Marginal Energy: Concepts and Applications

• Marginal energy is defined as the incremental cost or energy required for an additional unit, derived from the derivative of system cost functions and capturing operational, investment, and environmental factors.
• It serves as a pricing signal in networked systems by incorporating losses, congestion, and asset degradation, thus guiding optimal dispatch and infrastructure investment decisions.
• Applications extend to non-renewable resource modeling, optimal transport, and embodied energy economics, informing market design, decarbonization strategies, and economic price-embodied energy relations.

Marginal energy, in both engineering and economic contexts, refers to the incremental cost, value, or physical cost supply of an additional unit of energy delivered, transformed, transported, or consumed within an energy system. This concept arises in a wide range of disciplines, from power system operation and planning to resource economics and optimal transport, and admits a rigorous mathematical formulation as the derivative of a system cost function, value function, or physical energy measure with respect to a marginal change in energy injection, withdrawal, generation, or transformation at a specified location and time.

1. Core Definitions and Mathematical Formulation

Marginal energy, also called the marginal cost of energy, can be formulated as a partial derivative: $$\lambda_{i,t} := \frac{\partial C}{\partial p_{i,t}},$$ where $C$ is the total system cost (which may include fuel, operations, asset aging, losses, congestion, or environmental externalities) and $p_{i,t}$ is net energy injection at location $i$ and time $t$ (Andrianesis et al., 2019).

In networked energy systems (electricity, gas, hydrogen), the marginal energy price at a specific node and time is the shadow price (dual variable) of the relevant energy balance constraint in the economic optimization of system operation. For example, in AC Optimal Power Flow (OPF) and distribution network planning, the marginal energy price is the Distribution Locational Marginal Cost (DLMC), internalizing not just bulk energy acquisition but network losses, congestion, and short-run asset degradation (Andrianesis et al., 2019, 1811.09001, Andrianesis et al., 2021).

In extraction economics, the marginal energy cost is the price at which the highest-cost unit is supplied to meet aggregate demand, reflecting the cost of the "marginal" (last) resource extractable at each moment (Mercure et al., 2012).

2. Power and Energy Systems: Long-Run and Short-Run Marginals

Marginal energy in power markets is distinguished between short-run marginal cost (SRMC) and long-run marginal cost (LRMC) (Homaei et al., 2024):

• Short-Run Marginal Cost (SRMC): The change in total variable cost from producing one more unit of output, holding infrastructure fixed. For a power generator, SRMC typically equals its variable (fuel+O&M) cost when it is the marginal unit.
• Long-Run Marginal Cost (LRMC): The increase in total cost—including investment, replacement, and operation—from a permanent one-unit increase in output demand, allowing optimal system expansion. LRMC incorporates both fixed and variable costs of new capacity.

A generalized capacity-expansion model can rigorously define LRMC as the dual variable to the flow-balance constraint when all investment and operations can be optimized, and SRMC as the dual when capacity is exogenous and only short-run operational variables are adjustable. Divergence between LRMC and SRMC leads to the "missing-money" problem, where SRMC-based market prices under-recover total system costs unless uplift mechanisms are introduced (Homaei et al., 2024).

3. Marginal Energy Prices in Networked Systems

Power Systems

In transmission and distribution systems, nodal marginal energy prices (LMPs and DLMCs) are determined via the KKT stationarity conditions (Lagrange multipliers) of the OPF problem: $$\lambda^P_{j,t} = c^P_t + \sum_{lines} \text{loss, congestion components} + \sum_{assets} \text{degradation}$$ where $c^P_t$ is the wholesale energy price at the substation, loss terms capture line resistive losses, congestion terms include duals for voltage/ampacity constraints, and degradation terms reflect (primarily transformer) thermal aging (Andrianesis et al., 2019, Andrianesis et al., 2019, 1811.09001).

Transformer degradation is a critical component: the marginal cost of asset aging is modeled via the sensitivity of future hot-spot temperature and loss-of-life to current injection, leading to intertemporal coupling (Andrianesis et al., 2019).

Gas and Hydrogen Networks

Locational marginal energy pricing generalizes to gas and hydrogen infrastructure, notably when considering blending and decarbonization incentives (Sodwatana et al., 2023). Prices at each node are determined as the dual multipliers to flow balance, pressure, chemical composition, and carbon incentive constraints, resulting in an energy-weighted price: $$\lambda^e_j = \frac{(1-\gamma_j)\lambda^{NG}_j + \gamma_j\lambda^{H_2}_j}{R(\gamma_j)}$$ where $\gamma_j$ is the hydrogen fraction and $R(\gamma_j)$ is the composite calorific value at node $C$0.

4. Marginal Energy in Economic Resource Extraction

Marginal energy cost is central to non-renewable resource economics. Let $C$1 denote the initial cost-density function of extractable reserves, $C$2 the remaining density at time $C$3, and $C$4 the market price. The system evolves according to

$C$5

with $C$6 a smooth extraction probability function. The total extraction flow is

$C$7

The inverse problem, central to price/discovery analysis, solves for the price path so that the extraction flow matches exogenous demand, with the marginal cost being the unique price that clears the market at each $C$8 (Mercure et al., 2012). Dynamic evolution is path-dependent, with hysteresis—historical depletion influences current marginal cost.

This framework is implemented in technology-diffusion integrated assessment models like FTT:Power to simulate endogenous resource/technology substitution and emission outcomes (Mercure et al., 2012).

5. Marginal Energy Transfer in Embodied Energy Economics

Marginal energy provides a thermodynamically grounded metric for opportunity cost in production. The marginal energy transfer $C$9 required to produce one more unit of good $p_{i,t}$0 in period $p_{i,t}$1 is (Leiva, 2018): $p_{i,t}$2 where $p_{i,t}$3 is the direct energy transferred (via all prime movers), and $p_{i,t}$4 is the power-scarcity cost reflecting capital/prime-mover bottlenecks. Under exchange, the marginal energy transfer ratio between goods determines the equilibrium "barter price": $p_{i,t}$5 and when money is present, nominal prices become social representations of marginal energy transfers, up to inflation and prime-mover scarcity effects.

Average embodied energy is related but distinct, representing the mean historical energy input per unit, while marginal energy transfer reflects the incrementally required energy at the margin (Leiva, 2018).

6. Marginal Energy in Optimal Transport and Statistical Physics

In the continuous-time optimal transport setting, the minimum-energy velocity field $p_{i,t}$6 solves

$p_{i,t}$7

subject to the continuity equation $p_{i,t}$8 and prescribed family of marginals $p_{i,t}$9 for all $i$0. This $i$1 represents the field of minimum marginal energy consumption required to push forward the full path $i$2, generalizing the Benamou--Brenier formulation to all-time constraints (Nakano, 27 Apr 2026).

Practically, mesh-free kernel methods estimate $i$3 directly from samples, embedding the weak PDE residual in a reproducing kernel Hilbert space and penalizing the RKHS norm, ensuring convergence to the true minimum-energy solution.

7. Applications and Policy Implications

Marginal energy concepts underpin market pricing, optimal scheduling, and system planning in power, gas, and multi-energy networks:

• Electricity Markets: Marginal energy prices (LMPs or DLMCs) provide dispatch signals, guide investment, and determine cost recovery (Homaei et al., 2024, Andrianesis et al., 2019, Andrianesis et al., 2019).
• DER Scheduling: Locational and temporal marginal prices support decentralized optimal scheduling of distributed energy resources—EV charging, PV inverters—ensuring asset preservation and maximal hosting capacity (Andrianesis et al., 2019, Andrianesis et al., 2021).
• Resource Depletion Modeling: Marginal extraction cost models inform global resource exhaustion timelines, technology transition, and policy assessment (Mercure et al., 2012).
• Mobile Energy Storage: The marginal value of storage in networks, expressible via LMP gradients over space-time, determines relocation and operation strategies for mobile assets (Agwan et al., 2023).
• Decarbonization: Blended commodity networks (natural gas/hydrogen) use marginal energy pricing with carbon incentives to operationalize green transitions (Sodwatana et al., 2023).
• Economic Theory: Marginal energy transfer as a metric rationalizes observed price-embodied energy proportionality and formally incorporates thermodynamic limits into opportunity cost (Leiva, 2018).

Marginal energy represents a synthetic concept that merges optimization duality, physical network dynamics, resource economics, and thermodynamic foundations—providing the rigorous link between micro-level energy flow and system-level cost or scarcity signals.