Locational Marginal Pricing (LMP)

In wholesale electricity markets, prices aren't uniform — they vary by location. The Locational Marginal Price (LMP) at each bus is the cost of serving one additional megawatt of load at that location.

LMP is the economic signal that drives efficient generation dispatch, transmission investment, and demand response.

Why Prices Differ by Location

Two physical phenomena cause price separation:

1. Transmission Losses

Delivering power costs energy — wires have resistance, and $I^2R$ losses dissipate power as heat.

Example: To deliver 100 MW to a distant load when losses are 5%, you need to generate 105 MW. The price at the load bus should reflect these losses.

2. Transmission Congestion

When a transmission line reaches its thermal limit, cheap generation on one side can't reach load on the other side. More expensive local generation must run instead.

Example: A 50 MW line connects cheap coal ($20/MWh) to expensive gas ($60/MWh). When flow hits 50 MW:

On the coal side: additional load could use more cheap coal → LMP ≈ $20
On the gas side: additional load needs expensive gas → LMP ≈ $60

Congestion creates price separation — different LMPs on either side of a congested line.

The LMP Formula

LMP at bus $i$ decomposes into three components:

$$\text{LMP}i = \lambda + \lambda{\text{loss},i} + \lambda_{\text{cong},i}$$

where:

$\lambda$ = Energy component (system marginal cost)
$\lambda_{\text{loss},i}$ = Loss component (marginal cost of losses to serve bus $i$)
$\lambda_{\text{cong},i}$ = Congestion component (shadow prices of binding constraints)

Energy Component

The base cost of energy, equal to the most expensive generator currently dispatched (the marginal unit). Same at all buses.

Loss Component

Reflects how losses change when load at bus $i$ increases:

$$\lambda_{\text{loss},i} = \lambda \times \frac{\partial P_{\text{loss}}}{\partial P_i}$$

Buses electrically distant from generation have higher loss components.

Congestion Component

Reflects binding transmission constraints:

$$\lambda_{\text{cong},i} = \sum_k \mu_k \times \text{PTDF}_{k,i}$$

where:

$\mu_k$ = shadow price (dual variable) of constraint $k$
$\text{PTDF}_{k,i}$ = sensitivity of flow on line $k$ to injection at bus $i$

LMP from Optimal Power Flow

LMP emerges naturally from the OPF problem. Consider the simplified DC-OPF:

$$\min \sum_g c_g P_g$$

Subject to:

$$\sum_g P_g = \sum_d P_d + P_{\text{loss}} \quad (\lambda)$$ $$P_{\text{line},k} \leq P_{\max,k} \quad (\mu_k)$$ $$P_g^{\min} \leq P_g \leq P_g^{\max}$$

The Lagrange multipliers (dual variables) in parentheses have economic meaning:

$\lambda$ = marginal cost of serving one more MW of total load
$\mu_k$ = marginal cost of one more MW of transmission capacity on line $k$

LMP at bus $i$ equals the change in total cost if load at bus $i$ increases by 1 MW:

$$\text{LMP}i = \frac{\partial \text{Cost}}{\partial P{d,i}} = \lambda + \sum_k \mu_k \cdot \text{PTDF}_{k,i}$$

A Simple Example

Three buses, two generators, one constrained line:

   Gen A ($30/MWh)          Gen B ($50/MWh)
        [1]───────────────────[2]
                100 MW limit   │
                              [3] Load: 150 MW

Uncongested case (load = 80 MW):

Gen A supplies all 80 MW (cheapest)
LMP everywhere = $30/MWh

Congested case (load = 150 MW):

Gen A supplies 100 MW (line limit)
Gen B supplies 50 MW (expensive but unconstrained)
LMP at bus 1 = $30 (could use more cheap Gen A)
LMP at buses 2, 3 = $50 (marginal source is Gen B)

The $20 difference is the congestion rent — revenue collected from the price separation.

Economic Interpretation

For Generators

LMP tells you the value of your output:

Generate when LMP > your marginal cost
Don't generate when LMP < your marginal cost

Revenue = LMP × MWh generated

For Loads

LMP is what you pay for electricity at your location:

Cost = LMP × MWh consumed
High LMP locations pay more (incentive to reduce demand or relocate)

For Transmission Owners

Congestion creates financial transmission rights (FTRs):

Revenue = (LMP_sink - LMP_source) × MW
When congestion exists, transmission rights are valuable

LMP Components in Action

Bus	Energy	Loss	Congestion	Total LMP
Gen A	$30	$0	-$5	$25
Hub	$30	$2	$0	$32
Load	$30	$5	$15	$50

Interpretation:

Gen A gets $25 (below system price due to negative congestion component — can't export all it wants)
Hub is roughly at system price with small losses
Load pays $50 — premium for losses and congestion

Market Settlement

Wholesale electricity markets settle based on LMPs:

Day-Ahead Market:

Generators offer supply curves (price vs. quantity)
Loads bid demand (or take price)
Market operator runs SCOPF to clear market
LMPs emerge as dual variables

Real-Time Market:

Balances deviations from day-ahead schedules
5-minute LMPs reflect real-time conditions
Deviations settled at real-time LMP

Example Settlement

Generator at $30/MWh cost, dispatched for 100 MW:

LMP at their bus = $35/MWh
Revenue = 100 MW × $35 = $3,500/hour
Profit = $3,500 - $3,000 = $500/hour

This profit margin incentivizes efficient generation.

Negative Prices

LMPs can go negative when:

Renewable generation exceeds demand
Generators have minimum output constraints (must-run)
Transmission is congested (can't export surplus)

Negative LMP means: Pay someone to take your power!

Wind farms with production tax credits may still profit at negative prices, which has led to occasional -$100 to -$300/MWh prices in windy, uncongested areas.

LMP in GAT

GAT's OPF produces LMPs as part of the solution:

gat opf network.arrow --method dc

Output includes:

lmp.arrow: LMP at each bus
Decomposition into energy, loss, congestion components
Binding constraints and shadow prices

You can visualize LMP patterns:

gat viz network.arrow --color-by lmp

Geographic Patterns

In real markets, LMPs show characteristic patterns:

Load pockets: Urban areas with limited transmission have higher LMPs due to congestion.

Generation pockets: Remote wind/solar farms may have lower (even negative) LMPs when they can't export.

Hub prices: Major interconnection points tend toward system average.

Seasonal variation: Summer peaks drive congestion in AC-heavy regions; winter peaks in heating regions.

Key Takeaways

LMP = Energy + Losses + Congestion — price varies by location
Congestion creates price separation across constrained lines
LMPs emerge from OPF as dual variables (shadow prices)
Markets settle at LMP — generators paid, loads charged by location
Investment signal — high LMPs indicate where generation or transmission is needed