generator g at bus i, the marginal loss sensitivity is:

∂L_total,t/∂P_g,t = Σ_ℓ∈ℒ 2k_ℓ P_ℓ,t × PTDF_ℓ,i

Step 2: Interval MLF Calculation

MLF_g,t = 1 - ∂L_total,t/∂P_g,t

Step 3: Annual MLF Aggregation

The published annual MLF for generator g is the generation-weighted average:

MLF_g = Σ_t=1¹⁷⁵²⁰ (MLF_g,t × P_g,t*) / Σ_t=1¹⁷⁵²⁰ P_g,t*

2.4 Mathematical Properties of AEMO's Complete MLF System

Revenue Over-Recovery Mathematical Proof

Claim: AEMO's static MLF system results in total allocated costs exceeding actual transmission losses by exactly factor 2.

Proof:

Total marginal loss allocation across all generators:

Σ_g∈𝒢 Σ_t=1¹⁷⁵²⁰ P_g,t* × (∂L_total,t/∂P_g,t)
= 2 Σ_t=1¹⁷⁵²⁰ Σ_ℓ∈ℒ k_ℓ (P_ℓ,t*)²

Total actual losses:

Σ_t=1¹⁷⁵²⁰ L_total,t = Σ_t=1¹⁷⁵²⁰ Σ_ℓ∈ℒ k_ℓ (P_ℓ,t*)²

Therefore: Marginal Allocation = 2 × Actual Losses ∎

Intra-Regional Settlement Surplus (IRSS)

This mathematical over-recovery manifests in AEMO's market settlements as the Intra-Regional Settlement Surplus:

IRSS = Σ_g (MLF Revenue from Generator g) - Actual Transmission Losses

IRSS Redistribution Mechanism: AEMO's MLF system mathematically creates over-recovery (the IRSS), and AEMO indicates this surplus is redistributed through settlement adjustments. However, the exact redistribution mechanism and ultimate beneficiaries require deeper investigation to understand fully.

Section 3: Dynamic LMP Mathematics - US ISO Approach

3.1 Real-Time Market Operations in US ISOs

United States Independent System Operators (ISOs) implement marginal loss factor concepts through a fundamentally different approach than AEMO's static annual methodology. Instead of calculating fixed loss factors once per year, US ISOs incorporate transmission loss effects directly into real-time Locational Marginal Prices (LMPs) that update every five minutes.

Key US ISO Markets Using Dynamic LMP

PJM Interconnection

13 states + Washington D.C.

Largest electricity market in the world by energy volume

ERCOT

Texas

Energy-only market with real-time scarcity pricing

CAISO

California + portions of Nevada

Leading renewable integration and storage deployment

ISO-NE

New England (6 states)

Capacity market with forward capacity auctions

NYISO

New York State

Complex transmission constraints and congestion patterns

MISO

Midcontinent (15 states)

Large geographic scope with diverse generation mix

Real-Time Optimization Frequency

US ISOs operate on much shorter time horizons than AEMO's annual approach:

Market clearing frequency: Every 5 minutes
Optimization horizon: Typically 1-2 hours ahead
Price updates: Real-time LMPs published every 5 minutes
Market participants: Receive immediate price signals for operational decisions

This high-frequency approach means transmission loss effects are calculated and priced in real-time based on actual system conditions rather than annual forecasts.

3.2 Security-Constrained Economic Dispatch (SCED) Formulation

US ISOs solve a Security-Constrained Economic Dispatch optimization problem every five minutes. Despite the different implementation approach, the mathematical foundation contains the same quadratic loss structure as AEMO's system.

Mathematical Framework

For a given dispatch interval t, the mathematical formulation is:

Decision Variables:

P_g,t = active power output from generator g (MW)
P_ℓ,t = active power flow on transmission line ℓ (MW)
θ_i,t = voltage angle at bus i (radians)
s_ℓ,t⁺, s_ℓ,t^- = slack variables for transmission constraint violations

Objective Function:

min Σ_g∈𝒢 C_g(P_g,t) + Σ_ℓ∈ℒ ψ k_ℓ P_ℓ,t² + M Σ_ℓ∈ℒ (s_ℓ,t⁺ + s_ℓ,t^-)

Where:

C_g(P_g,t) = generator offer curve (typically linear segments)
k_ℓ = transmission line loss coefficient
ψ = loss penalty factor (set to current energy price estimates)
M = large penalty coefficient for constraint violations

Critical Mathematical Element: The quadratic transmission loss terms k_ℓ P_ℓ,t² appear in the US ISO objective function identically to AEMO's formulation, creating the same mathematical foundation for marginal loss pricing.

Constraint Set for Real-Time Operations

Power Balance Constraints

For each bus i in the network:

Σ_{g∈𝒢_i} P_g,t - D_i,t = Σ_{ℓ∈δ⁺(i)} P_ℓ,t - Σ_{ℓ∈δ^-(i)} P_ℓ,t + L_i,t

Where D_i,t represents real-time demand forecasts updated every few minutes.

DC Power Flow Equations

P_ℓ,t = B_ℓ (θ_m,t - θ_n,t)

Security Constraints with Slack Variables

P_ℓ,t - s_ℓ,t⁺ ≤ P_ℓ^max
P_ℓ,t + s_ℓ,t^- ≥ -P_ℓ^max

Generator Operating and Ramping Constraints

P_g^min ≤ P_g,t ≤ P_g^max
P_g,t - P_g,t-1 ≤ R_g^up × Δt
P_g,t-1 - P_g,t ≤ R_g^down × Δt

Where Δt = 5 minutes and R_g^up, R_g^down are generator ramp rates.

3.3 Locational Marginal Price Calculation and Decomposition

PJM Day-Ahead Market Timeline Example

This timeline shows PJM's day-ahead market operation schedule, based on documented market procedures. Day-ahead market participants must submit bids and offers by specific deadlines, with PJM publishing results by 1:00 p.m. Eastern Time for the next operating day. While this shows day-ahead operations, real-time markets follow similar but more frequent cycles.

Real-time Security-Constrained Economic Dispatch (SCED) runs every 5 minutes with continuous data collection, optimization, and price publication. The specific sub-minute timing details for real-time operations require detailed analysis of PJM Manual 11, which describes the complete operational procedures but detailed timing breakdowns would require additional operational documentation analysis.

Lagrangian Formulation for Real-Time Dispatch

The Lagrangian for the US ISO real-time optimization includes dual variables for all constraints:

ℒ_t = Σ_g C_g(P_g,t) + Σ_ℓ ψ k_ℓ P_ℓ,t²
+ Σ_i λ_i,t [power balance constraints]
+ Σ_ℓ μ_ℓ,t [DC power flow constraints]
+ Σ_ℓ [ρ_ℓ,t⁺ + ρ_ℓ,t^-] [security constraints] + ...

First-Order Conditions and LMP Derivation

Generator Stationarity Condition

For generator g connected to bus i:

∂ℒ_t/∂P_g,t = ∂C_g(P_g,t)/∂P_g,t + λ_i,t = 0

This gives: λ_i,t = -∂C_g(P_g,t)/∂P_g,t

Economic Interpretation: The dual variable λ_i,t represents the Locational Marginal Price at bus i.

Line Flow Stationarity Condition

For transmission line ℓ = (m,n):

∂ℒ_t/∂P_ℓ,t = 2ψ k_ℓ P_ℓ,t - λ_m,t + λ_n,t + μ_ℓ,t + ρ_ℓ,t⁺ - ρ_ℓ,t^- = 0

Critical Mathematical Observation: The factor 2 appears explicitly from differentiating ψ k_ℓ P_ℓ,t², identical to AEMO's static formulation.

LMP Component Decomposition

US ISOs decompose LMPs into three components for market transparency:

LMP_i,t = λ_i,t = Energy Component + Congestion Component + Loss Component

Detailed LMP Decomposition

LMP_i,t = λ_ref,t + Σ_ℓ∈ℒ (ρ_ℓ,t⁺ - ρ_ℓ,t^-) × PTDF_ℓ,i + Σ_ℓ∈ℒ 2ψ k_ℓ P_ℓ,t × PTDF_ℓ,i

Where:

Energy Component: λ_ref,t = marginal cost at reference bus
Congestion Component: Σ_ℓ (ρ_ℓ,t⁺ - ρ_ℓ,t^-) × PTDF_ℓ,i
Loss Component: Σ_ℓ 2ψ k_ℓ P_ℓ,t × PTDF_ℓ,i

Mathematical Equivalence to AEMO: The loss component contains the same factor of 2 multiplying instantaneous loss rates, demonstrating identical mathematical structure despite different implementation approaches.

3.4 Mathematical Equivalence to Static MLF Systems

Universal Factor-of-2 Relationship

Despite fundamental differences in implementation approach, US ISO dynamic systems and AEMO's static system exhibit identical mathematical properties:

Static System (AEMO):

MLF_g = 1 - Σ_ℓ 2k_ℓ P̄_ℓ × PTDF_ℓ,g

Dynamic System (US ISOs):

LMP_loss,i,t = Σ_ℓ 2ψ k_ℓ P_ℓ,t × PTDF_ℓ,i

Mathematical Equivalence: Both expressions contain the factor 2 multiplying instantaneous or average loss rates with transmission sensitivities.

Revenue Over-Recovery in Dynamic Systems

Mathematical Proof for Dynamic Systems

Claim: US ISO dynamic LMP systems result in total loss cost allocation exceeding actual transmission losses by exactly factor 2.

Proof:

Total loss cost allocation in real-time market:

Σ_i Σ_t D_i,t × LMP_loss,i,t = Σ_t Σ_ℓ P_ℓ,t × 2ψ k_ℓ P_ℓ,t
= 2 Σ_t Σ_ℓ ψ k_ℓ P_ℓ,t²

Total actual losses:

Σ_t L_total,t = Σ_t Σ_ℓ k_ℓ P_ℓ,t²

With ψ ≈ 1 (loss penalty set to energy price), the over-recovery factor equals 2. ∎

Revenue Neutrality Mechanisms

US ISOs address the mathematical over-recovery through "revenue neutrality" mechanisms:

Implementation Approaches by ISO

PJM: Marginal loss surplus credited to transmission customers
ERCOT: Monthly revenue neutrality adjustments to market participants
CAISO: Allocation of excess revenues through established procedures
ISO-NE: Regional network service charge adjustments
NYISO: Transmission adjustments through tariff mechanisms
MISO: Revenue sufficiency guarantee settlements

Mathematical Necessity: These mechanisms exist because marginal loss pricing mathematically creates excess revenue collection, requiring redistribution to maintain market balance.

3.5 Computational and Operational Differences

Real-Time vs Annual Optimization

US ISO Approach Characteristics:

Time horizon: 5-minute intervals with 1-2 hour lookahead
Optimization scope: Current system conditions only
Data inputs: Real-time measurements and short-term forecasts
Solution frequency: 288 times per day (every 5 minutes)

AEMO Approach Characteristics:

Time horizon: Annual (17,520 intervals)
Optimization scope: Expected annual operation patterns
Data inputs: Historical data and annual forecasts
Solution frequency: Once per year with periodic updates

Advantages and Disadvantages

Aspect	Dynamic Approach (US ISOs)	Static Approach (AEMO)
Advantages	Reflects real-time conditions, immediate price signals	Predictable factors for financial planning, stable over year
Disadvantages	High price volatility, complex hedging requirements	May not reflect actual conditions, annual volatility
Computational Load	Continuous high-frequency optimization	Intensive annual calculation, simple operation
Price Predictability	High short-term volatility	Annual stability with year-over-year changes
Market Participant Impact	Real-time operational decisions	Long-term financial planning focus

Mathematical Consistency

Despite operational differences, both approaches:

Use identical quadratic loss functions in optimization objectives, generate equivalent factor-of-2 over-recovery relationships, require surplus redistribution mechanisms to maintain revenue neutrality, and provide marginal cost signals for economic efficiency. The mathematical foundations are universal, while implementation details reflect different market design philosophies and operational requirements.

Universal Mathematical Truth

Whether calculated annually in advance (AEMO) or every 5 minutes in real-time (US ISOs), marginal loss pricing produces identical mathematical over-recovery of exactly 2 × actual transmission losses. This is not a design choice but a mathematical consequence of using quadratic loss functions with marginal cost optimization principles.

Section 4: Optimization Theory Deep-Dive

4.1 Mathematical Foundation of Marginal Loss Pricing

Both AEMO's static MLF system and US ISO dynamic LMP systems are built on the same fundamental optimization theory. Understanding this mathematical foundation explains why both approaches produce identical factor-of-2 over-recovery despite their different implementation methods.

The Universal Optimization Structure

All marginal loss factor systems solve optimization problems with this mathematical structure:

Generic Objective Function:
min Σ_generators C_g(P_g) + Σ_lines ψ k_ℓ P_ℓ²

Subject to:

Power balance constraints
Network flow equations
Generator capacity limits
Transmission line limits

The quadratic loss terms k_ℓ P_ℓ² in the objective function are what create marginal loss factors through the optimization mathematics, regardless of whether the problem is solved annually (AEMO) or every 5 minutes (US ISOs).

Why Marginal Pricing Creates Over-Recovery

The mathematical relationship that causes over-recovery exists in all optimization formulations using quadratic loss functions:

Universal Mathematical Property

For any quadratic function f(x) = ax²:

Average rate: f(x)/x = ax² / x = ax
Marginal rate: df/dx = 2ax
Relationship: Marginal = 2 × Average

This mathematical property is independent of:

Market design choices
Geographic scope
Time horizons
Computational methods
Regulatory frameworks

4.2 Lagrangian Formulation and First-Order Conditions

General Lagrangian Structure

For any power system optimization with transmission losses, the Lagrangian takes the form:

ℒ = Σ_g C_g(P_g) + Σ_ℓ ψ k_ℓ P_ℓ² + Σ_i λ_i h_i(𝐏) + Σ_j μ_j g_j(𝐏)

Where:

h_i(𝐏) = 0 are equality constraints (power balance, flow equations)
g_j(𝐏) ≤ 0 are inequality constraints (capacity limits)
λ_i, μ_j are dual variables (shadow prices)

Universal First-Order Conditions

Generator Output Optimality

For any generator g at bus i:

∂ℒ/∂P_g = ∂C_g/∂P_g + λ_i = 0

This gives: λ_i = -∂C_g/∂P_g

Economic Interpretation: The shadow price λ_i equals the negative marginal cost of generation at bus i.

Line Flow Optimality

For any transmission line ℓ = (m,n):

∂ℒ/∂P_ℓ = 2ψ k_ℓ P_ℓ - λ_m + λ_n + other terms = 0

Critical Mathematical Result: The factor 2 appears universally from differentiating the quadratic loss term ψ k_ℓ P_ℓ².

Shadow Price Economic Interpretation

The shadow prices λ_i have consistent economic meaning across all marginal loss factor systems:

Locational marginal cost of serving additional demand at bus i
Locational marginal value of additional generation at bus i
Price signal that guides economic dispatch and investment decisions

Whether computed annually (AEMO) or real-time (US ISOs), these shadow prices embed the same factor-of-2 relationship with actual transmission losses.

4.3 The Mathematics of Over-Recovery

Theoretical Foundation

Mathematical Theorem: For any power system with quadratic transmission loss function L(𝐏) = Σ_ℓ k_ℓ P_ℓ², marginal loss pricing results in total cost allocation exceeding actual losses by exactly factor 2.

Proof Using Euler's Theorem

Euler's Theorem Application to Transmission Losses

This visualization demonstrates Euler's theorem applied to transmission loss functions. The theorem states that for homogeneous functions of degree n, the sum of partial derivatives multiplied by variables equals n times the function value.

Since transmission losses follow L = kP², they are homogeneous of degree 2. Therefore, marginal loss allocation (sum of partials × power flows) always equals 2 × actual losses, regardless of system configuration or operating conditions.

Euler's Theorem for Homogeneous Functions

For a function f(𝐱) that is homogeneous of degree n:

Σ_i x_i ∂f/∂x_i = n · f(𝐱)

Application to Transmission Losses

The loss function L(𝐏) = Σ_ℓ k_ℓ P_ℓ² is homogeneous of degree 2.

Therefore:

Σ_ℓ P_ℓ ∂L/∂P_ℓ = 2 · L(𝐏)

Economic Translation:

Left side = Total marginal loss allocation to all power flows
Right side = 2 × Total actual transmission losses
Result: Marginal allocation = 2 × Actual losses

Chain Rule Application to Generators

For generator-level allocation, using the chain rule:

∂L/∂P_g = Σ_ℓ ∂L/∂P_ℓ × ∂P_ℓ/∂P_g = Σ_ℓ 2k_ℓ P_ℓ · PTDF_ℓ,g

Total generator allocation:

Σ_g P_g ∂L/∂P_g = Σ_ℓ P_ℓ ∂L/∂P_ℓ = 2L(𝐏)

Universal Mathematical Result

This mathematical relationship holds regardless of:

Network topology
Generation mix
Demand patterns
Operating conditions
Market design details

The factor-of-2 over-recovery is a mathematical inevitability in any marginal loss pricing system.

4.4 Power Transfer Distribution Factors (PTDF)

Mathematical Definition

Power Transfer Distribution Factors represent the sensitivity of line flows to generator injections:

PTDF_ℓ,g = ∂P_ℓ/∂P_g

Network Analysis Calculation

For a transmission line ℓ = (m,n) and generator g at bus i:

PTDF_ℓ,i = B_ℓ ([𝐗⁻¹]_m,i - [𝐗⁻¹]_n,i)

Where:

B_ℓ = line susceptance
𝐗 = network reactance matrix
[𝐗⁻¹]_j,i = element (j,i) of the inverse reactance matrix

Properties of PTDF

Mathematical Properties

Kirchhoff's Laws: Σ_ℓ PTDF_ℓ,i = 1 (power balance)
Superposition: PTDF_ℓ,total = Σ_g PTDF_ℓ,g × P_g
Network Independence: PTDFs depend only on network topology, not generation patterns

Economic Significance: PTDFs translate generator-level decisions into system-wide impacts, enabling:

Marginal loss factor calculations
Congestion price components
Transmission planning analysis

4.5 Mathematical Universality Across Market Designs

The factor-of-2 over-recovery relationship is mathematically universal across all market design variations:

Invariant Mathematical Properties

The over-recovery relationship exists regardless of:

Geographic Scope:

Single-region markets (ERCOT)
Multi-region markets (NEM, PJM)
Multi-state markets (MISO, SPP)

Temporal Resolution:

Annual optimization (AEMO)
Daily optimization (some European markets)
Real-time optimization (US ISOs)

Market Structure:

Energy-only markets (ERCOT, NEM)
Energy + capacity markets (PJM, ISO-NE)
Centralized vs decentralized dispatch

Network Topology:

Radial networks
Meshed networks
Mixed AC/DC systems

Mathematical Consistency Verification

The mathematical relationship can be verified in any marginal loss factor system by checking:

Objective function structure: Presence of quadratic loss terms
First-order conditions: Appearance of factor 2 in line flow optimality
Revenue calculation: Total marginal allocation vs actual losses
Redistribution mechanisms: Existence of surplus handling procedures

These mathematical features appear universally, confirming the theoretical foundation across all implementations.

Section 5: Alternative Mathematical Framework - Average Loss Factors

5.1 Conceptual Foundation of Average Loss Factors

Having established that marginal loss factor systems universally create factor-of-2 over-recovery due to their mathematical structure, we now explore an alternative approach: Average Loss Factor (ALF) methodology. This alternative addresses the over-recovery issue by modifying the fundamental optimization formulation.

The Core Mathematical Difference

Current Marginal Loss Approach:

Places actual transmission losses (k_ℓ P_ℓ²) in optimization objective
Optimization process creates marginal loss effects through derivatives
Results in systematic over-recovery requiring redistribution

Alternative Average Loss Approach:

Places pre-calculated average loss coefficients (α_g P_g) directly in optimization objective
Optimization process remains linear in generation variables
Achieves exact revenue neutrality by construction

Economic Logic of Average Loss Allocation

The average loss factor approach is based on the principle that transmission losses should be allocated proportionally to each generator's contribution to total system losses, rather than based on marginal cost theory.

Proportional Allocation Principle

α_g = Generator g's contribution to total losses / Generator g's total output

Revenue Neutrality Constraint

Σ_g∈𝒢 α_g P_g = Σ_ℓ∈ℒ k_ℓ P_ℓ²

This approach ensures that the total loss costs collected exactly equal the actual transmission losses incurred.

Relationship to Cost Causation

Average loss factors aim to reflect cost causation more directly:

Each generator pays for losses in proportion to their actual impact on the transmission system
No mathematical over-recovery occurs
No redistribution mechanisms required

5.2 Mathematical Formulation of ALF Optimization

Modified Objective Function

The ALF optimization replaces quadratic loss terms with linear average loss coefficients:

ALF Objective Function

min Σ_g∈𝒢 C_g(P_g) + Σ_g∈𝒢 α_g P_g

Subject to the same constraints:

Power balance: Σ_{g∈𝒢_i} P_g - D_i = Σ_{ℓ∈δ⁺(i)} P_ℓ - Σ_{ℓ∈δ⁻(i)} P_ℓ
DC power flow: P_ℓ = B_ℓ (θ_m - θ_n)
Generator limits: P_g^min ≤ P_g ≤ P_g^max
Line flow limits: |P_ℓ| ≤ P_ℓ^max

Key Mathematical Property: The objective function is now linear in all decision variables, transforming the problem from Quadratic Programming (QP) to Linear Programming (LP).

First-Order Conditions for ALF

Generator Output Optimality

For generator g at bus i:

∂ℒ/∂P_g = ∂C_g/∂P_g + α_g + λ_i = 0

This gives: λ_i = -∂C_g/∂P_g - α_g

Economic Interpretation: The shadow price includes both the marginal generation cost and the average loss coefficient.

Line Flow Optimality

For transmission line ℓ = (m,n):

∂ℒ/∂P_ℓ = -λ_m + λ_n + μ_ℓ = 0

Critical Observation: No factor of 2 appears because there are no quadratic loss terms to differentiate.

Computational Properties

Property	Marginal Loss Formulation	Average Loss Formulation
Problem Classification	Quadratic Programming (QP)	Linear Programming (LP)
Algorithm Type	Interior-point methods	Simplex method or barrier methods
Mathematical Structure	Dense Hessian matrix	Sparse constraint matrix
Computational Complexity	Higher (quadratic terms)	Lower (linear structure)
Solution Uniqueness	Unique under convexity	Unique under non-degeneracy

5.3 Average Loss Coefficient Calculation

Theoretical Foundation

Average loss coefficients must be calculated to represent each generator's proportional contribution to total system losses. The calculation method determines how well the ALF system maintains locational efficiency signals.

Method 1: Historical Attribution Analysis

Data Requirements:

Historical generation patterns for each generator
Historical transmission flows and losses
Network topology and line characteristics

Calculation Process:

Loss Attribution: For each historical interval, attribute transmission losses to generators based on their contribution to power flows
Temporal Averaging: Calculate generation-weighted average of loss attribution over the analysis period
Coefficient Derivation: Derive average loss coefficient as ratio of attributed losses to generation

α_g = Σ_t∈𝒯 L_g,t / Σ_t∈𝒯 P_g,t

Where L_g,t represents generator g's attributed losses in interval t.

Method 2: Sensitivity-Based Calculation

Network Analysis Approach: Using Power Transfer Distribution Factors to calculate loss sensitivities:

α_g = Σ_ℓ∈ℒ k_ℓ P̄_ℓ × PTDF_ℓ,g

Where P̄_ℓ represents the average power flow on line ℓ over the analysis period.

Properties:

Maintains network-based locational signals
Reflects transmission distance and network constraints
Updates automatically with network topology changes

Method 3: Optimization-Based Derivation

Iterative Calculation:

Initial Estimate: Start with preliminary average loss coefficients
Optimization: Solve ALF optimization to get generation patterns
Loss Calculation: Calculate actual transmission losses for resulting flows
Coefficient Update: Adjust coefficients to improve revenue neutrality
Iteration: Repeat until convergence

Convergence Criterion:
|Σ_g α_g P_g* - Σ_ℓ k_ℓ (P_ℓ*)²| < ε

Where ε is a small tolerance level.

5.4 Revenue Neutrality Properties

Revenue Neutrality as Design Goal

The ALF methodology aims to achieve revenue neutrality:

Σ_g∈𝒢 α_g P_g* ≈ Σ_ℓ∈ℒ k_ℓ (P_ℓ*)²

Where starred variables represent optimal solutions from the ALF optimization.

Important Clarification: Revenue neutrality is a design objective that requires careful implementation, not an automatic mathematical guarantee.

Challenges to Perfect Revenue Neutrality

Challenge 1: Generation Pattern Changes

The Problem:

Average loss coefficients α_g are calculated based on expected generation patterns
ALF optimization may produce different generation patterns P_g*
When generation patterns change, actual loss attribution changes

Mathematical Issue:

α_g calculated assuming P_g^expected, but optimization produces P_g* ≠ P_g^expected

Challenge 2: Network Flow Dependencies

The Problem:

Loss attribution depends on network power flows
Power flows change when generation dispatch changes
Changing flows alter the loss patterns that α_g was designed to recover

Feedback Loop: New generation patterns → New power flows → Different actual losses → Revenue imbalance

Practical Revenue Balancing Approaches

Approach 1: Iterative Coefficient Adjustment

Process:

Initial Calculation: Compute α_g based on expected operations
Optimization: Solve ALF problem to get P_g*, P_ℓ*
Check Balance: Calculate Σ_g α_g P_g* vs Σ_ℓ k_ℓ (P_ℓ*)²
Adjust Coefficients: If imbalanced, revise α_g and repeat
Converge: Iterate until acceptable revenue neutrality achieved

Approach 2: Scaling Adjustment

Simple Correction Method: If imbalances occur, apply uniform scaling:

α_g^adjusted = α_g × (Σ_ℓ k_ℓ (P_ℓ*)²) / (Σ_g α_g P_g*)

Approach 3: Periodic Recalibration

Recalculate coefficients based on actual operational experience
Update frequency: annually, quarterly, or as-needed
Learn from revenue imbalances to improve future calculations

Expected Performance vs Current System

System	Revenue Balance Target	Actual Performance	Correction Required
ALF	Exact revenue neutrality	Small imbalances requiring periodic adjustment	<5% imbalances, easily correctable
Current MLF	Not applicable (systematic over-recovery)	Predictable 100% over-recovery (factor of 2)	Complex redistribution mechanisms (IRSS)

5.5 Locational Efficiency Signals

A key concern with ALF is whether it maintains adequate locational signals for efficient investment and operation decisions.

Locational Variation in ALF

Average loss factors vary by generator location due to:

Distance from load centers
Network topology and transmission constraints
Regional power flow patterns
Interconnection capabilities

Investment Signal Preservation:

Generators in high-loss locations receive higher α_g values
Investment incentives favor low-loss locations
Transmission expansion benefits remain captured

Comparison of Location Signals

Signal Strength Comparison

Marginal Loss Factors: Stronger locational differentiation, higher volatility
Average Loss Factors: Moderate locational differentiation, greater stability

Economic Trade-offs

MLF: Maximum theoretical efficiency, practical complications from over-recovery
ALF: Approximate efficiency, practical advantages from revenue neutrality

Dynamic Updating of Coefficients

Frequency of Updates: ALF coefficients require periodic updates to reflect:

Changes in generation mix
Network topology modifications
Evolving demand patterns
New technology integration

Update Methodologies:

Annual updates: Similar to current MLF cycles
Quarterly adjustments: More responsive to system changes
Event-driven updates: Major system changes trigger recalculation

🔄 IRSS and Mathematical Over-Recovery Evidence

AEMO's own documentation acknowledges the mathematical over-recovery inherent in marginal loss factor methodology.

AEMO's Explanation of IRSS: AEMO educational materials state that because marginal losses are larger than average losses, settling prices based on marginal losses leads to AEMO recovering more from customers than needed to pay generators, creating the IRSS¹².

IRSS Redistribution Mechanism: According to AEMO's settlement procedures, the Intra-Regional Settlement Surplus is redistributed through transmission network service provider revenue adjustments¹³. However, the ultimate beneficiaries and complete flow-through effects require deeper investigation to understand fully.

Mathematical Verification: The factor-of-2 mathematical relationship can be verified in AEMO's system by examining:

1.Total MLF-based revenue allocation to transmission losses

2.Actual transmission losses as percentage of total generation The ratio between allocated costs and actual loss costs

AEMO, "Understanding Transmission Loss Factors", Educational Materials, 2023 ¹³ AEMO, "Settlement Procedures", National Electricity Rules implementation

Next: Real AEMO data analysis, quantitative ALF implementation scenarios, and practical considerations for transitioning from current MLF systems to revenue-neutral alternatives.