Marginal Value of Energy in an Automated Economy

Research question

The research question is narrow:

Under what assumptions should the next feasible unit of electricity be assigned to a particular automated workload, node, and interval?

The strongest defensible answer is conditional. If workload benefit functions are concave, costs are evaluated at fixed price-taking terms, uncertainty enters through a declared risk adjustment, and all operational constraints are convex, then an optimum has a marginal equalization rule. The rule is not a claim that markets always satisfy those conditions. It is a benchmark against which nonconvexities, strategic bidding, missing prices, and measurement gaps can be named.

Contributions

The paper contributes the following:

1. a node-indexed and time-indexed allocation model that separates market energy prices from congestion, losses, reserves, reliability, workload service loss, and marginal emissions;

2. a KKT equalization theorem under explicit convexity, price-taking, information, and feasibility assumptions;

3. a clean distinction among average energy intensity, private marginal cost, nodal system opportunity cost, and social marginal cost;

4. linear representations of capacity, interface, reserve, ramp, deadline, and reliability restrictions;

5. a small standard-library solver with shadow-price, finite-difference, feasibility, and complementary-slackness checks;

6. scenario results whose synthetic status is stated in every interpretation.

Nonclaims

This paper does not estimate national demand response for AI, manufacturing, transport, or cryptographic settlement. It does not prove that flexible computing benefits a grid. It does not treat a distribution-network tariff as a wholesale nodal price. It does not model voltage, reactive power, transient stability, or unit commitment. It does not rank sectors without matched response data. Those limitations are part of the result.

Marginal cost, welfare, and peak load

The welfare logic is older than modern electricity markets. Marginal-cost pricing links an incremental decision to the opportunity cost of the resources it consumes. Samuelson’s treatment of constrained optimization and welfare provides the general economic frame [1]. Boiteux’s peak-load pricing work showed why electricity costs depend on when capacity is required, not merely on total annual energy [2]. Schweppe, Caramanis, Tabors, and Bohn developed a location-sensitive and time-sensitive account of electricity spot pricing [3]. Stoft connects the same pricing principles to practical electricity-market design and scarcity [17].

These references do not imply that every observed price equals social marginal cost. Retail rates can average costs across time and geography. Wholesale markets contain uplift, scarcity rules, capacity mechanisms, and administrative interventions. External damages may be unpriced. Consumers may not see real-time prices. Borenstein and Holland show that time-invariant retail prices can break the welfare correspondence that competitive wholesale pricing might otherwise provide [4]. The present model therefore keeps the price vector, environmental term, and workload benefit functions separate. Borenstein’s review of renewable electricity also distinguishes private incentives from public benefits and costs [18].

Locational marginal pricing and congestion

FERC’s 2024 Energy Markets Primer states that organized market software matches supply and demand while respecting physical resource and transmission limits. When a transmission constraint binds, the least-cost generator for the whole region may be unable to serve an additional load at a constrained node. A more expensive local or redispatched unit then affects the nodal price [5]. The LMP decomposition is commonly written $$\operatorname{LMP}_{n,t} = p^{E}_{t}+p^{C}_{n,t}+p^{L}_{n,t},$$ where $p^{E}_{t}$ is the reference energy component, $p^{C}_{n,t}$ is congestion, and $p^{L}_{n,t}$ is the marginal-loss component.

The formula has a precise domain. It describes a market result produced by a particular network model and dispatch. It does not show that an end user paid the LMP. It also does not make a workload’s internal deadline disappear. Our allocation model treats an observed or scenario LMP as exogenous. It then adds internal and external terms without claiming to reproduce the system operator’s dispatch engine.

Reserve, ramping, and reliability

Energy balance alone does not keep a power system reliable. FERC distinguishes energy from ancillary services and notes that regulation follows short-term load changes, while operating reserves must be available within specified times [5]. NERC defines and maintains reliability terminology and standards used by balancing authorities [6]. A load allocator must not consume capacity that has already been withheld for reserve or a declared reliability margin.

Ramping matters on both sides of the meter. A generator cannot always change output instantly. An automated workload may also have a maximum change between intervals because of thermal cycling, queue stability, material continuity, charging limits, or restart overhead. The mathematical form can be similar even though the engineering causes differ.

Reliability enters the model in two places. First, system operators may withhold deliverable capacity. Second, a workload may value scheduled energy less when service delivery can fail and may face an expected failure loss. A full reliability model would treat correlated outages and state-dependent recourse. The reference implementation uses an explicit reliability factor and loss coefficient. It is an audit-friendly approximation.

Marginal emissions

An average emissions rate divides total emissions by total generation. A marginal emissions factor asks how emissions change under an incremental load or resource change. Siler-Evans, Azevedo, and Morgan estimated regional and hourly marginal factors for the United States and showed substantial variation [7]. Holland and coauthors demonstrate that the environmental effect of electric driving can vary by location because marginal generation and damages vary [8]. Callaway, Fowlie, and McCormick make the same general point for renewable energy and demand-side resources: location and timing change marginal value [9]. Recent work by Holland and coauthors treats marginal-emissions estimation as a high-dimensional identification problem and uses economic restrictions to regularize the estimate [19].

EPA’s AVERT tool estimates emissions impacts from load or clean-energy changes using historical hourly generation and emissions patterns. The current EPA description says that AVERT’s statistical module uses observed generation and emissions data, while its main module applies regional scenario changes [10]. EPA also publishes approximate marginal emission rates for AVERT regions [12]. These are more relevant than annual average mix factors for many incremental studies, but they remain model-based regional estimates. They are not exact node-level causal derivatives.

Convex optimization

The necessary conditions trace to Karush and to Kuhn and Tucker [15, 16]. The mathematical result uses standard convex duality. Boyd and Vandenberghe show that KKT conditions are sufficient for convex programs and are necessary under suitable constraint qualifications [13]. Rockafellar provides the broader convex-analysis foundation [14]. The economic interpretation of a multiplier as a local value of relaxing a constraint follows from the value function and envelope theorem.

This paper does not contribute a new KKT theorem. Its contribution is the disciplined mapping of automated workload value, nodal electricity opportunity cost, emissions, service constraints, and energy boundaries into one model, together with a reference implementation that checks the claimed conditions.

Energy convention

The denominator in this paper is marginal grid energy delivered to an incremental workload at node $n$ and interval $t$. It is not annual primary energy, device telemetry, facility electricity, or lifecycle energy. If a facility-level analysis adds cooling or conversion overhead, the analyst must state whether the nodal price applies before or after that overhead. If embodied energy changes at the margin because a capacity investment is triggered, the short-run model is no longer sufficient.

Private value and social value

Let $V_i$ be a declared benefit function for workload $i$. It may describe private surplus or a social welfare component. Those are not interchangeable. A private allocator might maximize expected profit after energy charges. A social planner might add consumer surplus, local pollution damages, reliability externalities, and distributional weights. The equation can accommodate either, but the label must say which.

This paper does not infer welfare from revenue. Revenue is a transfer between parties unless it proxies a separately defined surplus. Likewise, a token, hash, manufactured unit, or vehicle-mile is an output quantity, not a common value unit. Cross-sector allocation requires a declared value functional or a vector report.

Average and marginal quantities

For a workload with total outcome $W(E)$, average value per unit is $$\operatorname{AV}(E)=\frac{W(E)}{E},$$ while marginal value is $$\operatorname{MV}(E)=W'(E).$$ They coincide only for a linear function through the origin. With diminishing returns, average value can remain high after marginal value has fallen below the opportunity cost. Allocating the next unit by average ratios can therefore send energy to the wrong workload even inside a perfectly measured model.

The same distinction applies to cost and emissions: $$\operatorname{AC}(E)=\frac{C(E)}{E}, \qquad \operatorname{MC}(E)=C'(E),$$ $$\operatorname{AEF}(E)=\frac{G(E)}{E}, \qquad \operatorname{MEF}(E)=G'(E).$$ The derivatives are local to a counterfactual operating point. They may change when a constraint binds, a generator starts, or an investment is triggered.

Indices and decisions

Let $i\in\mathcal{I}$ index workloads, $n\in\mathcal{N}$ nodes, and $t\in\mathcal{T}$ intervals. The decision $$x_{i,n,t}\geq 0$$ is incremental MWh assigned to workload $i$ at node $n$ in interval $t$. Define aggregate workload energy $$q_{i,t}=\sum_{n\in\mathcal{N}}x_{i,n,t}.$$ Variables can represent alternatives. For example, one batch may run at either of two data centers, subject to a shared completion requirement.

Let $\omega$ denote uncertain workload and system conditions. The declared risk-adjusted workload benefit is $$B_i(x_i) = \mathbb{E}_{\pi}\!\left[V_i(x_i,\omega)\right] - \rho_i R_i(x_i),$$ where $R_i$ is a risk functional and $\rho_i\geq 0$ is a declared risk weight. In the reference code, $$B_{i,n,t}(x) = a_{i,n,t} \left(v_{i,n,t}x-\frac{\kappa_{i,n,t}}{2}x^2\right) - (1-a_{i,n,t})f_{i,n,t}x.$$ Here $a$ is service reliability, $v$ the intercept of marginal gross value, $\kappa>0$ curvature, and $f$ failure loss per MWh. For the separable quadratic implementation, the workload-level object in the main objective is $$B_i(x_i) = \sum_{n\in\mathcal{N}} \sum_{t\in\mathcal{T}} B_{i,n,t}(x_{i,n,t}).$$

Nodal opportunity cost

The declared external cost per MWh is $$\begin{align} c_{n,t} ={}& p^{E}_{n,t} +p^{C}_{n,t} +p^{L}_{n,t} +p^{R}_{n,t} \nonumber\\ &+\chi_t g_{n,t} +d_{n,t}. \label{eq:nodalcost} \end{align}$$ The components are:

• $p^{E}$: marginal energy charge;

• $p^{C}$: congestion component;

• $p^{L}$: marginal transmission-loss component;

• $p^{R}$: reserve or reliability service adder allocated to the load;

• $g$: marginal emissions in kg per MWh;

• $\chi$: declared damage, tax, or shadow value per kg;

• $d$: another declared marginal damage or network term.

When an observed LMP already contains energy, congestion, and losses, those terms must not be added twice. A capacity payment or fixed network charge belongs in Equation (linked) only if the incremental decision changes it. Average bill allocation is not enough. Energy, congestion, and loss price components may be negative. The reference code permits signed market components and keeps any sign restriction on an environmental damage value explicit.

Capacity, reserve, and reliability margin

Let $\bar C_{n,t}$ be incremental deliverable capacity before load-side deductions. Let $r_{n,t}$ be capacity withheld for operating reserve and $m_{n,t}$ a declared reliability margin. Then $$\sum_{i\in\mathcal{I}}x_{i,n,t} \leq \bar C_{n,t}-r_{n,t}-m_{n,t}. \label{eq:capacity}$$ This is a load-allocation constraint, not a statement about the full bulk system reserve requirement. The terms must be based on the same node, interval, and capacity convention.

Congested interfaces

A linear interface approximation is $$\sum_{i,n,t}h_{\ell,i,n,t}x_{i,n,t} \leq \bar F_{\ell}, \qquad \ell\in\mathcal{L}. \label{eq:interface}$$ The coefficients $h$ may be simple usage shares or distribution factors from a separate network model. The reference code accepts arbitrary signed coefficients and a limit. It does not derive them and does not perform power flow. If the coefficients are invented, the resulting shadow price is only a scenario artifact.

Bounds and ramp limits

Each workload option has $$0\leq x_{i,n,t}\leq \bar x_{i,n,t}. \label{eq:bounds}$$ Aggregate workload ramp limits are $$-D_i \leq q_{i,t}-q_{i,t-1} \leq U_i. \label{eq:ramp}$$ A historical initial level $q_{i,t_{\mathrm{start}}-1}$ must be fixed when the first modeled interval is ramp constrained. The reference code accepts this value as initial_energy_mwh. If it is omitted, ramp limits apply only between modeled intervals and no claim is made about the transition into the first one. A computing workload may have small $U_i$ because abrupt queue changes violate a service target. A continuous industrial process may have small $D_i$ because rapid curtailment damages output. A proof-of-work workload may be electrically curtailable but still face restart, pool, or revenue effects. The coefficients are application data, not properties of the sector label.

Deadlines and minimum service

Let $d_{i,k}$ be a deadline and $Q_{i,k}$ the minimum cumulative energy required for service milestone $k$: $$\sum_{n\in\mathcal{N}} \sum_{t\leq d_{i,k}} x_{i,n,t} \geq Q_{i,k}. \label{eq:deadline}$$ The constraint can represent energy needed to finish a batch, charge a fleet, or meet a production target. Energy is only a valid completion proxy when the workload has a stable conversion between scheduled energy and service. If efficiency changes with power, temperature, or state of charge, a richer service constraint is required.

Chance-constrained reliability extension

Suppose delivered energy is $A_{i,n,t}x_{i,n,t}$, where $A$ is random availability. A service reliability constraint can be written $$\mathbb{P}_{\pi}\left[ \sum_{n\in\mathcal{N}} \sum_{t\leq d_{i,k}} A_{i,n,t}x_{i,n,t} \geq Q_{i,k} \right] \geq 1-\alpha_{i,k}. \label{eq:chance}$$ Equation (linked) is not automatically convex. It can have convex deterministic equivalents under specific distributional assumptions, or it can be replaced by a conservative robust set. The theorem below applies only when the chosen representation keeps the feasible set convex.

Objective

The allocator solves $$\begin{equation} \begin{aligned} \max_{x\in\mathcal{X}}\quad F(x) &= \sum_{i\in\mathcal{I}}B_i(x_i) - \sum_{i,n,t}c_{n,t}x_{i,n,t},\\ \mathcal{X} &= \{x:\text{Equations link 1 below to link 2 below and any convex reliability constraints hold}\}. \end{aligned} \label{eq:program} \end{equation}$$(link 1: equation and link 2: equation) The objective is incremental surplus relative to the declared counterfactual. Fixed costs cancel only when the decision does not change them. Startup, shutdown, hardware expansion, and discrete staffing can make the objective or feasible set nonconvex.

What “equalization” does not mean

The theorem does not say that gross value per MWh is equal across sectors. It concerns marginal, risk-adjusted benefit after accounting for every binding constraint. It does not say that two workloads at different nodes face the same opportunity cost. It does not say that a workload at a lower average tariff should run first. It also does not claim that an observed market allocation is socially optimal.

The theorem is local. If an extra unit triggers a discrete hardware purchase, starts a generator, crosses a demand-charge threshold, or changes a market price, the derivative can jump. A mixed-integer or equilibrium model is then needed.

Two-workload example

Suppose two interior workloads share 10 MWh at one node. Their gross marginal value curves are $$80-4x_A \quad\text{and}\quad 70-2x_B$$ in declared currency units per MWh. The nodal energy price is 20, and there are no other terms. Capacity binds, so $x_A+x_B=10$. Equalization gives $$80-4x_A-20 = 70-2x_B-20 = \lambda.$$ Solving yields $$x_A=5, \qquad x_B=5, \qquad \lambda=40.$$ The two workloads receive the same allocation even though workload A starts with the higher intercept, because its marginal value falls faster. A sector table based on one average ratio would miss this diminishing-return effect.

Finite-difference verification

Let $F^\star(C)$ be optimized surplus under capacity $C$. For a small $\epsilon>0$, $$\widehat{\lambda}_{\epsilon} = \frac{F^\star(C+\epsilon)-F^\star(C)}{\epsilon}.$$ The software test uses $\epsilon=10^{-5}$ MWh and compares $\widehat{\lambda}_{\epsilon}$ with the KKT multiplier. Agreement within numerical tolerance checks both the dual sign convention and the optimizer. It does not validate the economic inputs.

Energy component

The energy component is the cost of supplying another unit at the reference location before congestion and marginal losses. In an organized market the component comes from security-constrained economic dispatch. In a bilateral or vertically integrated setting, the relevant short-run avoided cost may come from a different calculation. The notation $p^E$ does not make the sources interchangeable.

Congestion component

Congestion is a scarcity effect. When a line or interface is binding, an incremental withdrawal at one node can require redispatch elsewhere. In the allocator, congestion may appear once in the observed nodal price, or through an explicit interface constraint whose multiplier is allocated by a shift coefficient. Including both without reconciliation double counts the same constraint.

The explicit interface form is useful for an internal campus or feeder problem. Suppose a feeder serves two buildings and has 8 MWh of incremental headroom. The constraint $$x_{\mathrm{north},t}+x_{\mathrm{south},t}\leq 8$$ generates one shadow value. If the two loads have different electrical effects, the coefficients change. Any physical claim then depends on the source of those coefficients.

Loss component

Marginal losses differ from average losses because line losses are nonlinear in current. A nodal price can contain a marginal-loss component. If an analyst instead starts from generator-side energy and converts to delivered load, the loss convention must be explicit. Multiplying an LMP by an additional average loss factor is usually inconsistent.

Reserve and reliability terms

Reserve can be represented as a capacity deduction, a price, or both when the two objects cover distinct services. The capacity deduction protects physical headroom. The price represents the cost of procuring or consuming a reserve product. A single undifferentiated “reliability adder” can conceal overlap.

The model uses $r_{n,t}$ for withheld capacity, $m_{n,t}$ for a reliability margin, and $p^R_{n,t}$ for a marginal reserve charge. Each is optional. If they come from the same requirement, the analyst must provide a bridge showing why there is no duplicate cost.

Intertemporal opportunity cost

The true opportunity cost of running now can be larger or smaller than the current electricity price. Running now may consume a limited daily energy budget, make a later ramp infeasible, or satisfy a deadline that would otherwise become expensive. Ramp and deadline multipliers carry this intertemporal value through Equation (linked).

This is why “move all computation to the cheapest hour” is not a theorem. The cheap interval may lack network capacity. The workload may need an earlier partial run to respect a ramp-up limit. Delay may reduce completion probability. The cost ranking is only one input.

A derivative tied to a counterfactual

Let $G_{n,t}(\delta)$ be total system emissions when incremental load $\delta$ is added at node $n$ and time $t$, relative to a named dispatch counterfactual. The local marginal factor is $$g_{n,t} = \left. \frac{\partial G_{n,t}(\delta)}{\partial\delta} \right|_{\delta=0}.$$ Its unit might be kg CO$_2$ per MWh. A damage value or carbon price $\chi_t$ gives the term $\chi_tg_{n,t}$ in Equation (linked).

This scalar is already a compression. Separate pollutants have different locations, damages, and time profiles. A serious social-cost application should report a vector before applying monetary weights: $$g_{n,t} = \left( g^{\mathrm{CO_2}}_{n,t}, g^{\mathrm{NO_x}}_{n,t}, g^{\mathrm{SO_2}}_{n,t}, g^{\mathrm{PM}}_{n,t} \right).$$

What AVERT supplies

EPA describes AVERT as a tool for evaluating emissions changes associated with energy policies, programs, and projects [11]. The statistical module uses historical hourly generation and emissions data. EPA’s published avoided-emission-rate products provide regional approximations for specified resource categories [12]. These data can support a bounded scenario or screening calculation.

AVERT does not supply a universal node-level factor for any future workload. Its regions are larger than market nodes. Historical relationships may not transport to a system with different fuel prices, retirements, transmission, storage, or weather. A large new load may alter investment, not merely dispatch. The paper therefore labels an AVERT-based term with its region, data year, scenario class, and uncertainty.

Average-factor ranking reversal

Consider two nodes. Node West has a low electricity price but a high marginal emissions factor. Node East has a higher price but a low factor. With no emissions value, a flexible batch prefers West. Once $\chi g$ exceeds the price difference, East becomes cheaper in social marginal terms. The code test constructs exactly this reversal.

The example is algebraic. It does not assert that any named region has those numbers. A credible empirical ranking would require matched nodal prices, marginal emissions, workload performance, data-movement energy, and response curves over the same intervals.

Expected value is not enough in every application

An expected-surplus objective treats upside and downside through their probabilities and values. That may be adequate for many repeated, small jobs. It may be inadequate for a safety-critical industrial batch or a vehicle fleet that must be ready by a fixed hour. A deadline chance constraint or distributionally robust requirement can be more appropriate.

Three choices must remain separate:

1. forecast uncertainty about prices, emissions, availability, and demand;

2. risk preference, such as expected shortfall or a chance constraint;

3. model uncertainty about whether the response curve or probability law is correct.

A confidence interval for an estimated price forecast does not automatically describe tail risk. A robust set does not automatically have a frequency interpretation. The manifest must name the choice.

Two-stage extension

Let $x$ be a day-ahead schedule and $y(\omega)$ real-time recourse. One extension is $$\max_{x} \left\{ B^{DA}(x)-c^{DA\top}x + \mathbb{E}_{\pi} \left[ \max_{y(\omega)\in\mathcal{X}(x,\omega)} B^{RT}(y,\omega)-c^{RT}(\omega)^\top y \right] \right\}.$$ The formulation captures cancellation, migration, or emergency curtailment after uncertainty resolves. It also makes clear that a deterministic shadow price is conditional on the scenario and information set.

Reliability value versus reserve value

Reliability can produce private and system value. A workload may avoid a service failure. The grid may preserve operating headroom. These are different effects. The reference code’s failure_loss_per_mwh parameter enters workload benefit. reliability_margin_mwh reduces node capacity. Neither parameter estimates loss of load probability or the value of lost load.

Correlation

Independent availability assumptions can materially understate risk when many workloads depend on one substation, network carrier, cloud region, or weather event. A common-shock model is needed for portfolio reliability. The small solver does not model correlation. Treating its reliability factors as independent probabilities would be an unsupported extension.

Scenario status

The scenarios in this section are synthetic. They test model behavior and expose missing data. The coefficients are not estimates of national or firm response. In particular, the paper has no credible common-scale marginal value curves for AI, cryptographic settlement, manufacturing, and transport. Any sector ranking is therefore obstructed.

Scenario A: one node, two smooth workloads

The quadratic example in Section 7 verifies the basic equalization condition. Both workloads are interior and share one binding capacity constraint. The result is a clean theorem fixture. It has almost no sector realism.

Scenario B: reserve and reliability deductions

Consider three intervals with 8 MWh of gross incremental availability in each. One MWh is withheld for reserve and one MWh for a reliability margin, leaving 6 MWh dispatchable. A batch workload has a 9 MWh completion requirement, a 6 MWh option bound per interval, and a 1 MWh ramp limit between intervals. A shared feeder caps total batch use at 12 MWh.

The executable fixture checks:

• every interval remains below the 6 MWh dispatchable limit;

• adjacent allocations differ by no more than 1 MWh;

• cumulative energy meets the 9 MWh deadline;

• the feeder limit is respected;

• all KKT residuals are below tolerance.

The point is not that 1 MWh is a sensible reserve for a real grid. The point is that reserve, reliability, ramp, and deadline are distinct constraints with distinct multipliers.

Scenario C: emissions-sensitive location

Two nodes have identical workload response curves. West costs 20 currency units per MWh and has a scenario marginal factor of 800 kg per MWh. East costs 35 and has 100 kg per MWh. An 8 MWh shared compute budget links the choices. Without a carbon value, West receives more energy. With $\chi=0.1$ currency units per kg, East receives more, and modeled marginal emissions fall.

This is a rank reversal caused by a declared social-cost term. It is not a finding about a real West or East region. Using an annual average factor could produce a different and potentially misleading ranking.

Scenario D: reliability-adjusted workload value

A fragile service has a high gross value intercept but only 0.5 reliability and a failure loss. A reliable service has a lower gross intercept but a higher risk-adjusted marginal benefit. Under shared capacity, the reliable service receives more energy, while the fragile service remains positive in the chosen fixture. The test confirms that expected failure loss appears separately in the objective decomposition.

Scenario E: infeasible deadline

A job requires 3 MWh by the only available interval, but both its option bound and node capacity are 2 MWh. The solver returns infeasible. This result matters: an optimization pipeline must not silently violate a service contract because its objective value looks attractive.

Why no sector league table appears

A sector league table would require common monetary or welfare units, compatible energy boundaries, matched locations and intervals, response curves at the relevant margins, reliability losses, and uncertainty. The current evidence does not supply these data. Assigning plausible-looking curves would replace the research question with invented answers.

The paper instead supplies a protocol. A future empirical study can populate one sector at a time, publish the response data, and report where cross-sector comparison remains obstructed.

Step 1: state the decision

Write the intervention as a quantity, location, interval, and alternative. For example: “allocate an additional 0.5 MWh to batch inference at node N17 between 02:00 and 02:15, rather than defer the batch to the next admissible window.” An annual sector label is not a decision specification.

Step 2: measure the response curve

Estimate workload value as a function of incremental energy over the relevant range. Randomized throttling, queue experiments, staggered price exposure, or engineering response tests may identify part of the curve. Record quality, failure, latency, and rework. Do not regress revenue on electricity and call the coefficient causal without a design.

For AI, the service unit should be a verified task outcome or another declared quality measure, not tokens alone. For proof of work, revenue and settlement assurance must remain separate. For manufacturing, record scrap and quality changes. For transport, record service readiness and battery effects.

Step 3: obtain marginal electricity terms

Collect interval-matched energy, congestion, and loss components where they exist. State whether the load actually faces those prices or whether they are social opportunity-cost inputs. Reconcile reserve, capacity, demand, and fixed charges. A bill total divided by MWh is an average.

Step 4: model physical constraints

Record deliverable headroom, feeder or interface limits, ramp capability, location restrictions, downtime, restart cost, and deadlines. Verify that energy is a valid service proxy. If not, express constraints in the actual service unit and link energy through a measured conversion function.

Step 5: estimate marginal emissions

Choose a method matched to scale and horizon. AVERT may support regional screening based on historical dispatch patterns. Unit-level econometric, dispatch, or production-cost models may be needed for other questions. Investment-induced emissions require a long-run model. Publish the pollutant vector and uncertainty before monetization.

Step 6: solve and audit

Report primal decisions, objective decomposition, every constraint slack, and every shadow price. Check KKT residuals for a convex model. Perturb each important right-hand side and compare finite differences with its multiplier. For a mixed-integer model, report the integrality gap and local sensitivity without labeling a discrete jump as a derivative.

Step 7: test sensitivity

At minimum vary:

• workload value-curve parameters and failure loss;

• marginal rather than average price and emissions inputs;

• reserve and reliability deductions;

• deadline and ramp bounds;

• price-taking versus endogenous price response;

• short-run versus capacity-expansion horizons;

• environmental weights and pollutant scope.

If a ranking changes under credible alternatives, report the range or Pareto set. Do not hide the reversal in an appendix.

Objects and units

The module joule_standard.marginal_allocation contains:

• NodalPrice, with energy, congestion, losses, reserve, marginal emissions, and emissions-price components;

• NodeInterval, with available capacity, reserve, and reliability deductions;

• WorkloadSlice, with node, interval, concave benefit, upper bound, reliability, and failure loss;

• DeadlineRequirement, RampLimit, and InterfaceLimit;

• solve_allocation, the active-set solver;

• finite_difference_shadow_price and check_kkt.

Energy is in MWh. Monetary coefficients use one declared currency base per MWh. Emissions use kg per MWh, and the emissions value uses currency per kg. The code checks finiteness, nonnegativity where required, matching node-time keys, unique slice identifiers, and valid constraint references.

Active-set enumeration

Bounds and operational requirements are compiled into $Ax\leq b$. For each candidate active set with at most $J$ linearly independent constraints, the solver:

1. forms the Gram matrix in Equation (linked);

2. solves for candidate multipliers;

3. rejects negative multipliers;

4. reconstructs the allocation;

5. rejects primal violations;

6. keeps the feasible point with largest surplus.

The routine stops if the number of candidate sets exceeds a declared cap. This guard prevents an apparently small example from turning into an accidental combinatorial workload.

KKT audit

The audit computes four residuals: $$\begin{align*} r_{\mathrm{primal}} &= \max_k [A_kx-b_k]_+,\\ r_{\mathrm{dual}} &= \max_k[-\lambda_k]_+,\\ r_{\mathrm{stationarity}} &= \left\|Qx-r+A^\top\lambda\right\|_\infty,\\ r_{\mathrm{complementarity}} &= \max_k \left|\lambda_k(b_k-A_kx)\right|. \end{align*}$$ All must lie below the requested tolerance.

Test contract

The focused test file contains ten tests:

1. interior marginal values equal the binding capacity shadow price;

2. the capacity multiplier matches a finite-difference value derivative;

3. reserve, reliability margin, ramp, deadline, and interface constraints are feasible;

4. an impossible deadline raises an infeasibility error;

5. marginal emissions pricing reverses a two-node dispatch;

6. reliability and failure loss change the allocation;

7. binding and slack constraints satisfy complementary slackness;

8. slice-level decisions aggregate correctly by workload;

9. signed energy, congestion, and loss components are accepted;

10. an optional historical initial ramp state constrains the first modeled interval.

The tests establish computational evidence for the implementation. They do not prove that a real plant, fleet, mine, or data center has the scenario coefficients.

Market power

A large flexible load may influence the market price or anticipate its effect on scarcity. Then $c_{n,t}$ depends on $x$, and price taking fails. A strategic bidder maximizes against a residual supply curve, not a fixed vector. The KKT machinery can still apply to a suitably convex private problem, but the welfare interpretation changes.

Nonconvex operations

Unit commitment, minimum-run constraints, startup costs, integer servers, minimum batch sizes, and all-or-nothing deadlines create nonconvexities. An interior marginal equalization rule can fail because no nearby feasible point exists. Convex-hull prices, uplift, or mixed-integer methods may be needed.

Endogenous capacity

The short-run model treats node capacity as given. Persistent automated load can trigger generation, transmission, distribution, or data-center investment. Long-run marginal cost then includes capital and embodied effects. A short-run LMP should not be extrapolated into an investment theorem.

Distribution network limits

A wholesale node may hide distribution constraints. Thermal feeder limits, voltage, phase imbalance, and protection settings can determine whether an incremental load is deliverable. A single interface proxy does not validate those conditions.

Dynamic stability

The reference model does not simulate frequency, voltage, inverter controls, or transient behavior. Even a full DC optimal power flow would not prove dynamic stability. Claims about grid support require separate engineering analysis.

Baseline error

Marginal value and emissions require a counterfactual. A deferred batch might run later rather than disappear. Curtailed mining may shift to another site. Delayed manufacturing may incur overtime. Vehicle charging may move into a more polluting interval. Counting only the immediate reduction overstates the effect if the load reappears.

Telemetry and attribution

Facility meters may combine many loads. Allocating shared cooling or network energy by server time, peak power, or direct metering can change the result. The attribution rule must be published. An unexplained proportional allocation is not a measurement.

A benchmark, not an automatic mandate

The equalization rule describes an optimum inside a declared model. It does not tell a regulator which value functional to adopt or how to weight pollution, reliability, employment, privacy, or distribution. Those choices enter the numerator, constraints, or prices.

The model can still clarify policy disagreements. If two analysts disagree because one uses an average retail rate and the other uses nodal marginal cost, that is an estimand disagreement. If they disagree about the carbon weight, that is a valuation disagreement. If one includes an industrial deadline and the other does not, that is a feasible-set disagreement. These should not be blended into one unexplained score.

Flexible load programs

A flexible-load program can improve dispatch only if the load responds when and where flexibility is useful, respects telemetry and control rules, and does not shift costs or emissions outside the measured window. Compensation should be tied to an auditable baseline and delivered service. The paper does not design that baseline or settlement rule; Paper 8 in the program addresses market design.

Environmental dispatch

Adding $\chi g$ to nodal cost is one transparent way to represent marginal emissions damages. Other approaches include emissions caps, clean-energy constraints, or multi-objective frontiers. A monetized scalar can be useful, but the underlying pollutant vector and weights should remain visible.

Reliability and essential service

Some workloads may receive minimum allocations because they provide essential service. That policy appears as a lower bound or deadline, and its shadow price reports the local opportunity cost. The multiplier does not decide whether the requirement is legitimate. It makes the resource tradeoff inspectable.

Estimating workload response

The central empirical task is to estimate marginal service value under controlled energy variation. For AI workloads, randomized queue throttling can measure completion, latency, quality, and rework. For manufacturing, plant experiments need product-quality safeguards. For fleet charging, route readiness and battery degradation belong in the outcome. Proof-of-work studies must distinguish private mining revenue from network assurance.

Linking flexible loads to grid nodes

Public workload data rarely identify electrical nodes. Data-center region labels are too coarse for congestion studies. Partnerships with utilities, system operators, and large users could produce privacy-preserving data that join meter location, interval load, price components, constraints, and workload service.

Long-run investment

Persistent automated demand changes generation, transmission, and data-center capacity. A long-run model should include discrete investment, lead times, embodied energy, and demand uncertainty. Short-run and long-run marginal value should be reported separately.

Distribution and governance

An automated allocation rule can move pollution, congestion, and economic activity across communities. A vector account should report local emissions, reliability exposure, consumer effects, and workload outcomes before any weighted scalar is formed.

Robust optimization

Marginal emissions, prices, and workload value curves are estimated. Robust sets or distributionally robust methods can show which allocations survive credible misspecification. Their conservatism should be calibrated against held-out operating periods, not chosen for aesthetic smoothness.

Bounds

The upper bound $x_j\leq\bar x_j$ is a row of $A$. The lower bound $x_j\geq 0$ is written $-x_j\leq 0$. If neither binds, both multipliers are zero. If $x_j=0$, the lower-bound multiplier can absorb a negative unconstrained marginal surplus. Thus zero allocation is compatible with the KKT rule and does not require equal marginal value with an interior workload.

Deadline sign convention

A minimum requirement $$\sum_{j\in D}x_j\geq Q$$ is compiled as $$-\sum_{j\in D}x_j\leq -Q.$$ Relaxing the compiled right side means lowering the required minimum. The associated multiplier is the local increase in optimized surplus from that relaxation. Conversely, tightening the physical deadline requirement by one MWh reduces surplus by the same first-order amount.