Time-Varying Electricity Pricing

• Time-varying electricity pricing is a dynamic rate mechanism where prices fluctuate based on supply costs, demand changes, and ancillary system requirements.
• It integrates energy costs with marginal ancillary costs, using both current and past consumption data to incentivize more efficient load shifting.
• Simulations reveal that these mechanisms can lower retail prices, reduce peak loads, and achieve significant welfare gains over traditional static pricing models.

Time-varying electricity pricing refers to mechanisms in which the unit price of electricity changes over time, often reflecting underlying supply costs, system constraints, market dynamics, or operational externalities. Unlike static or flat pricing, these schemes aim to expose consumers and market participants to prices that better capture marginal production costs, demand fluctuations, integration of renewables, and ancillary services requirements.

1. Principles of Time-Varying Electricity Pricing

Traditional electricity pricing has relied on static rates or real-time marginal cost pricing, where the energy price in period $t$ is set as the derivative of the energy cost function with respect to aggregate demand, $C'(A_t, s_t)$. This approach internalizes the instantaneous marginal production cost but is generally insufficient when supplier costs are driven by the variability and ramp rates of demand—particularly as the share of renewables increases or when ancillary services become more costly (Tsitsiklis et al., 2012).

An essential insight is that when supplier cost depends not only on the current aggregate demand $A_t$ but also changes between periods (e.g., due to reserve adjustments or ramping), setting prices purely at the instantaneous marginal cost fails to achieve social optimality. Dynamic pricing mechanisms must instead include terms that price the externality imposed by a consumer’s actions on future ancillary costs, effectively encouraging demand trajectories that lower overall system costs over time.

Dynamic pricing can be formalized via a multi-component price at each stage $t$:

• $p_t = C'(A_t, s_t)$: marginal energy cost.
• $$w_t = \partial H(A_{t-1}, A_t, \overline{s}_t)/\partial A_t$$: marginal ancillary cost with respect to current demand.
• $$q_t = \partial H(A_{t-1}, A_t, \overline{s}_t)/\partial A_{t-1}$$: marginal impact of previous period's demand on current ancillary cost.

This results in the individual consumer's stage-$t$ payment:

$$U_t(x_{i,t}, s_t, a_{i,t}) - (p_t + w_t)a_{i,t} - q_t a_{i,t-1}$$

Negative $q_t$ acts as a rebate on previous consumption and incentivizes intertemporal load shaping (Tsitsiklis et al., 2012).

2. Mathematical Formulation and Dynamic Game-Theoretic Analysis

In a dynamic setting, the social cost function typically involves not only the sum of energy costs $C(A_t, s_t)$ but also an explicit ancillary cost $H(A_{t-1}, A_t, s_{t-1}, s_t)$ capturing the costs associated with demand swings:

$$\sum_t \left[ C(A_t, s_t) + H(A_{t-1}, A_t, s_{t-1}, s_t) \right]$$

Consumers solve an intertemporal optimization, anticipating both the immediate price and the delayed penalty or reward incurred via $q_{t+1}$ for past actions. The corresponding equilibrium, under the "dynamic oblivious equilibrium" (DOE) concept, ensures that as the number of consumers grows, the individual impact on aggregate demand vanishes, and the pricing rule becomes asymptotically socially optimal (asymptotic Markov equilibrium) (Tsitsiklis et al., 2012).

Key assumptions:

• Cost functions $C$ and $H$ are convex and differentiable.
• $H$ depends only on current and previous aggregate demand.
• Consumers are price-takers, base their action on internal state and exogenous stochastic state history.
• Uniform equicontinuity and bounded derivatives guarantee small actions induce only moderate price changes.

Theoretical results show that such mechanisms, under DOE, achieve social welfare maximum and lead to load profiles with reduced peakiness, as demonstrated in detailed equilibrium conditions and numerical examples.

3. Implications for Demand Response and Ancillary Services

Internalizing ancillary costs via time-varying pricing mechanisms realigns consumer incentives to promote load shifting and reduction of demand swings. When $q_t < 0$, consumers are effectively rewarded for shifting their consumption away from periods causing high ramping or reserve costs. The mechanism thus provides tangible economic motivation to flatten the demand curve, reducing the necessity for costly peaking units.

Simulations from (Tsitsiklis et al., 2012) show that with standard parameter choices (e.g., substitutability parameter $E$ indicating consumer flexibility), the proposed pricing mechanism yields:

• Average retail price substantially lower than marginal cost pricing.
• Off-peak loads that increase only slightly (e.g., from 1.00 to 1.09) resulting in an effective price that can be zero for some periods.
• Welfare gains up to 50% higher over the marginal cost pricing baseline, especially at high substitutability.
• Moderate but meaningful reductions in peak load, thus deferring or eliminating the need for new peaking investments.

The effect is to allocate the cost of fluctuations directly to users responsible for causing those fluctuations, correcting the misalignment inherent in static pricing models.

4. Dynamic Program and Equilibrium Characterization

A two-stage and generalized multi-stage dynamic game structure is analyzed:

• In the two-stage case, the equilibrium is characterized by the conditions:

$$U_0'(a_0) = C'(a_0) - H'(a_1 - a_0) \ U_1'(a_1) = C'(a_1) + H'(a_1 - a_0)$$

so that the ancillary cost term $H'$ is internalized into the price paid for each action.

• The total payment combines both the marginal energy costs and ancillary terms affecting both current and past actions.
• For a general $T$-period game, each consumer’s strategy is a function of their own state and exogenous history, with the aggregate price sequence adjusting via the DOE concept.

Convexity and regularity of cost and utility functions ensure the uniqueness and stability of equilibrium, and the dynamic price profile converges to a welfare-maximizing solution as the number of agents increases.

5. Numerical Illustration and Welfare Outcomes

The paper provides case studies in which:

• Under flat rate (reference) pricing, off-peak demand is at baseline.
• Under marginal cost pricing only, consumers react by a limited amount to price signals, with little load shifting.
• Under the proposed mechanism, off-peak consumption rises but is met with a negative $q_1$ that rebates previous consumption, leading to average cost reduction.
• For instance, calculated prices in one example were $p_0 + w_0 ≈ 4.44$, $p_1 + w_1 ≈ 6.76$, $q_1 ≈ -4.44$.
• With higher flexibility ($E$ large), the welfare gain from implementing the dynamic pricing mechanism (compared to marginal cost pricing) increases significantly, and the reduction in peak demand becomes more pronounced.

These results highlight the mechanism’s ability to incentivize socially beneficial load shifting and reduce the reliance on capacity that is used only during short peaks.

6. Limitations, Assumptions, and Implementation Considerations

The formulation presumes:

• Aggregate supplier cost comprises only immediate energy and ancillary cost depending on consecutive demand levels.
• All consumers are rational, forward-looking, and price-taking, with perfect knowledge of their own states and price paths.
• The dynamic pricing rule is implementable and can be disseminated to consumers at the requisite temporal granularity.

In real-world settings, implementation would necessitate:

• Advanced metering infrastructure capable of real-time (or at least high-frequency) data collection and pricing signal communication.
• Well-designed market interfaces that transparently present $q_t$, $w_t$, and $p_t$ to end-users.
• Mechanisms for estimating or learning $H$, the ancillary cost function, from data on system ramping and reserve requirements.

Potential limitations involve consumer understanding, delays in response, institutional constraints on tariff structures, and regulatory acceptance. Moreover, secondary effects such as strategic load reshaping by aggregators or non-convexities in demand response may require further analysis.

7. Significance in the Context of Modern Power Systems

As volatile renewables increase the variability of net load, internalizing fluctuation-induced costs via time-varying pricing becomes essential. The mechanism laid out in (Tsitsiklis et al., 2012) anticipates grid-wide economic and operational benefits, including:

• Improved efficiency through better-aligned incentives, social cost minimization, and reduced ancillary service procurement.
• Enhanced robustness against renewable intermittency by aligning consumption with system capability, not merely instantaneous marginal cost.
• Clear theoretical justification for regulatory and market design changes that explicitly price demand-side variability.
• Empirical evidence suggesting material welfare gains and peak shaving can be achieved with practical, implementable price schedules—provided appropriate technology and market infrastructure are in place.

The rigorous dynamic game-theoretic analysis and detailed equilibrium characterization provide a template for future dynamic pricing policies in high-renewable penetration power systems, situating this body of work as foundational for ongoing research and policy development in electric power markets.