Unlocking Seasonal Capacity Value: A Sub-Annual Capacity Market for Economic Robustness

Abstract

As variable renewable energy penetration increases, resource adequacy becomes strongly seasonal, while annual accreditation can mask temporal reliability differences. This paper proposes a Sub-Annual Capacity Market and compares it with an Annual Capacity Market and an uncapped Energy-Only benchmark. Capacity credits are calculated using a marginal ELCC formulation based on Expected Energy Not Served and embedded into phase-specific clearing constraints. Using a Shanxi case study, we examine both deterministic and stochastic settings with 151 jointly perturbed load and renewable scenarios. Results show that ACM and SubACM can both approximate EO outcomes when parameters are well calibrated, but SubACM yields more stable economic performance under uncertainty, with 29% lower cost-deviation standard deviation and 67% fewer tail-risk scenarios, as confirmed by formal dispersion tests. The main benefit of sub-annual design is improved temporal alignment between capacity payments and physical reliability contribution, rather than guaranteed large average cost reductions.

1. Introduction

1.1. Motivation

Capacity adequacy is fundamental to the secure and reliable operation of power systems. As wind and solar photovoltaic (PV) penetration increases, reliability assessment is shifting from a deterministic, peak-load-centered paradigm to a probabilistic, weather-driven framework. In capacity markets, prices are intended to reflect the value of reliability and to incentivize a diverse resource portfolio that meets established standards [1], such as Loss of Load Expectation (LOLE) and Expected Energy Not Served (EENS). Under the prevailing annual design, Independent System Operators (ISOs) typically set the Installed Reserve Margin (IRM) to satisfy reliability targets and procure capacity against the forecast annual peak load. In addition, capacity accreditation and procurement are largely based on a resource’s annual average contribution, implicitly assuming that adequacy risks are concentrated in a narrow set of extreme-load hours and that resources provide time-invariant reliability value throughout the year [2].

However, this assumption is increasingly inconsistent with the operational realities of high-VRE systems. Adequacy risk cannot be fully represented by a single annual snapshot; instead, it varies across seasons and operating conditions [3]. For example, summer midday peaks driven by cooling demand may be partially offset by strong solar output, whereas winter nighttime peaks driven by heating demand may coincide with low renewable generation. In coal-dominated systems such as China, these seasonal differences are further amplified during the heating season by thermo-electric coupling constraints in CHP units [4]. Such seasonally asymmetric patterns are not unique to China; dual-peak load profiles driven by electrified heating and cooling are also common in parts of Europe and North America. Consequently, a traditional ACM can become structurally misaligned, overcompensating resources that are redundant in non-critical seasons while undervaluing resources that provide substantial reliability benefits in specific periods but are not available year-round. This inefficiency raises procurement costs and distorts long-term investment signals, motivating a transition toward sub-annual or seasonal market designs [5]. Recognizing this challenge, PJM, MISO, and ISO-NE have begun exploring or adopting seasonal capacity constructs.

1.2. Literature Review

The evolution of capacity remuneration mechanisms has been a focal point of academic and industrial research for decades. Among these mechanisms, capacity markets are the most widely implemented approach [6,7,8]. As power systems decarbonize, the relevant literature has gradually moved from asking whether a separate capacity mechanism is needed to asking how its product definition, accreditation rules, and temporal granularity should adapt to weather-dependent resources and seasonally shifting adequacy risks.

First, extensive research addresses the economic rationale and core design of capacity markets. Foundational studies frame capacity remuneration as a response to the “missing money” problem in energy-only (EO) markets and examine how centralized procurement and demand-curve-based auctions can support long-term adequacy [9,10,11]. More recent work emphasizes that the value of capacity is increasingly shaped by the broader energy transition, because price caps, technology heterogeneity, and policy-driven resource turnover alter investment incentives and the composition of the resource mix [12]. However, much of this literature still treats the capacity product as an annual obligation, implicitly assuming that resources provide relatively stable adequacy value throughout the year.

Second, capacity accreditation—especially for VRE and storage—has received growing attention. Since Garver’s seminal formulation of ELCC [13], a large body of work has shown that probabilistic accreditation methods are better suited than static derating factors for valuing weather-dependent and energy-limited resources. Recent studies further demonstrate that accreditation outcomes depend on system conditions, resource penetration, operational constraints, and the methodological choice between marginal and average ELCC [14,15,16]. This literature has substantially improved the technical measurement of capacity value, yet in many applications the resulting credit is still condensed into a single annual metric, which can mask intra-annual variation in reliability contribution.

Third, adequacy risk itself is not temporally uniform. Studies on obligation periods and resource adequacy modeling argue that when both load and resource availability exhibit strong seasonality, annual procurement can become inefficient because it forces season-specific resources to fit a year-round product [17]. Related resource adequacy analyses further show that monthly or seasonal risk patterns may differ markedly owing to temperature-sensitive outages, changing net-load conditions, and shifting scarcity hours, implying that procurement targets and accreditation rules should not necessarily be uniform across the year [18]. These insights are increasingly reflected in policy and market reform discussions in PJM, MISO, and ISO-NE, where seasonal constructs are being explored or implemented to better align procurement with seasonal reliability needs [5,19,20,21].

Taken together, the literature provides a strong rationale for moving beyond annual average constructs, but the existing strands remain only partially connected. Market-design studies typically discuss sub-annual obligation periods at a conceptual level, accreditation studies concentrate on estimating ELCC values rather than embedding them in alternative market-clearing models, and institutional reform documents rarely benchmark seasonal procurement against an ideal EO adequacy outcome within a unified analytical framework. In addition, limited work has quantitatively examined whether sub-annual procurement remains advantageous when realized load and VRE conditions deviate from the ex-ante assumptions used to set market parameters. This gap motivates the present study, which links phase-specific capacity accreditation, annual and sub-annual market clearing, and stochastic robustness assessment in one consistent framework.

1.3. Contributions

To address these challenges, this paper develops an analytical framework to evaluate sub-annual capacity market design in power systems with high renewable penetration and pronounced seasonal asymmetry. The main contributions are as follows:

• Feasibility Analysis of a Sub-Annual Capacity Market. We establish a unified workflow that starts from an uncapped Energy-Only benchmark representing the cost-minimizing adequacy outcome under ideal price signals. Based on this benchmark, we compute phase-specific capacity credits using a marginal ELCC formulation and embed these credits into ACM and SubACM clearing models for a consistent comparison under price-capped settings. Relative to prior studies that examine accreditation or market design separately, this framework links reliability valuation and procurement design within one model structure.
• Economic Efficiency Assessment of Phase-Based Design. We explicitly account for uncertainty by constructing a large ensemble of stochastic scenarios that jointly perturb load, wind, and solar profiles, and using them to evaluate economic performance under ex-ante parameter mismatch. This setting goes beyond the predominantly conceptual discussion in the seasonal-capacity literature and is used to verify whether SubACM remains more robust than ACM when realized operating conditions deviate from forecasts.
• Empirical Evaluation with Real-World Data. Using data from the Shanxi power system, we quantify how phase-based procurement changes accreditation outcomes, investment composition, and system cost performance. The results show that SubACM improves temporal value recognition and delivers more stable economic outcomes under stochastic conditions, thereby providing quantitative and policy-relevant evidence for market reform in high-renewable systems with pronounced seasonal asymmetry.

The remainder of this paper is organized as follows. Section 2 presents the methodology, including the EO benchmark, capacity credit calculation, and market-clearing formulations for ACM and SubACM. Section 3 introduces the case setup and reports the main simulation results under deterministic and uncertainty scenarios. Section 4 discusses institutional implementation considerations and modeling limitations. Section 5 concludes the paper.

2. Methodology

2.1. Overall Framework

Ideally, an EO market without price caps can provide efficient incentives for resource investment, ensuring adequate generation capacity to maintain system reliability [22]. However, in real-world markets, the presence of regulatory price caps suppresses scarcity pricing signals, giving rise to the “missing money” problem that undermines long-term resource adequacy. To establish an ideal economic benchmark, we develop an uncapped EO market model (Section 2.2) that co-optimizes investment and operational dispatch to minimize total system cost, drawing on [15] while adapting it to our case design and assumptions. By assigning a sufficiently high Value of Lost Load (VOLL), set to CNY 10,000/MWh here, this model endogenously determines the economically optimal capacity mix and reliability level.

The resulting EO solution serves two purposes in the analytical framework. It provides the techno-economic optimum against which the efficiency of capacity market designs is measured, and it supplies the data from which CCs and IRMs are calibrated as fixed inputs to the ACM and SubACM clearing models. This sequential workflow, in which the benchmark is established first, auction parameters are derived second, and market clearing is performed last, mirrors standard practice in operational capacity markets. In PJM’s Reliability Pricing Model, for instance, accreditation and reliability parameters are published in the Planning Period Parameters for Base Residual Auction document prior to each auction, and the capacity auction is subsequently organized on the basis of these pre-determined parameters. Our calibration follows the same logic: CCs and IRMs are derived once from the EO equilibrium and then held fixed throughout all subsequent market-clearing evaluations.

As shown in Figure 1, we start from the EO benchmark and then quantify each resource’s reliability contribution using the ELCC method introduced in Section 2.3. Capacity credits (CCs) are evaluated at two temporal resolutions: annual CCs for ACM and phase-specific CCs for SubACM, defined over distinct operating periods (e.g., heating and non-heating phases, or four seasonal phases). These CC estimates are then embedded in the market-clearing models in Section 2.4, where ACM and SubACM apply adequacy constraints that are calibrated to the EO reliability target under price-capped settings. Section 3 then uses the Shanxi system as a case study to compare the economic performance and reliability robustness of ACM and SubACM under different load scenarios.

2.2. Energy-Only Market Model

In the EO model, generation resources and ESSs are treated as price takers, and demand is represented as perfectly inelastic load. To capture uncertainty in load and renewable output, we use multi-year historical time-series data. The model minimizes total annualized system cost and identifies the optimal capacity mix for a target year. It is a static planning framework, so it does not endogenously represent construction timing or multi-year expansion pathways. The model adopts a single-node network representation, so that transmission constraints, inter-regional power exchanges, and locational price differentials are not modeled; all generation, storage, and demand are aggregated at a single system-level bus, and the results therefore apply to a system-level analysis without spatial disaggregation. We also adopt a Greenfield setting in which all resources are modeled as new assets, so that the EO solution serves as an analytical benchmark for mechanism comparison rather than a realistic generation expansion plan for any specific power system. This ensures that the resulting capacity mix is determined entirely by techno-economic fundamentals rather than by legacy fleet composition or sunk-cost effects, thereby isolating the mechanism design comparison from confounding factors related to fleet history. Constraints associated with heterogeneous financing conditions and exogenous policy interventions are likewise excluded.

In capacity planning and dispatch models, thermal unit start-up/shut-down decisions and storage charging/discharging states are typically discrete variables, which can make long-horizon optimization computationally demanding. Following [23], we convexify thermal commitment by relaxing these binary decisions into continuous variables, which yields a tractable convex formulation. For storage, we do not impose strict mutual exclusivity between charging and discharging [24]. When the storage decision variable represents an aggregated fleet or a battery array composed of multiple cells, a portion of the fleet may charge while another portion discharges simultaneously through plant-level coordination. In such aggregate representations, the complementarity between charging and discharging power is not a hard physical constraint but an economic outcome: incorporating cycling degradation costs into the objective function penalizes energy throughput and thereby discourages the optimizer from scheduling simultaneous charging and discharging whenever such behavior would be economically irrational [25]. We have verified the optimization results across all scenarios and confirmed that no instance of simultaneous charging and discharging occurs in the solution.

The specific objective function and constraints of the model are as follows.

2.2.1. Objective Function

The model minimizes the total annualized system cost, encompassing investment, operation, and energy unserved penalties:

$$\begin{array}{cc}\hfill min& \sum _{t\in \mathcal{T}}\sum _{g\in {\mathcal{G}}^{\mathrm{th}}}\left(C_g^{\mathrm{var}}{p}_{g,t}+C_g^{\mathrm{su}}{v}_{g,t}\right)+\sum _{g\in \mathcal{G}}\left(C_g^{\mathrm{inv}}{P}_g^{max}{x}_g\right)+\sum _{s\in \mathcal{S}}\left(C_s^{\mathrm{inv}}{E}_s^{\mathrm{cap}}\right)\hfill \\ \hfill & +\sum _{t\in \mathcal{T}}\sum _{s\in \mathcal{S}}{C}_s^{\mathrm{var}}\left(p_{s,t}^{\mathrm{dch}}+p_{s,t}^{\mathrm{ch}}\right)+\sum _{t\in \mathcal{T}}\sum _{b\in \mathcal{B}}\left(V^{\mathrm{OLL}}{l}_{b,t}^{\mathrm{shed}}\right)\hfill \end{array}$$

(1)

where $\mathcal{T}$, $\mathcal{G}$, ${\mathcal{G}}^{\mathrm{th}}$, $\mathcal{S}$, and $\mathcal{B}$ represent the sets of time periods, all generators, thermal generators, ESSs, and buses, respectively. For the cost parameters, $C_g^{\mathrm{var}}$ and $C_s^{\mathrm{var}}$ denote the variable costs for generator g and storage s; $C_g^{\mathrm{su}}$ is the start-up cost for thermal unit g; and $C_g^{\mathrm{inv}}$ and $C_s^{\mathrm{inv}}$ represent the annualized investment cost per MW for generation capacity and storage capacity. $V^{\mathrm{OLL}}$ denotes the Value of Lost Load (VOLL), i.e., the penalty (in CNY/MWh) applied to load shedding. Decision variables include the power output $p_{g,t}$, the integer number of invested units $x_g$, the installed storage energy capacity $E_s^{\mathrm{cap}}$, storage charge/discharge power $p_{s,t}^{\mathrm{ch}}$ and $p_{s,t}^{\mathrm{dch}}$, thermal unit start-up action $v_{g,t}$, and the amount of load shed $l_{b,t}^{\mathrm{shed}}$. $P_g^{max}$ is the nameplate capacity of g.

2.2.2. System Constraints

Supply–Demand Balance.Under the single-node representation adopted above, the system-wide supply must balance aggregate demand at each time step:

$$\sum _{g\in \mathcal{G}}{p}_{g,t}+\sum _{s\in \mathcal{S}}(p_{s,t}^{\mathrm{dch}}-p_{s,t}^{\mathrm{ch}})+\sum _{b\in \mathcal{B}}{l}_{b,t}^{\mathrm{shed}}=\sum _{b\in \mathcal{B}}{D}_{b,t},\forall t\in \mathcal{T}$$

(2)

where $D_{b,t}$ is the demand at bus b and time t.

Spinning Reserve Requirement. The sum of available generator headroom, storage reserve capability, and charged storage capacity must satisfy the reserve margin relative to system load:

$$\sum _{g\in {\mathcal{G}}^{\mathrm{th}}}{r}_{g,t}+\sum _{s\in \mathcal{S}}(r_{s,t}+p_{s,t}^{\mathrm{ch}})\ge \rho \sum _{b\in \mathcal{B}}{D}_{b,t},\forall t\in \mathcal{T}$$

(3)

where $r_{g,t}$ and $r_{s,t}$ are the spinning reserve contributions from generator g and storage s, respectively. $\rho$ represents the required reserve margin ratio relative to the total system load.

2.2.3. Thermal Generation Constraints

Thermal Power Output. The power output is bounded by the minimum stable level and the available capacity:

$${\alpha }_g{P}_g^{max}{u}_{g,t}\le p_{g,t},\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in \mathcal{T}$$

(4)

$$p_{g,t}+r_{g,t}\le (1-{\varphi }_g)P_g^{max}{u}_{g,t},\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in \mathcal{T}$$

(5)

where $u_{g,t}\in [0,1]$ is the relaxed commitment status (1 if fully online, 0 if offline, and fractional values allowed). ${\alpha }_g$ represents the minimum stable output factor, and ${\varphi }_g$ denotes the forced outage rate, effectively derating the maximum capacity. In this study, ${\alpha }_g$ is time-invariant and set to 40% for all thermal units; the heating-season reduction in electrical flexibility is captured separately through a maximum output derating to 85% of nameplate capacity (Section 3), rather than through a time-varying minimum output factor.

Investment Consistency. The number of units committed at any hour cannot exceed the total installed capacity.

$$u_{g,t}\le x_g,\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in \mathcal{T}$$

(6)

State Transition Constraints. The commitment status is linked to start-up ($v_{g,t}$) and shut-down ($w_{g,t}$) variables:

$$u_{g,t}-u_{g,t-1}=v_{g,t}-w_{g,t},\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in \{2,\dots ,|\mathcal{T}\left|\right\}$$

(7)

where $v_{g,t},w_{g,t}\in [0,1]$ represent start-up and shut-down actions, respectively. The initial commitment state $u_{g,1}$ is down.

Minimum Up/Down Time. Units must respect physical minimum online ($T_g^{\mathrm{on}}$) and offline ($T_g^{\mathrm{off}}$) durations:

$$u_{g,t}\ge \sum _{\tau =t-T_g^{\mathrm{on}}+1}^t{v}_{g,\tau },\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in [T_g^{\mathrm{on}}+1,\left|\mathcal{T}\right|]$$

(8)

$$1-u_{g,t}\ge \sum _{\tau =t-T_g^{\mathrm{off}}+1}^t{w}_{g,\tau },\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in [T_g^{\mathrm{off}}+1,\left|\mathcal{T}\right|]$$

(9)

Ramp Rate Constraint. The change in power output is limited by the ramping capability $R_g^{\mathrm{up}}$ and $R_g^{\mathrm{down}}$:

$$(p_{g,t}+r_{g,t})-p_{g,t-1}\le R_g^{\mathrm{up}}(1-{\varphi }_g)P_g^{max}{u}_{g,t},\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in \{2,\dots ,|\mathcal{T}\left|\right\}$$

(10)

$$(p_{g,t-1}+r_{g,t-1})-p_{g,t}\le R_g^{\mathrm{down}}(1-{\varphi }_g)P_g^{max}{u}_{g,t},\forall g\in {\mathcal{G}}^{\mathrm{th}},t\in \{2,\dots ,|\mathcal{T}\left|\right\}$$

(11)

2.2.4. Renewable Constraints

Renewable Power Output. The power output of renewable resources is bounded by their installed capacity multiplied by the time-dependent availability factor. Moreover, this model does not impose a mandatory renewable energy consumption rate, allowing for wind and solar curtailment. Curtailment is costless in this formulation and is determined endogenously by the optimizer to minimize total system cost; no mandatory consumption obligations or curtailment penalties are imposed.

$$0\le p_{g,t}\le x_g{P}_g^{max}{\xi }_{g,t},\forall g\in {\mathcal{G}}^{\mathrm{ren}},t\in \mathcal{T}$$

(12)

where ${\mathcal{G}}^{\mathrm{ren}}$ is the set of renewable generators. ${\xi }_{g,t}\in [0,1]$ is the availability factor for resource g at time t, derived from historical meteorological data.

2.2.5. Energy Storage Constraints

$$s_{s,t}=s_{s,t-1}+\eta p_{s,t}^{\mathrm{ch}}-{\displaystyle \frac{1}{\eta }}{p}_{s,t}^{\mathrm{dch}},\forall s\in \mathcal{S},t\in \{2,\dots ,|\mathcal{T}\left|\right\}$$

(13)

$$p_{s,t}^{\mathrm{dch}}-p_{s,t}^{\mathrm{ch}}+r_{s,t}\le P_s^{max},\forall s\in \mathcal{S},t\in \mathcal{T}$$

(14)

$$p_{s,t}^{\mathrm{dch}}+p_{s,t}^{\mathrm{ch}}\le P_s^{max},\forall s\in \mathcal{S},t\in \mathcal{T}$$

(15)

$$s_{s,t}\le E_s^{\mathrm{cap}},\forall s\in \mathcal{S},t\in \mathcal{T}$$

(16)

$$s_{s,t}\ge {\displaystyle \frac{1}{\eta }}\left(p_{s,t}^{\mathrm{dch}}+{\tau }^{\mathrm{res}}{r}_{s,t}\right),\forall s\in \mathcal{S},t\in \mathcal{T}$$

(17)

where Equation (13) is the state transition constraint for energy storage; $s_{s,t}$ represents the state of charge (MWh). $P_s^{max}$ is the maximum charging/discharging power of storage unit s, and $\eta$ is the charging/discharging efficiency. Equation (14) specifies the power limit for storage reserve provision. Equation (15) enforces the charging/discharging power limit, and Equation (16) imposes the energy capacity limit. Equation (17) ensures energy adequacy for reserve provision, where ${\tau }^{\mathrm{res}}$ denotes the time requirement for reserve activation.

2.3. Capacity Credit Calculation

CC quantifies the reliability value contributed by a resource. Existing studies generally use two major approaches, ELCC and Effective Firm Capacity (EFC) [14]. In this study, we use marginal reliability impact (MRI), which is not conceptually separate from ELCC, but a marginal implementation under a selected reliability metric. Specifically, CC is computed as the ratio between the reliability improvement from a 1 MW increment of resource g and that from a 1 MW increment of a perfect resource, which is consistent with marginal ELCC interpretation. The difference is therefore methodological rather than economic. We adopt EENS as the reliability metric rather than frequency-based indices such as LOLE for two reasons. First, in systems with energy storage, EENS provides a more stable basis for marginal CC computation. Because storage is fundamentally a time-shifting resource, the temporal allocation of its discharge can be varied without changing the total unserved energy, yet different dispatch schedules can yield different numbers of loss-of-load hours and hence different LOLE values. EENS, by contrast, reflects the aggregate energy shortfall regardless of how storage is dispatched across hours, and therefore produces more stable marginal reliability impact values when computing CCs through the perturbation approach used here. Second, EENS weights scarcity events by their severity, so that a resource reducing the depth of the most severe shortfalls receives a higher marginal CC than one that merely eliminates a shallow scarcity hour.

Following this marginal ELCC formulation, the capacity credit of resource g, denoted as $C{C}_g$, is defined as:

$$C{C}_g={\displaystyle \frac{{\mathit{EENS}}_0-{\mathit{EENS}}_g}{{\mathit{EENS}}_0-{\mathit{EENS}}_{\mathrm{perfect}}}}$$

(18)

where ${\mathit{EENS}}_0$ is the baseline EENS under the optimal capacity mix from the EO model, ${\mathit{EENS}}_g$ is the EENS after a marginal 1 MW increment of resource g, and ${\mathit{EENS}}_{\mathrm{perfect}}$ is the EENS after a marginal 1 MW increment of a perfect resource. Under this normalization, $C{C}_g$ measures the relative reliability contribution of resource g against the perfect-capacity benchmark. All three EENS values are computed under the same optimal capacity mix from the EO model, with only the specified marginal increment varied; the remaining fleet composition is held fixed across all perturbation evaluations.

It is worth noting that in optimization models incorporating unit commitment constraints and energy storage dynamics, the EENS metric does not vary smoothly with installed capacity; rather, it behaves as a piecewise convex function. As a consequence, marginal CC values can change abruptly at the breakpoints of this function, leading to discontinuities in accreditation. Because linear programming solvers select extreme-point solutions, these discontinuities tend to coincide with the equilibrium capacity levels identified by the EO model. A single set of CCs computed from only one perturbation direction may therefore be insufficient to ensure that the capacity market solution exactly reproduces the EO outcome; the market-clearing model could converge to a slightly different capacity mix with marginally higher EENS. To mitigate this issue, this study computes CCs using both increment and decrement perturbations of a perfect generator at the EO equilibrium point. The two resulting CC sets capture the range of marginal capacity values on either side of the breakpoint.

Conventional CC estimation is commonly based on year-round simulations or representative scenarios, producing a single annual metric. This aggregation can obscure temporal heterogeneity in adequacy conditions, because both electricity demand and renewable generation are strongly seasonal. The issue is particularly salient in Northern China, where winter electrified heating drives demand to annual peaks, while coal-fired thermal units face binding output constraints due to heat supply obligations, thereby intensifying supply–demand imbalances during winter stress periods.

Accordingly, an annual-average CC alone cannot adequately represent seasonal variation in resource scarcity value. To account for temporal heterogeneity in load-resource alignment and to improve the economic efficiency of capacity-market design, this study introduces phase-specific granularity into CC estimation. The two approaches are distinguished as follows:

• Annual CC (applied in the ACM): Computed using full-year simulation data, yielding a single, static reliability value for each technology. This value is applied uniformly throughout the year, implicitly smoothing out seasonal variations in resource adequacy contributions.
• Phase-specific CC (applied in the SubACM): Computed independently for each defined operational phase (e.g., heating season vs. non-heating season). This approach captures the time-varying reliability value of resources under distinct operating constraints (such as heating-season thermal output limitations) and fluctuating renewable availability, thereby providing a more precise signal for capacity remuneration.

2.4. Capacity Market Models

Building upon the EO benchmark, we construct two capacity market variants. These mechanisms are designed to address the “missing money” problem caused by administrative price caps in the energy market, by restoring resource adequacy through explicit capacity payments.

2.4.1. Annual Capacity Market

The ACM represents the conventional design paradigm, enforcing a single, annualized system adequacy constraint. It requires that the total accredited capacity of the system meets or exceeds the annual peak demand plus a mandated reserve margin. This formulation assumes that reliability risks are concentrated around a single annual peak and that resource availability is relatively constant. The clearing condition is formulated as:

$$\sum _{g\in \mathcal{G}}{P}_g^{max}·x_g·C{C}_g^{annual}+\sum _{s\in \mathcal{S}}{P}_s^{max}·C{C}_s^{annual}\ge L_{max}^{annual}·(1+IR{M}^{annual})$$

(19)

where $L_{max}^{annual}$ denotes the system annual peak load, $IR{M}^{annual}$ is the annual installed reserve margin, and $C{C}_g^{annual}$ and $C{C}_s^{annual}$ denote the technology-specific annual capacity credit for generator g and storage s, respectively, computed as described in Section 2.3.

2.4.2. Sub-Annual Capacity Market

The SubACM extends the conventional annual capacity market by disaggregating the annual adequacy requirement into multiple phase-specific constraints. The rationale for this disaggregation rests on the observation that capacity adequacy is not a time-invariant system property but rather a state jointly shaped by the temporal correlation between generation resource availability and system load. When the supply–demand balance and the resulting pattern of adequacy stress differ materially across periods, there exists a clear physical basis for treating those periods as separate capacity products, each with its own accreditation schedule and reserve margin requirement.

Under this design, each operating year is partitioned into distinct phases, and each phase constitutes an independent capacity product with its own adequacy constraint, capacity credit schedule, and performance obligation. The choice of phase boundaries should be guided by the underlying adequacy risk structure rather than by arbitrary calendar conventions. In our case study, we adopt a two-phase partition aligned with China’s administrative heating season, November through March, and non-heating season, April through October. This particular partition is well suited to the study context for two reasons. First, the heating and non-heating boundary corresponds to regulatory definitions that already govern thermal unit dispatch obligations, heat supply contracts, and grid operation protocols across northern China, so that a phase-based capacity product can be implemented within the existing institutional framework without introducing new administrative categories. Second, the heating season represents the most pronounced source of seasonal adequacy heterogeneity in coal-dominated northern Chinese systems, where winter heat supply obligations substantially reduce the electrical output available from the thermal fleet and fundamentally alter the generation mix contributing to system reliability. The two-phase design therefore captures the dominant seasonal contrast while keeping the mechanism simple enough to isolate the effect of temporal disaggregation from confounding factors that would arise with a more complex partitioning scheme.

In contrast, the ACM treats the entire year as a single capacity product with uniform accreditation, implicitly assuming that each resource’s reliability contribution is time-invariant. This averaging can overcompensate resources that are redundant in non-critical periods while undervaluing those with strong but phase-concentrated adequacy contributions.

The market clears according to phase-specific constraints:

$$\sum _{g\in \mathcal{G}}{P}_g^{max}·x_g·C{C}_{g,k}+\sum _{s\in \mathcal{S}}{P}_s^{max}·C{C}_{s,k}\ge P{D}_k·(1+IR{M}_k),\forall k\in \mathcal{K}$$

(20)

where k indexes operational phases, $\mathcal{K}$ is the set of operational phases, $P{D}_k$ is the peak demand observed during phase k, $IR{M}_k$ is the phase-specific reserve margin derived from the EO model’s seasonal capacity utilization, and $C{C}_{g,k}$ is the capacity credit of resource g during phase k, computed independently for each phase as described in Section 2.3.

A central feature of SubACM is that it reduces the temporal mismatch inherent in annual procurement. Whereas the ACM applies a single annual reliability target that can mask seasonal differences in resource availability, SubACM evaluates phase-specific capacity credits against phase-specific adequacy needs. The phase-specific reserve margin $IR{M}_k$, derived from the EO equilibrium capacity mix, serves two purposes: it avoids over-crediting resources that contribute primarily in one season, and it aligns procurement targets with seasonal stress conditions, thereby reducing the annual-averaging bias in which a system appears adequate in aggregate but remains exposed in specific phases. We note that the formulation in Equation (20) is fully generalizable to any number of phases $\left|\mathcal{K}\right|$, and the computational framework imposes no structural restriction on the number of segments. Both the annual $IR{M}^{annual}$ and the phase-specific $IR{M}_k$ values are calibrated by substituting the EO-optimal capacity mix and the corresponding CCs into Equations (19) and (20) and solving for the IRM that makes each constraint bind. This ensures that, under perfect calibration, both ACM and SubACM exactly reproduce the EO benchmark capacity mix and system cost, providing a consistent basis for the subsequent comparison under stochastic conditions.

3. Case Study

To evaluate the economic performance of the proposed sub-annual capacity market, we conduct a case study based on the Shanxi provincial power system. Shanxi hosts China’s first officially operated electricity spot market and features a generation mix dominated by coal, alongside rapidly growing renewable penetration. In addition to conventional district heating supplied by the coal-fired thermal fleet, the province has substantial electric heating demand, which creates a dual-peak load pattern. This section describes the data, scenario design, and simulation results.

3.1. Case Study Data

The dataset includes five consecutive years of historical load and renewable generation data. Load series are normalized and then scaled to a peak of 5000 MW to form hourly profiles for each year, a scale chosen for computational tractability while preserving proportionality with the original profiles; results can be interpreted as representative of a subregional system or scaled proportionally to full provincial dimensions. The scaling preserves the original diurnal characteristics. For renewable resources, we select two wind farms and two PV stations from different sub-regions. Their output profiles show clear seasonal heterogeneity and complementarity. The two wind farms have average capacity factors of 0.26 and 0.31, with stronger output in winter. The two PV stations have average capacity factors of 0.12 and 0.18, with stronger output in summer. This seasonal contrast provides a suitable basis for comparing the economic performance of the proposed market designs. The case study uses a representative set of technologies and profiles rather than the full Shanxi fleet, primarily to maintain computational tractability given the large-scale optimization involved and to work within the constraints of publicly available data. Although the case study draws on real Shanxi load and renewable data to provide empirical grounding, the Greenfield assumption means the modeled system does not represent the actual Shanxi generation fleet with its sunk assets, contractual obligations, and institutional path dependencies; the case study demonstrates the methodology in a realistic data environment rather than prescribing outcomes for the actual system. Despite these simplifications, the selected profiles preserve the essential seasonal heterogeneity needed to evaluate the temporal disaggregation of capacity procurement.

For thermal generation, we use a representative 660 MW ultra-supercritical unit, reflecting the dominant technology in Shanxi’s thermal fleet. Cost parameters are calculated from [26]. For annualization, we adopt a discount rate of 6.5% and an economic lifetime of 20 years for coal-fired generation. To represent heating-season operational constraints on the thermal fleet, the maximum available output of coal units is reduced to 85% of nameplate capacity during heating periods, based on publicly available regulatory data on thermal unit output compression during winter heating obligations. This is a system-level approximation rather than a unit-specific CHP model. The annual minimum output ratio is set to 40%, and the average forced outage rate is set to 3%. Minimum up-time and down-time are both 24 h, and the reserve requirement is 10%, which is consistent with typical spinning reserve practices in Chinese provincial power systems and is held fixed across all scenarios for comparability. We impose a 10,000 MW installed-capacity cap on both thermal and renewable resources. Energy storage is modeled with a 2-h duration at rated power and a round-trip efficiency of 95%; these specifications are consistent with typical utility-scale lithium-ion battery systems and are adopted as reference values for the analysis. Detailed cost parameters are reported in

Table 1. Cost parameters of generation resources.

Resource Type	Investment Cost (CNY/MW/Year)	Variable Cost (CNY/MWh)	Start-Up Cost (CNY/MW)
Coal-fired	531,530	256	1000
Wind	462,090	/	/
PV	327,000	/	/
ESS	99,216	200	/

Note: “/” indicates that the cost component is not applicable or is set to zero in the model.

.

3.2. Scenario Definitions

To quantify the benefits of the proposed mechanism, we define three core simulation scenarios (

Table 2. Definition of simulation scenarios.

Name	Description
Energy-Only Market (EO)	A generation expansion planning model integrated with system dispatch operations. The absence of price caps facilitates a theoretical optimal planning solution, yielding the minimum system cost. The generation expansion outcome in the EO scenario is used to calculate the CCs of generation resources.
Annual Capacity Market (ACM)	Using CCs calculated from the EO case, the annual IRM is derived from EO planning outcomes and CCs by solving Equation (19). On this basis, ACM sets VOLL to CNY 1500/MWh and clears the market to determine the installed capacity configuration and system cost.
Sub-Annual Capacity Market (SubACM)	Every operational year is partitioned into two distinct phases: Non-heating(April–October) and Heating (November–March), aligned with official heating season definitions. After solving phase-specific CCs and IRM, SubACM sets VOLL to CNY 1500/MWh and clears the market under constraints Equation (20) to determine the installed capacity configuration and system cost.

).

The EO benchmark uses VOLL = CNY 10,000/MWh to approximate unconstrained scarcity pricing, whereas the ACM and SubACM scenarios set VOLL = CNY 1500/MWh to reflect the administrative price cap typically imposed in energy markets. This difference is an intentional design choice: it creates the “missing money” gap that capacity payments are designed to remedy.

To evaluate the proposed framework, we design two case studies aligned with the scenarios above.

(1)

Validation of Market Effectiveness

The first case examines whether SubACM can reproduce the EO benchmark outcome under well-calibrated inputs. We solve the EO model to obtain the reference generation expansion plan and system cost, compute annual and phase-specific CCs from that solution, and derive the corresponding IRMs. With these inputs, both ACM and SubACM are expected to converge to the same capacity mix and total cost as the EO case, confirming internal consistency of the calibration and the theoretical validity of SubACM.

(2)

Economic Efficiency under Uncertainty

Although ACM and SubACM can approximate EO outcomes under perfect calibration, real-world capacity auctions are conducted years ahead of delivery, so deviations between ex ante parameters and realized conditions are unavoidable. To assess comparative robustness, we fix the CCs and IRMs at their ex ante values and evaluate both mechanisms across an ensemble of stochastic scenarios in which realized load and renewable output deviate from the calibration basis. We employ a multivariate VAR(1) model to generate 151 such scenarios through Monte Carlo simulation. For each scenario, we evaluate planning outcomes and system cost while keeping ex ante CCs and IRMs fixed. This setup quantifies the cost efficiency and robustness of SubACM relative to ACM under realistic parameter mismatch.

3.3. Case Study Results

3.3.1. Capacity Credit Accreditation

A critical distinction between the SubACM and ACM lies in their capability to accurately evaluate the effective contribution of generation resources to system capacity adequacy. As discussed in Section 2.3, CCs are computed using both increment and decrement perturbations to address discontinuities in marginal reliability values. The results presented below are based on the increment perturbation set, which reflects the marginal reliability contribution of adding capacity at the EO equilibrium point. Figure 2 illustrates the capacity credits of resources under different mechanisms.

The accredited capacity factors reveal qualitatively distinct characteristics of resource adequacy contributions across temporal structures. Under the ACM, coal-fired generation (G1) demonstrates a capacity credit of approximately 87.3%, while wind resources (W1, W2) exhibit relatively modest credits ranging from 4.2% to 10.2%. PV resources (P1, P2) show capacity credits of 7.8% and 7.2% respectively, and energy storage (S1) achieves approximately 57.5%. These annual-averaged metrics, while providing a unified baseline, inherently obscure the temporal heterogeneity of resource adequacy value.

In contrast, the SubACM unveils substantial seasonal differentiation. During the heating phase, characterized by evening load peaks coinciding with low solar availability, thermal units maintain relatively stable capacity credits around 82.0%, while wind resources demonstrate slightly enhanced contributions (W1: 12.3%, W2: 4.5%) compared to their annual averages. Solar PV resources, however, exhibit notably diminished winter capacity credits (P1: 4.9%, P2: 3.8%), reflecting their limited alignment with the evening net load peak. Energy storage shows a moderate winter credit of approximately 41.8%, constrained by the prolonged duration of heating load plateaus that challenge limited-duration storage assets.

During the non-heating phase, EENS occurs during the noon peak hours. Wind resources show reduced capacity credits (W1: 6.6%, W2: 3.9%) due to seasonal output patterns, while solar PV resources achieve substantially elevated credits (P1: 13.1%, P2: 13.4%), capitalizing on strong coincidence with afternoon cooling demand peaks. Energy storage attains a high capacity credit of 85.7% in non-heating phase, effectively providing firm capacity during the shorter, more pronounced peak demand periods. Coal-fired generation approaches its technical availability limit (approximately 97.0%). The lower coal credit during the heating phase compared to the non-heating phase reflects the reduced electrical flexibility of CHP units operating in heat-led mode, where heat supply obligations constrain their available electrical output and thereby limit their marginal reliability contribution.

These seasonal patterns underscore a fundamental limitation of annual averaging mechanisms: resources with strong temporal complementarity, particularly solar PV and storage, receive capacity credits that neither reflect their peak-season value nor appropriately discount their off-peak contributions. The SubACM framework, by disaggregating reliability requirements across seasonal contexts, provides a more physically grounded representation of each resource’s adequacy contribution. Since the decrement-based CC values are quantitatively similar and exhibit the same seasonal patterns, their effect is reflected in the market-clearing results discussed next.

3.3.2. Market Clearing Results

To examine the effect of dual CC sets on market-clearing outcomes, Figure 3 presents the results under five configurations: the EO benchmark, Annual CM with increment-based CCs (Annual CM Increase) and decrement-based CCs (Annual CM Decrease), and Phased CM with increment-based CCs (Phased CM Increase) and decrement-based CCs (Phased CM Decrease).

All five configurations converge to similar aggregate installed capacity levels of approximately 11,600 MW. Thermal coal generation remains highly stable across scenarios, ranging narrowly from 3704 to 3706 MW. Wind resources collectively contribute approximately 5075–5079 MW, with only marginal redistribution among wind units. Solar Unit 2 capacity varies within a narrow band of 1771–1778 MW, while energy storage investment ranges from 1066 to 1075 MWh. The EO benchmark yields a total system cost of 8771 million CNY, and all four capacity market configurations converge to 8781 million CNY, corresponding to a cost deviation of only 0.11%.

More importantly, the increment- and decrement-based results within each market design are closely aligned. For the ACM, the two CC sets produce nearly identical capacity mixes, with the largest component-level difference being only 9 MW in Solar Unit 2 (1775 vs. 1771 MW) and 6 MWh in energy storage (1066 vs. 1072 MWh). The SubACM exhibits a similar pattern, with component-level variations of no more than 8 MW for Solar Unit 2 (1778 vs. 1774 MW) and 5 MWh for energy storage (1070 vs. 1075 MWh). This consistency between the two perturbation directions confirms that, although the EENS function is piecewise convex and marginal CC values can change at breakpoints, the resulting numerical sensitivity is limited in practice. The dual CC approach effectively bounds the range of marginal capacity values around the EO equilibrium point, and both sets lead to economically comparable outcomes.

Overall, the close agreement among all five configurations demonstrates that both ACM and SubACM can achieve market effectiveness comparable to the EO benchmark when CCs and IRMs are properly calibrated. The cost premium of approximately 0.11% relative to the uncapped EO solution reflects the inherent efficiency loss from price-cap-induced adequacy constraints rather than methodological deficiency, supporting the economic rationality of both capacity market designs.

3.3.3. Economic Efficiency Under Uncertainty

In practice, capacity market parameters are calibrated ex ante, yet realized load and renewable output are inherently stochastic. To evaluate the comparative economic robustness of the SubACM versus the ACM under such uncertainty, we construct a multivariate scenario ensemble using a first-order Vector Autoregressive (VAR(1)) model that jointly represents the uncertainty in load demand, wind generation (two representative stations), and solar generation (two representative stations), yielding a five-dimensional system with variables indexed by $i\in \{L,W_1,W_2,P_1,P_2\}$.

To remove seasonal trends, hourly time series for each variable are first aggregated into daily means ${\overline{x}}_i(y,d)={\displaystyle \frac{1}{24}}{\sum }_{h=1}^{24}{x}_i(y,d,h)$. A climatological mean ${\mu }_i\left(d\right)={\displaystyle \frac{1}{N_Y}}{\sum }_{y=1}^{N_Y}{\overline{x}}_i(y,d)$ is then computed for each calendar day across all $N_Y$ historical years, and the multiplicative residual ${\delta }_i(y,d)={\overline{x}}_i(y,d)/{\mu }_i\left(d\right)-1$ isolates weather-driven percentage deviations from the seasonal norm. For solar variables, days with ${\mu }_i\left(d\right)$ below a minimum threshold are excluded by setting ${\delta }_i=0$ to avoid division by near-zero denominators.

The five-dimensional residual vector $\delta \left(d\right)={[{\delta }_L\left(d\right),{\delta }_{W_1}\left(d\right),{\delta }_{W_2}\left(d\right),{\delta }_{P_1}\left(d\right),{\delta }_{P_2}\left(d\right)]}^{\top }$ is modeled as a regime-switching VAR(1) process. To maintain consistency with the seasonal adequacy framework, the autoregressive coefficient matrix and innovation covariance are estimated separately for the heating season ($s=\mathrm{H}$, November–March) and the non-heating season ($s=\mathrm{NH}$, April–October):

$$\mathit{\delta }\left(d\right)={\mathbf{A}}_{s\left(d\right)}·\mathit{\delta }(d-1)+\mathit{\epsilon }\left(d\right),\mathit{\epsilon }\left(d\right)\sim \mathcal{N}\left(\mathbf{0},{\mathbf{\Sigma }}_{s\left(d\right)}\right)$$

(21)

where $s\left(d\right)\in \{\mathrm{H},\mathrm{NH}\}$ denotes the regime determined by the calendar date of day d. The coefficient matrix ${\mathbf{A}}_s\in {\mathbb{R}}^{5\times 5}$ captures within-regime temporal persistence and lagged cross-variable dependence, while ${\mathbf{\Sigma }}_s$ encodes the contemporaneous correlation structure. This formulation accommodates the distinct meteorological–electrical coupling patterns across seasons, such as the stronger wind–load co-movement during winter cold spells and the solar-dominated supply–demand alignment in summer.

For each regime, parameters are estimated via ordinary least squares. Consecutive- day sample pairs for which both days belong to regime s are used to obtain ${\widehat{\mathbf{A}}}_s=\left({\mathbf{R}}_{\mathrm{curr},s}^{\top }{\mathbf{R}}_{\mathrm{prev},s}\right){\left({\mathbf{R}}_{\mathrm{prev},s}^{\top }{\mathbf{R}}_{\mathrm{prev},s}\right)}^{-1}$, and the innovation covariance ${\mathbf{\Sigma }}_s$ is computed from the corresponding residuals. A Cholesky decomposition ${\mathbf{\Sigma }}_s={\mathbf{L}}_s{\mathbf{L}}_s^{\top }$ yields the regime-specific lower-triangular factor ${\mathbf{L}}_s$ for correlated sampling. Based on the estimated parameters, K independent 365-day perturbation trajectories are generated by the recursion in Equation (22):

$${\delta }^{\left(k\right)}\left(d\right)={\widehat{\mathbf{A}}}_{s\left(d\right)}·{\delta }^{\left(k\right)}(d-1)+{\mathbf{L}}_{s\left(d\right)}·{\mathit{\omega }}^{\left(k\right)}\left(d\right),{\mathit{\omega }}^{\left(k\right)}\left(d\right)\sim \mathcal{N}(\mathbf{0},{\mathbf{I}}_5)$$

(22)

where ${\mathit{\omega }}^{\left(k\right)}\left(d\right)$ is an i.i.d. standard normal vector and the regime index $s\left(d\right)$ switches deterministically at season boundaries. Each scenario is initialized with an independent random seed to ensure reproducibility.

The daily perturbation ${\delta }_i^{\left(k\right)}\left(d\right)$ is then applied multiplicatively to the original hourly profiles: $D_{d,h}^{\left(k\right)}=D_{d,h}^{\mathrm{base}}·(1+{\delta }_L^{\left(k\right)}\left(d\right))$ for load, and ${\xi }_{g,d,h}^{\left(k\right)}=\mathrm{clip}\left[{\xi }_{g,d,h}^{\mathrm{base}}·(1+{\delta }_i^{\left(k\right)}\left(d\right)),0,1\right]$ for wind and solar capacity factors. The load profile is renormalized to a 1.0 p.u. annual peak. Because all 24h within the same day share a single perturbation coefficient, the intra-day profile shape is preserved while the daily level varies stochastically.

As shown in Figure 4, the generated scenario ensemble closely reproduces the pairwise correlation structure of the historical data, with all pairwise Pearson correlation coefficient differences (in absolute value) remaining within 0.05. Using this stochastic framework, we generate an ensemble of 151 scenarios (150 stochastic perturbations plus the original baseline), each containing correlated perturbations of load, wind, and solar profiles. For each scenario, we evaluate the system cost under the Energy-Only benchmark, and the system costs under both the SubACM and ACM frameworks using the fixed capacity credit factors and installed reserve margins derived in the previous subsection on capacity accreditation. The comparative performance is assessed through the relative cost deviation of each capacity market mechanism relative to the Energy-Only benchmark across the entire scenario ensemble. All optimization models are formulated as linear programs using the Gurobi 11.0.3 solver interfaced through Python 3.10.9. Computations are performed on a workstation equipped with an i7-13700KF CPU and 64 GB of RAM, with the entire scenario ensemble completed in approximately 722 min. As demonstrated by the results below, this sample size proves sufficient for the reported statistical inferences where all hypothesis tests achieve significance at conventional levels and the ensemble captures pronounced tail behavior that enables meaningful differentiation of the robustness profiles of the two market designs.

Since each stochastic scenario s simultaneously produces $C_{\mathrm{EO}}\left(s\right)$, $C_{\mathrm{ACM}}\left(s\right)$, and $C_{\mathrm{SubACM}}\left(s\right)$, the three cost observations are inherently paired. All location tests therefore employ paired nonparametric methods—the Friedman test for omnibus comparison and the Wilcoxon signed-rank test for pairwise follow-ups—rather than independent-sample alternatives such as Kruskal–Wallis and Mann–Whitney U, to fully exploit the within-scenario correlation and maximize statistical power.

Figure 5 presents the system cost distributions of the three market mechanisms across 151 stochastic scenarios, and

Table 3. Descriptive statistics of annualized system cost (million CNY) under 151 stochastic scenarios.

Statistic	EO	ACM	SubACM
Mean $\mu$	8079	8196	8194
Std. dev. $\sigma$	568	656	615
Median	8119	8137	8172
95% CI	[7988, 8170]	[8090, 8301]	[8095, 8293]
IQR	878	991	947

summarizes the corresponding descriptive statistics. The mean annualized system cost is 8079 million CNY for EO, 8196 million CNY for ACM, and 8194 million CNY for SubACM. A Friedman test detects a highly significant omnibus difference (${\chi }^2=234.5$, $p<0.001$), and pairwise Wilcoxon signed-rank tests with Bonferroni correction reveal that both capacity market mechanisms incur systematically higher costs than the EO benchmark (EO vs. ACM: $W=0$, $p_{\mathrm{adj}}<0.001$, $r=0.87$; EO vs. SubACM: $W=0$, $p_{\mathrm{adj}}<0.001$, $r=0.87$), while the two CM designs are statistically indistinguishable from each other (ACM vs. SubACM: $p_{\mathrm{adj}}=0.209$, $r=0.15$). The $W=0$ statistic indicates that the CM cost exceeds the EO cost in every one of the 151 scenarios without exception. Nonetheless, the average cost premium is only approximately 1.4%, suggesting that the introduction of capacity remuneration imposes a modest and practically acceptable additional cost in exchange for enhanced resource adequacy assurance.

Although the two capacity market designs yield comparable mean costs, they differ substantially in the cross-scenario stability of their cost deviations from the EO benchmark. Figure 6 plots the distribution of the relative cost deviation $\delta =(C_{\mathrm{CM}}-C_{\mathrm{EO}})/C_{\mathrm{EO}}\times 100\%$, and

Table 4. Dispersion comparison and statistical tests for relative cost deviation between ACM and SubACM.

	ACM	SubACM	Ratio (ACM/SubACM)
Dispersion metrics
Std. dev. $\sigma$ (%)	2.04	1.44	1.41
IQR (%)	1.28	0.89	1.44
Range (%)	11.77	8.61	1.37
Variance	4.15	2.08	2.00
	Statistic	Two-Tailed $\mathit{p}$	One-Tailed $\mathit{p}$
Hypothesis tests for equal dispersion
Levene (median-based)	$F=2.943$	0.087	0.044 *
Ansari–Bradley	$C=\mathrm{10,523}$	0.012 *	0.006 **

*: p < 0.05, **: p < 0.01.

reports the corresponding dispersion metrics and statistical tests.

The mean deviations are nearly identical (ACM: 1.40%; SubACM: 1.40%), confirming that both mechanisms introduce a comparable average cost premium. However, SubACM exhibits consistently lower variability across all dispersion measures: its standard deviation is 29% lower than that of ACM (1.44% vs. 2.04%), its IQR is 31% narrower (0.89% vs. 1.28%), and the sample variance of ACM is approximately twice that of SubACM (variance ratio $=2.00$). This dispersion difference is also clearly visible in Figure 6, where ACM displays a longer upper tail with more extreme outliers.

To quantify the practical implications,

Table 5. Tail-risk comparison of relative cost deviation between ACM and SubACM.

Metric	ACM	SubACM	Ratio
Scenarios with $\delta >3\%$	13.2% (20/151)	7.9% (12/151)	1.67
Scenarios with $\delta >5\%$	6.6% (10/151)	4.0% (6/151)	1.67
Maximum deviation	11.84%	8.72%	1.36

compares the tail-risk profiles of the two designs. ACM produces cost deviations exceeding 3% in 13.2% of scenarios (20/151), compared to 7.9% (12/151) for SubACM—a 67% higher incidence. At the 5% threshold, the ratio is similar (6.6% vs. 4.0%). The worst-case deviation is 11.8% for ACM versus 8.7% for SubACM, a gap of more than three percentage points.

The dispersion difference is confirmed by formal hypothesis tests. The Ansari–Bradley test, which is particularly sensitive to differences in distributional tails, rejects the null hypothesis of equal scale at the two-tailed 5% level ($C=\mathrm{10,523}$, $p=0.012$). The median-based Levene test is marginally non-significant at the two-tailed level ($F=2.943$, $p=0.087$), in line with its lower sensitivity to tail-driven dispersion differences. SubACM calibrates capacity credits on a seasonal basis, which provides a structural, a priori rationale for expecting lower cost variability than under ACM; the directional alternative $H_1:\mathrm{Var}\left(\mathrm{ACM}\right)>\mathrm{Var}\left(\mathrm{SubACM}\right)$ is therefore well motivated, and both tests reach significance under this one-sided formulation (Levene $p=0.044$; Ansari–Bradley $p=0.006$). Nevertheless, Table 4 reports two-tailed and one-tailed results for transparency.

As shown in Figure 7, the notably higher unit-level variability under SubACM, particularly for energy storage, whose non-heating CC ($86\%$) is more than double its heating-season value ($42\%$), is a structural consequence of the phase-specific capacity credit design. Unlike ACM, which assigns a single annual CC and thus produces a nearly uniform capacity portfolio across scenarios, SubACM allows the optimizer to tailor the capacity mix to the specific seasonal constraint that is binding in each scenario. When the non-heating season drives the adequacy requirement, storage and solar receive larger allocations; when the heating season dominates, investment shifts toward wind and thermal resources. This scenario-adaptive reallocation explains the wide unit-level capacity distributions observed in Figure 7. Although individual asset capacities vary more across scenarios, the resulting portfolio in each case is better aligned with the prevailing seasonal adequacy needs, which is why SubACM achieves lower system-level cost variability despite exhibiting higher unit-level investment dispersion. This trade-off implies that a phased capacity market enhances cost predictability for the system as a whole, but may increase investment uncertainty for individual asset owners.

In summary, converging evidence from descriptive dispersion metrics, tail-risk statistics, and formal hypothesis tests consistently indicates that SubACM produces more concentrated cost outcomes across stochastic scenarios. This robustness advantage stems from the phase-specific alignment between capacity credits and seasonal reserve requirements under the SubACM design. By disaggregating reliability contributions across distinct seasonal periods, the mechanism values each resource according to its actual availability during the periods when capacity shortfalls are most likely to occur. This finer-grained pricing structure reduces the sensitivity of investment signals to any single scenario’s renewable output realization, thereby attenuating cost variability under load uncertainty.

4. Discussion

4.1. Institutional Implementation of Sub-Annual Capacity Products

Translating the SubACM framework from an analytical model into a functioning market mechanism requires addressing several institutional design dimensions. Phase-based adequacy constraints can be incorporated into existing centralized auction clearing mechanisms as additional linear constraints, analogous to the treatment of locational deliverability constraints in operational capacity markets such as PJM and ISO-NE. A single forward auction can simultaneously clear products for all phases within the existing contracting timeline, or separate sequential auctions can be held for individual phases. The contracting horizon can follow existing annual capacity market timelines, with phase-specific obligations embedded within the same forward commitment period. Resources cleared for a given phase are required to demonstrate availability during designated assessment hours within that phase, with non-performance penalties triggered by shortfalls during phase-specific scarcity events. This compliance structure follows existing annual mechanisms but is applied independently to each phase, ensuring that performance obligations are temporally aligned with the periods in which they are needed.

A notable advantage of the phase-based structure is the natural treatment of seasonal performance asymmetry. Resources with strong seasonal variation in output or availability, such as solar PV and thermal units subject to heating-season constraints, receive differentiated capacity credits across phases, ensuring that capacity remuneration reflects actual reliability contributions rather than annual averages. This differentiated accreditation provides more accurate investment signals for resources with phase-concentrated adequacy value. Phase-based products can also coexist with bilateral contracts by crediting bilateral positions against phase-specific obligations, similar to self-supply provisions in existing capacity markets.

Product segmentation may increase market concentration in individual phase markets and reduce per-product liquidity, as the pool of eligible resources for each phase may be smaller than for the annual product. These risks can be mitigated through market monitoring, offer caps, and minimum participation requirements, which are standard safeguards in existing capacity market designs. Additionally, allowing cross-phase resource substitution under specified conditions could help maintain adequate participation in each phase market.

4.2. Limitations

This study adopts several simplifications that constrain the generalizability of its findings. The Greenfield, single-node formulation abstracts away legacy fleet composition, transmission constraints, and locational adequacy conditions. The case study uses a single representative thermal technology with an aggregate heating-season output derating rather than unit-level CHP feasibility regions. The storage formulation relaxes charge/discharge mutual exclusivity, which could in principle overstate ESS capacity value under extreme stress events, although cycling costs prevent simultaneous operation in all observed solutions. The stochastic scenario ensemble perturbs load, wind, and solar profiles jointly but at daily granularity, so sub-daily forecast errors, prolonged low-generation events, and spatially correlated renewable droughts are not captured; the five-year data window may also miss rare extreme weather events or long-term climate trends. On the methodological side, the piecewise convexity of the EENS function can cause discontinuities in marginal capacity credits, and the ex ante CCs are held fixed throughout the stochastic analysis without separate sensitivity quantification. The analytical framework imposes no structural restrictions on technology types, network nodes, or temporal phases; extending it to richer system representations remains an important direction for future work.

5. Conclusions

This paper develops a Sub-Annual Capacity Market framework that disaggregates annual adequacy requirements into phase-specific capacity products and compares it with an Annual Capacity Market and an uncapped Energy-Only benchmark through a Shanxi-based case study. The phase-specific capacity credit analysis reveals pronounced seasonal heterogeneity in resource adequacy value that annual averaging inherently masks: solar PV and energy storage credits vary by a factor of two to three between the heating and non-heating seasons, reflecting the time-varying alignment between resource availability and system stress. Under deterministic conditions with well-calibrated parameters, both ACM and SubACM reproduce the EO benchmark to within 0.11%, confirming the internal consistency and economic rationality of the proposed calibration workflow.

The distinguishing advantage of SubACM emerges under uncertainty. Across 151 stochastic scenarios with jointly perturbed load and renewable profiles, the two mechanisms yield comparable mean cost premiums of approximately 1.4% over the EO benchmark, yet SubACM produces significantly more concentrated cost outcomes, with 29% lower standard deviation in relative cost deviation and 67% fewer tail-risk scenarios. This robustness gain stems from the phase-specific alignment between capacity credits and seasonal reserve requirements, which allows the capacity portfolio to adapt to whichever seasonal constraint is binding rather than being locked into a single annual accreditation schedule. At the same time, the improved system-level cost stability comes with greater unit-level investment variability, a trade-off that warrants attention in practical market design.

Overall, the primary contribution of sub-annual capacity market design lies not in large average cost reductions but in improved temporal matching between capacity payments and physical reliability contributions, and in enhanced economic robustness under parameter mismatch. The framework is particularly relevant for systems with pronounced seasonal adequacy heterogeneity, such as coal-dominated northern Chinese grids where winter heating obligations materially constrain thermal fleet availability. From a policy perspective, as China advances its nationwide electricity market reform and establishes capacity remuneration mechanisms, regulators in regions with strong seasonal load, resource availability, or operational constraints may consider adopting phase-based capacity products as a complement to, or eventual replacement of, annual procurement, thereby better aligning market incentives with the physical reliability landscape shaped by heating-season constraints and rising renewable penetration.

Future work should extend the analysis to incorporate existing fleet constraints, retirement dynamics, and richer technology diversity with unit-level CHP modeling, as well as transmission-constrained deliverability assessment. More granular renewable uncertainty representations, such as sub-daily forecast errors and prolonged low-generation events, would further strengthen the robustness evaluation. In addition, data-driven phase partitioning methods and sensitivity analysis across different temporal granularities could help quantify the marginal benefit of finer phase resolution and guide practical market design choices.