A scenario-based framework for measuring time-varying generation costs in medium- and long-term electricity markets

Summary

Accurately measuring time-varying generation costs is essential for effective pricing and resource allocation in electricity markets. Conventional cost measures capture either long-term averages or short-run operating conditions, but fail to jointly reflect temporal variability and capacity cost recovery. This study proposes a scenario-based framework for measuring time-varying generation costs under China’s time-division trading mechanism, integrating accounting-based capacity costs with market-driven operational and opportunity costs across different supply-demand conditions. Using 8,760 h of data from Province H, the model reveals substantial hourly and seasonal cost variation, with cost spikes concentrated in short-lived peak and shortage periods and persistently low costs during renewable-rich hours. A comparative case study in Province J confirms that the framework robustly captures cost dynamics under different load profiles, providing practical cost signals for time-differentiated pricing and contract design in medium- and long-term electricity markets.

Graphical abstract

Highlights

• •Proposed a scenario-based framework for time-varying generation cost measurement
• •Estimated hourly generation costs across 8760 h in a provincial market
• •Revealed pronounced intra-day and seasonal heterogeneity in generation costs
• •Improved cost signal accuracy beyond static or average cost approaches

Applied sciences; Electrical engineering; Energy Resources

Introduction

China’s electricity market has undergone substantial reforms aimed at transitioning from administratively determined dispatch toward market-based pricing mechanisms.¹ Within this evolving framework, the medium- and long-term electricity market plays a central role by providing the primary platform for electricity transactions, price formation, and risk hedging for market participants.² In practice, medium- and long-term contracts account for the vast majority of electricity traded in China, forming the foundation upon which short-term spot markets and ancillary services operate.³ Unlike real-time markets, the medium- and long-term markets rely on contract-based transactions spanning annual, monthly, weekly, and increasingly shorter time horizons, making accurate cost signals particularly critical for effective pricing and resource allocation.

In recent years, China’s medium- and long-term electricity market has experienced a clear trend toward finer temporal segmentation.⁴ Trading products have gradually evolved from uniform flat-period contracts to differentiated peak, flat, and valley periods, and in some regions to hourly or near-hourly intervals.⁵ Time-division trading is intended to better reflect temporal variations in electricity value and system conditions, thereby enhancing allocative efficiency and encouraging demand-side flexibility.^6,7 However, in most provinces, the segmentation of trading periods remains largely experience-based and administratively defined, often inherited from traditional retail tariff structures.⁸ These fixed classifications are typically unchanged across seasons and years, despite substantial variation in system load, renewable generation, and operating conditions.⁹ As a result, existing time-division schemes may fail to reflect the intrinsic temporal fluctuations in generation costs associated with changing supply-demand conditions.¹⁰

The challenge of accurately measuring time-varying generation costs has become more pronounced with the rapid growth of renewable energy. High penetration of wind, solar, and other variable renewable resources has increased supply uncertainty and amplified temporal heterogeneity in system operation.^11,12 Generation costs are no longer uniform across time but vary significantly with residual load levels, dispatch constraints, and system stress conditions.¹³ Moreover, recent research has shown that medium- and long-term electricity markets are increasingly influenced by carbon constraints, transmission limitations, and interactions with related markets, further complicating cost formation mechanisms.¹⁴ Under these conditions, generation costs cannot be adequately characterized by static or uniform measures, underscoring the need for cost models that explicitly account for temporal and scenario-dependent system dynamics.^15,16,17 The absence of such high-resolution, economically consistent cost signals not only hinders efficient market design but also constrains downstream operational optimizations, such as distribution network reconfiguration with distributed resources or strategic bidding for generation companies, which rely on accurate temporal cost or price inputs to guide decision-making under uncertainty.^18,19

Existing approaches to electricity generation cost measurement can be broadly classified into static average-cost models and dynamic marginal-cost pricing mechanisms.^20,21,22 Static methods, such as the levelized cost of energy (LCOE), average costs over the project life cycle and neglect short-term operational variability and binding system constraints.^23,24 In contrast, dynamic pricing models, including locational marginal pricing and real-time pricing, capture short-run marginal costs and enhance dispatch efficiency in spot markets.²⁵ However, these models typically exclude the allocation and recovery of fixed capacity costs, which are essential for ensuring long-term system adequacy and investment incentives.^26,27 This limitation is particularly problematic in medium- and long-term electricity markets, where pricing mechanisms must reconcile short-term operating conditions with long-term capacity cost recovery.^28,29

Recent extreme events further highlight the inadequacy of existing cost measurement frameworks.³⁰ Scarcity-driven price spikes during supply shortages and prolonged periods of negative pricing under renewable oversupply reveal that generation costs are highly scenario-dependent.³¹ These contrasting yet increasingly frequent conditions indicate that generation cost formation differs fundamentally across system states, shaped by both scarcity and surplus dynamics.³² Capturing such asymmetric cost behavior requires a framework that integrates short-term operational signals with long-term capacity valuation in a consistent and economically meaningful manner.³³ Developing such a framework is a prerequisite for providing transparent cost benchmarks that can inform not only time-differentiated pricing but also a wide range of system operations and market strategies, from evaluating reliability costs in integrated energy systems20 to formulating profit-maximizing bids in competitive markets.³⁴

Against this backdrop, this study proposes a scenario-based framework for measuring time-varying generation costs in China’s medium- and long-term electricity market under the time-division trading mechanism. The framework identifies typical and recurring system states by classifying supply-demand conditions into four scenarios: balanced, shortage, oversupply, and extreme oversupply, which represent typical and recurring operating conditions observed in provincial power systems. For each scenario, a corresponding cost measurement approach is applied, allowing generation costs to reflect both operational conditions and economic scarcity or surplus. A key feature of the proposed method is the allocation of capacity costs according to utilization rates derived from the load duration curve, establishing a direct linkage between system operating conditions and capacity cost recovery. By integrating regulatory accounting-based costs with market-oriented economic cost metrics, the framework bridges short-term operational variability and long-term investment recovery within a unified modeling structure.

This study makes three main contributions. First, it develops a scenario-based model that captures the temporal and operational heterogeneity of generation costs in medium- and long-term electricity markets under time-division trading. Second, it integrates capacity cost allocation and scenario-specific cost metrics into a unified framework, addressing the disconnect between marginal pricing and long-term cost recovery. Third, it validates the proposed approach using hourly data from representative Chinese provinces with diverse load profiles, demonstrating its ability to reflect realistic temporal cost fluctuations. Overall, this research provides a systematic and policy-relevant approach to measuring time-varying generation costs, offering practical insights for contract pricing, bidding strategies, and market design in electricity systems undergoing rapid transformation.

Background and theoretical mechanism

Institutional features of China’s medium- and long-term electricity market

China’s power system dispatch historically relied heavily on centralized, manual operations conducted by grid companies, which limited market efficiency and flexibility.³⁵ To address these limitations, China introduced the medium- and long-term electricity market in the early 2000s as a core component of its power sector reform, gradually replacing mandatory dispatch mechanisms with contract-based market transactions.³⁶ This market segment has since evolved into the primary platform for price formation and risk allocation in China’s electricity system, playing a key role in stabilizing supply and ensuring reliable base-load delivery.³⁷ A defining feature of China’s medium- and long-term electricity market is its operation as a physical contract market, in which contracts are directly linked to actual electricity dispatch and require mandatory physical delivery.^38,39,40,41 As a result, contracted volumes directly influence generation schedules and system operations, causing generation costs and dispatch constraints to be tightly coupled.^42,43

China’s medium- and long-term market encompasses multiple trading horizons, including annual, monthly, and shorter-term contracts such as weekly and multi-day trades. These layered products allow generators, retailers, large consumers, and grid agents to hedge risks, secure supply, and coordinate operations across time scales.⁴⁴ Currently, transactions concluded through this market account for more than 90% of total electricity traded nationwide, underscoring its central role in China’s power market architecture.

Recent reforms emphasize the imperative of enhancing coordination between medium- and long-term contracts and spot market settlements.⁴⁵ In 2021, the National Development and Reform Commission (NDRC) and the National Energy Administration (NEA) introduced a time-division trading mechanism through the notice on signing medium- and long-term electric power contracts (document no. 1784).⁴⁶ This policy marked a shift from the traditional “flat price, full volume” model to a “time-differentiated, value-based” trading structure,⁴ aligning medium- and long-term contracts more closely with spot market operations.

Under time-division trading, the 24-h day is segmented into multiple trading intervals, each associated with independent pricing and transaction processes. By differentiating prices across time, this mechanism aims to align contract prices more closely with the temporal variation of system value and operating constraints, thereby enhancing demand-side responsiveness and generation scheduling efficiency. Provinces differ in their implementation strategies, ranging from simplified peak-flat-valley segmentation to finer, near-hourly divisions.

Despite these advances, challenges remain in the effective implementation of time-division trading. In particular, the definition of trading intervals and associated price differentials is still largely determined by administrative rules or historical experience, rather than by explicit representations of time-varying generation costs. Given that cost is the fundamental driver of price differentiation, the absence of a dynamic and economically grounded cost measurement framework constrains the ability of time-division trading to convey accurate market signals. This limitation motivates the development of a high-resolution cost framework capable of capturing hourly system conditions in medium- and long-term electricity markets.

Theoretical framework for time-varying generation cost formation

Electricity generation costs are inherently time-dependent, shaped by fluctuating supply-demand balances, operational constraints, and market conditions.⁴⁷ In power systems with the increasing penetration of variable renewable energy, residual load dynamics and dispatch constraints vary substantially across hours, amplifying temporal heterogeneity in generation costs. Accurately capturing this heterogeneity requires a cost framework that explicitly links physical system states to economic cost formation mechanisms. To address this challenge, this study conceptualizes generation cost formation through a scenario-based framework that maps system operating conditions to differentiated cost metrics. Rather than relying on ex ante time classifications (e.g., peak or valley hours), the framework identifies system states endogenously based on supply-demand relationships and dispatchable capacity constraints. Four representative supply-demand scenarios are defined: equilibrium, shortage, oversupply, and extreme oversupply, each corresponding to a distinct cost formation logic.

Supply-demand equilibrium scenario

When system demand can be met within the feasible operating range of dispatchable generation, generation costs are measured using an average cost framework that combines time-varying capacity costs and variable operating costs. Capacity costs are allocated based on utilization rates using the peak-load responsibility method, following the principle of “high utilization, low cost; low utilization, high cost.” This approach allocates long-term fixed costs according to both the magnitude and duration of system load, ensuring economically consistent cost recovery across hours. Variable costs reflect fuel consumption and normal operating conditions. By avoiding predefined peak-flat-valley classifications, this method preserves continuous cost variation across time.

Supply shortage scenario

When demand exceeds available dispatchable capacity, the system enters a shortage state characterized by binding capacity constraints. In this scenario, generation cost is represented by the customer-side value of lost load (VOLL), which measures the economic cost of unmet demand from the consumer perspective.⁴⁸ VOLL captures the opportunity cost associated with service interruptions, particularly during peak demand periods.⁴⁹ Incorporating customer-side VOLL enables the framework to internalize scarcity costs and reflect the rising marginal value of electricity as capacity constraints tighten.^50,51

Oversupply scenario

When generation exceeds demand but remains above the minimum stable output of dispatchable units, the system experiences oversupply without forcing unit shutdowns. In this case, generation cost is measured using marginal cost, reflecting short-run variable operating expenses of the marginal generating unit. Marginal cost pricing captures the declining system value of electricity during low-demand periods while excluding long-term capacity recovery costs, thereby discouraging inefficient overproduction.⁵²

Extreme oversupply scenario

In extreme oversupply conditions, residual load becomes negative and minimum stable output constraints bind, potentially forcing generators to curtail output or shut down. Under such circumstances, some generators may accept prices below marginal cost, or even negative prices, to avoid shutdown and associated restart costs.⁵³ Generation-side VOLL, also referred to as avoidable cost, is used to represent the minimum acceptable revenue required for generators to remain online, capturing the economic trade-off between continued operation and shutdown.

By mapping hourly system states to these four scenarios, the proposed framework establishes a transparent linkage between physical operating conditions and economically meaningful cost formation mechanisms. The joint consideration of utilization-based capacity cost allocation and scenario-specific opportunity costs allows the framework to capture both long-term capacity commitments and short-term scarcity or surplus effects. This integrated treatment provides consistent time-varying cost signals and forms the theoretical foundation for the model.

Results

Data sources

To verify the applicability and accuracy of the proposed time-varying generation cost model, we conducted an empirical analysis using full-year hourly operational data from Province H in central China for 2021. The dataset covers the complete 8,760 h of system operation and is derived from authorized provincial electricity market operation and settlement records, ensuring consistency with actual market and system conditions. Due to strict regulatory and confidentiality requirements in China’s electricity sector, detailed identifiers of the specific provincial authorities and original datasets are anonymized. All data used in this study have been processed through desensitization procedures prior to analysis, without affecting their internal consistency, temporal structure, or suitability for empirical validation.

Province H features a representative power supply structure that combines hydropower, thermal power, wind power, and photovoltaic power. Among them, hydropower accounts for 42.78%, thermal power for 38.25%, wind power for 8.16%, and photovoltaic power for 10.81%. The annual capacity cost of the approved system generating units is 2.08 × 10¹⁰ CNY, the standard coal consumption coefficient of the thermal power units is 0.311 kg/(kWh), and the average price of the power purchased from outside the province is 0.296 CNY/(kWh). The customer-side VOLL is set at 3 CNY/kWh, consistent with the maximum price cap currently permitted in the provincial electricity market, which is generally interpreted as the upper bound of users’ willingness to pay. The marginal variable cost coefficient of hydro, wind power, and photovoltaic power is 0 CNY/(kWh), and the wind speed of the wind turbine’s cut-in, rated, and cut-out wind speeds are 3, 12, and 22 m/s, respectively. Figure 1 presents the 2021 hourly load curve of Province H, covering the full 8,760 h of the year; Figure 2 depicts the relationship between monthly power generation and electricity demand, explicitly revealing the seasonal imbalance between power supply and load; Figure 3 illustrates the structure of participating entities in Province H. The proposed framework is applied to compute hourly generation costs for Province H across the entire year of 2021. The results are presented at three levels of temporal resolution: annual distribution, monthly seasonal variation, and representative daily profiles. These multi-resolution results provide a systematic link between high-frequency cost measurement and the practical design of time-varying pricing schemes in medium- and long-term electricity markets, and allow for the independent evaluation of the model’s internal logic and empirical performance despite data anonymization.

Figure 1.

Load curve and peak-valley of Province H in 2021

Figure 2.

The relationship between monthly generation capacity and electricity demand

Figure 3.

Structure of the provincial participating entities

Annual time-varying generation costing results

As shown in Figure 4, the results for the annual hourly time-varying generation costs show significant fluctuations throughout the year. The yearly average time-varying generation cost is 0.2313 CNY/kWh, while the maximum cost reaches 3.0 CNY/kWh, and the minimum falls to 0.0794 CNY/kWh. Table 1 illustrates the distribution of annual time-varying generation cost results. These fluctuations stem from temporal mismatches between net load and generation flexibility. During periods of high renewable output, particularly in spring and autumn, the cost remains low due to the predominance of zero-marginal-cost resources. However, in the summer months, system load peaks result in a sharp increase in costs, as thermal units raise their generation levels to meet demand. At the aggregate level, the estimated annual average generation cost is lower than the average prices observed in the medium- and long-term electricity markets of Province H (0.3844 CNY/kWh). This difference indicates that prevailing medium- and long-term transaction prices reflect not only generation cost recovery, but also additional components such as risk premiums, regulatory constraints, and limited temporal differentiation embedded in current pricing rules. More importantly, while the modeled peak-to-valley cost spread reaches nearly 2.92 CNY/kWh, the allowable price fluctuation range in the provincial medium- and long-term market is restricted to ±20%, which is substantially narrower than the intrinsic cost variation revealed by the model. This contrast highlights a structural mismatch between actual cost dynamics and the degree of price flexibility permitted under existing market arrangements.

Figure 4.

The result of the hourly time-varying generation costs in Province H

Table 1.

Distribution of annual time-varying generation cost results

Cost Range (CNY/kWh)	Count	Percentage
0.0–0.1	73	0.83%
0.1–0.2	3458	39.47%
0.2–0.3	3806	43.45%
0.3–0.4	993	11.34%
0.4–0.5	290	3.31%
0.5–1.0	130	1.49%
1.0–2.0	4	0.05%
2.0–3.0	5	0.06%

Statistical analysis indicates that the standard deviation reaches 0.1133 CNY/kWh, and the cost range spans nearly 2.92 CNY/kWh, revealing a high degree of volatility in marginal cost signals. These dynamics underscore the limitations of uniform pricing mechanisms and highlight the critical need to incorporate time-based cost signals into long-term contract pricing. A breakdown of cost spikes reveals that the majority of high-cost hours are concentrated in summer, especially during the province’s peak load day, accounting for 3 out of 5 h of power shortages observed throughout the year. These findings confirm that shortages in Province H are short-lived and peak-driven, rather than systemic. As such, the implementation of scarcity pricing based on customer-side VOLL should be limited to critical peak hours, rather than applied uniformly across broader time windows.

These simulation results effectively reflect the time-varying nature of system cost formation and are consistent with broadly observed patterns in real-world electricity markets. With respect to the spot market, it should be noted that Province H has not yet entered full-scale spot market operation, and the currently available pilot-stage data do not provide a sufficiently stable basis for a systematic quantitative comparison. Nevertheless, the proposed scenario-based framework is model-agnostic with respect to market type and can be directly extended to spot market environments once stable hourly price data becomes available.

Moreover, it should be noted that the generation costs in this paper do not take into account the plant’s self-consumption rate and rate of return. Under the assumptions of a 6% self-consumption rate and an 8% rate of return, the corresponding average cost of supplying electricity on a time-differentiated basis increases to 0.2657 CNY/kWh, which is closer to prevailing medium- and long-term transaction prices. These findings provide valuable references for developing dynamic pricing mechanisms and time-differentiated contracting strategies in China’s medium- and long-term electricity market.

Based on the fuzzy C-means (FCM) clustering analysis, the annual load profile was segmented into distinct time-of-use periods, yielding three representative load regimes: valley, flat, and peak. These regimes correspond to fundamentally different system operating conditions and generation cost formation environments.⁵⁴ The classification is applied to the hourly generation cost outcomes to compute average costs for each time-of-use period. The resulting time intervals and their associated average generation costs are reported in Table 2, providing a basis for examining the consistency between load conditions and cost differentiation.

Table 2.

Typical period of the year and the corresponding time-varying generation costs

Period	Time range	Time-varying generation cost (CNY/kWh)
Peak	11:00-12:00, 18:00-22:00	0.2934
Flat	8:00-10:00, 13:00-17:00, 23:00-0:00	0.2370
Valley	0:00-7:00	0.1646

The peak period, defined as 11:00-12:00 and 18:00-22:00, exhibited the highest average generation cost of 0.2934 CNY/kWh. The flat period, covering 08:00-10:00, 13:00-17:00, and 23:00-24:00, shows a moderately lower average cost of 0.2370 CNY/kWh. The valley period, from 01:00 to 07:00, corresponds to the lowest average generation cost of 0.1646 CNY/kWh. These results highlight a clear cost gradient aligned with daily load variations and resource availability. The wide variation in costs across time makes a strong case for implementing a time-differentiated pricing mechanism to encourage demand-side flexibility and optimize generation scheduling. More importantly, the delineation of distinct cost-based periods provides a robust basis for designing time-differentiated electricity pricing schemes and optimizing medium- and long-term electricity contracts. By mapping hourly generation costs into a small number of representative time-of-use periods, the clustering results transform detailed cost information into pricing intervals that can be directly adopted in time-differentiated tariffs and medium- and long-term contract design.

Monthly time-varying generation costing results

The monthly variation in time-varying generation costs throughout the year, from January to December, is shown in Figure 5. The solid line represents the average generation cost for each month, while the shaded area illustrates the range between the minimum and maximum hourly costs. Monthly points are color-coded by season: winter (blue), spring (green), summer (orange), and autumn (brown), highlighting seasonal variation patterns. It is clear from the figure that there is a significant spike in generation costs between July and August, with the maximum hourly cost reaching 3.00 CNY/kWh, which is much higher than in the other months. These sharp summer cost spikes are primarily driven by short-duration peak load conditions associated with cooling demand, rather than by persistent system-wide capacity shortages. During extreme temperature periods, net system load temporarily approaches or exceeds the flexible operating range of thermal generation, activating high-cost units and scarcity-related opportunity costs. As a result, generation costs rise sharply but remain highly concentrated in a limited number of peak hours. Furthermore, generation cost levels are lower and less volatile in the spring and fall, while average monthly costs are generally high and fluctuate significantly more in the summer. This pattern reflects heightened demand for system flexibility during high-load periods, rather than a uniform increase in generation costs across the entire season. At the same time, renewable energy generation plays an important mitigating role by suppressing cost levels during off-peak and shoulder hours, particularly in periods with strong photovoltaic output. However, renewable output is less effective in alleviating evening peak demand in summer, when cooling loads remain high, but solar generation rapidly declines, which explains the persistence of scarcity-driven cost spikes.

Figure 5.

The monthly statistics of time-varying generation costs in Province H

To illustrate the seasonal characteristics of generation costs, four representative months were selected for detailed analysis. These months capture the typical characteristics of spring, summer, autumn, and winter in Province H. Figures 6 and 7 collectively illustrate the seasonal variation of time-varying generation costs and the corresponding distribution of typical daily time-of-use periods in Province H. Figure 7 shows that summer consistently incurs the highest average generation costs across peak, flat, and valley periods, with peak costs reaching 0.4465 CNY/kWh. This reflects the combined effect of elevated cooling demand and tighter operational constraints on thermal generation during peak hours, rather than uniformly higher costs throughout the day.

Figure 6.

Seasonal average time-varying generation costs by period in Province H

Figure 7.

Typical periods across seasons in Province H

Conversely, valley periods maintain the lowest costs throughout the year, indicating substantial renewable energy contributions during off-peak hours. Complementing this, Figure 8 illustrates the temporal distribution of typical daily time-of-use periods across seasons. The duration and timing of peak, flat, and valley periods shift notably with seasonal changes. In particular, the extension of peak periods into summer evenings is consistent with sustained cooling demand after sunset, when renewable output declines and reliance on thermal generation increases. Such variations underscore the necessity of adopting seasonally adjusted time-of-use pricing schemes to accurately reflect underlying cost dynamics and incentivize demand-side flexibility. These findings provide valuable insights for designing more efficient and equitable medium- and long-term electricity contracts that reflect the true temporal variability of generation costs in Province H.

Figure 8.

Load and time-varying generation cost on the typical maximum load day

Typical day’s time-varying generation costing results

The maximum load day and minimum load day are selected as typical representatives to capture the operational extremes of the power system within Province H.

On the maximum load day, the load steadily increases from early morning, peaking around late evening with values exceeding 29,000 MW. During peak hours, the net system load approaches or exceeds the upper dispatchable limits of thermal units, indicating a tight supply-demand condition. Correspondingly, the generation cost exhibits significant hourly variation, rising sharply during peak hours between 20:00 and 22:00. Notably, generation costs reach extreme values of 3.00 CNY/kWh during the late evening peak. This sharp increase reflects the dispatch of higher-cost thermal units and the scarcity of available capacity under peak-load conditions, which is consistent with the cost formation mechanism embedded in the scenario-based framework. The elevated costs during these peak hours confirm that the model appropriately captures system stress and cost escalation when supply flexibility becomes constrained.

Conversely, the minimum load day displays a much flatter load profile, with maximum loads of approximately 17,000 MW, roughly half of those observed on the peak day. Throughout most hours of the day, the net load remains within the feasible operating range of thermal units or even approaches surplus conditions, indicating ample system capacity. The generation costs are correspondingly lower and more stable throughout the day, fluctuating between 0.1167 CNY/kWh and 0.2189 CNY/kWh. The absence of pronounced cost spikes reflects sufficient availability of low-cost renewable generation and reduced reliance on high-cost thermal capacity, which is consistent with the expected behavior of the system under low-load conditions.

The contrasting patterns observed in Figures 8 and 9, therefore, provide mutual verification of the proposed modeling approach. Higher load levels and tighter supply-demand conditions are associated with greater cost volatility and higher marginal generation costs, while lower load levels correspond to more stable and lower-cost outcomes. These results confirm that the time-varying generation costs produced by the model are economically intuitive and consistent with system operation principles.

Figure 9.

Load and time-varying generation cost on the typical minimum load day

Discussion

Sensitivity analysis

To assess the robustness of the proposed scenario-based framework with respect to key modeling assumptions, we conducted a targeted sensitivity analysis focusing on parameters that are most influential for generation cost formation. Rather than re-estimating the full 8,760-h cost series under all alternative settings, this analysis evaluates the stability of aggregate cost characteristics and scenario structures under plausible parameter variations, consistent with the intended application of the model.

We first examine sensitivity to fuel price assumptions, which primarily affect the variable cost component of thermal generation. The baseline fuel price level is adjusted upward and downward within a range consistent with recent fluctuations observed in China’s coal market, while the temporal profile of fuel prices is held constant. As illustrated in Figure 10, changes in fuel prices lead to a gradual shift in the annual mean generation cost, whereas the peak-valley cost spread remains virtually unchanged across all tested scenarios. This result indicates that fuel price variations mainly influence the overall cost level, but do not alter the temporal structure of generation costs. In particular, the relative differentiation among peak, flat, and valley periods remains stable, suggesting that the observed temporal cost volatility is driven primarily by underlying supply-demand conditions rather than by specific fuel price assumptions.

Figure 10.

Sensitivity of generation costs to fuel price assumptions

We next assess sensitivity to the specification of opportunity costs, which predominantly affect cost formation under tight supply-demand conditions. Alternative scaling factors are applied to the baseline opportunity cost proxy to reflect plausible variations in scarcity valuation under different system tightness conditions. As shown in Figure 11, higher opportunity cost levels increase peak-hour generation costs under tight supply-demand conditions, while having a limited impact on average cost levels. The timing and frequency of scarcity-driven cost spikes remain unchanged.

Figure 11.

Sensitivity of generation costs to opportunity cost assumptions

Overall, the sensitivity analysis indicates that while alternative parameter assumptions lead to changes in the absolute level of generation costs, the main empirical findings of this study remain stable. The results consistently exhibit pronounced temporal cost volatility, scarcity-driven peak behavior, and clear differentiation among peak, flat, and valley periods across all tested scenarios. Although other factors may also influence generation cost formation and are not explicitly examined here due to space limitations, the evidence presented suggests that the proposed scenario-based framework captures fundamental structural characteristics of cost formation under time-division trading, rather than being dependent on specific parameter calibrations.

Further empirical validation

To further validate the reasonableness and robustness of the proposed time-varying generation cost model, an additional case study is conducted using data from Province J in northern China. Unlike Province H, which exhibits significant peak-valley load variations, Province J features a relatively smaller peak-valley difference and a smoother load profile throughout the year. By selecting a region with such contrasting load characteristics, this analysis aims to strengthen the generalizability of the model’s conclusions and demonstrate its applicability across diverse regional power systems. The load profile of the J region is illustrated in Figure 12. Province J exhibits a notably smoother load profile with a smaller peak-valley difference compared to Province H, which is characterized by pronounced daily and seasonal load fluctuations. This difference in load profile shape plays a central role in explaining the contrast in generation cost volatility between the two provinces, within the unified cost measurement framework proposed in this study. In Province H, sharp and short-duration load peaks, particularly during summer cooling periods, frequently push net demand toward the upper bounds of thermal generation flexibility. These conditions activate scarcity-related cost components and result in pronounced hourly cost spikes. In contrast, Province J’s flatter demand profile substantially reduces the frequency and severity of tight supply-demand conditions, even under comparable aggregate load levels.

Figure 12.

Load curve and peak-valley of Province J in 2021

The annual time-varying generation cost results for Province J are illustrated in Figure 13, with the corresponding cost distribution summarized in Table 3. Province J exhibits notable temporal cost spikes, particularly during critical peak load periods, where hourly generation costs exceed 2 CNY/kWh. However, compared to Province H, these cost peaks are less frequent and less pronounced. This reflects the stabilizing effect of a smoother demand profile, which limits the activation of high-cost thermal units and scarcity-driven opportunity costs. The distribution of hourly generation costs in Province J is markedly concentrated, with nearly 90% of costs falling within a narrow range of 0.2–0.3 CNY/kWh. This tight clustering contrasts with the broader cost fluctuations observed in Province H. Differences in generation mix further reinforce, but do not dominate, this contrast. While higher renewable penetration in Province H contributes to lower costs during off-peak and valley periods, it also amplifies intra-day net load variability when renewable output fluctuates. By comparison, Province J’s more balanced generation structure and smoother net load evolution help dampen extreme cost movements. As a result, Province J exhibits lower overall cost volatility despite similar average cost levels.

Figure 13.

The result of the hourly time-varying generation costs in Province J

Table 3.

Distribution of annual time-varying generation cost results

Cost Range (CNY/kWh)	Count	Percentage
0.0–0.1	23	0.26%
0.1–0.2	298	3.40%
0.2–0.3	7843	89.54%
0.3–0.4	398	4.54%
0.4–0.5	84	0.96%
0.5–1.0	86	0.98%
1.0–2.0	25	0.29%
2.0–3.0	3	0.03%

These contrasting patterns have significant operational implications. Province J’s relatively stable load allows for simpler and more uniform pricing mechanisms, which can efficiently manage operational risks with less temporal granularity. In contrast, Province H’s pronounced load variability necessitates more sophisticated, time-sensitive pricing strategies that accurately signal operational risks and encourage demand-side flexibility. Importantly, the proposed time-varying generation cost model effectively captures these contrasting regional patterns, demonstrating robustness and adaptability. Its capability to represent both high variability and stable load scenarios confirms its suitability for diverse regional power systems. Overall, the consistent performance of the model across provinces with contrasting load patterns confirms that the proposed framework is not region-specific but structurally adaptable, making it suitable for application in electricity markets with heterogeneous demand characteristics. Its ability to distinguish cost dynamics driven by load profile shape and generation structure demonstrates that the framework is not region-specific, but structurally adaptable across heterogeneous provincial power systems.

Although the empirical analysis focuses on Provinces H and J, the two case studies already represent contrasting provincial system characteristics in China’s electricity market. Province H exhibits pronounced load peaks and tighter supply-demand conditions, while Province J features a flatter load profile and relatively smoother cost dynamics. These differences capture two representative system archetypes commonly observed across Chinese provinces.

For provinces with a higher coal generation share, the proposed framework would mainly reflect higher baseline variable costs, while preserving the same mechanism through which supply-demand tightness drives temporal cost volatility. Conversely, in provinces with ultra-high renewable penetration, the framework would yield more frequent low-cost periods and steeper scarcity-driven peaks, as renewable variability amplifies the mismatch between net load and dispatchable capacity. Importantly, the core structure of the model remains unchanged, with scenario classification determined by the relationship between net load and the operational flexibility of thermal generation.

Therefore, the framework is not tailored to specific provincial characteristics but is adaptable to different generation mixes through parameter calibration. A systematic multi-province expansion is left for future research, particularly as more consistent data becomes available under the ongoing development of China’s unified electricity market.

This study develops a scenario-based framework to measure time-varying generation costs. It is designed specifically for China’s medium- and long-term electricity market under time-division trading. The framework integrates capacity costs allocated based on utilization rates with variable operational and opportunity costs. It considers typical supply-demand scenarios to capture the dynamic and varied nature of generation costs throughout the year. We applied the model to 8,760 hourly data points from Province H. The results show significant fluctuations in generation costs. The yearly average cost is 0.2313 CNY/kWh, ranging from 0.0794 to 3.00 CNY/kWh. Costs spike sharply during peak load periods, especially in summer. In contrast, high renewable output in spring and autumn lowers costs. Cluster analysis further delineates peak, flat, and valley periods with distinct average costs of 0.2934, 0.2370, and 0.1646 CNY/kWh, respectively, validating the necessity of time-differentiated pricing mechanisms. These cost-based periods provide a quantitative foundation for designing time-differentiated contracts and identifying critical pricing windows in medium- and long-term electricity markets. We also compared results with Province J. Province J has a smoother demand profile and a narrower cost distribution, mostly between 0.2 and 0.3 CNY/kWh. This comparison shows the model can handle different load patterns effectively.

This framework overcomes the limits of traditional static and marginal cost models. It combines capacity cost allocation based on utilization rates with opportunity cost considerations. This approach provides a comprehensive and economically meaningful view of generation costs. It helps inform better operational and investment decisions. This advancement improves understanding of cost causality in segmented electricity markets. By explicitly distinguishing between normal operating costs and scarcity-driven cost spikes, the framework supports the design of flexible pricing mechanisms and targeted demand response programs that focus on high-cost and high-risk periods rather than uniform interventions across all hours. From a practical standpoint, the model offers clear insights for market participants and regulators. The detailed cost signals help optimize contract structures and strengthen demand response programs. These improvements boost market efficiency and aid renewable energy integration.

Based on the empirical results, this study supports the adoption of dynamic, time-differentiated pricing mechanisms that accurately reflect capacity utilization and system scarcity conditions. Importantly, the proposed approach does not imply the direct use of hourly price signals in the settlement of medium- and long-term electricity contracts. Instead, hourly generation cost information serves as an analytical basis for deriving aggregated and interpretable pricing benchmarks. In practice, high-resolution cost signals can be translated into representative time-of-use categories, such as peak, flat, and valley periods, and further aggregated at monthly or seasonal scales. These period-specific cost benchmarks can then be incorporated into medium- and long-term contracts through differentiated block prices, seasonal coefficients, or adjustment factors. Such an approach preserves key scarcity-related economic signals while maintaining the simplicity, stability, and risk-manageability required for contractual design. In this way, medium- and long-term contracts can provide stable, cost-reflective baselines, while spot markets, once fully implemented, continue to deliver short-term efficiency and operational balancing through ex-post price formation. Through this mechanism, time-differentiated pricing can be effectively applied to the structuring of medium- and long-term contracts, the definition of peak and critical peak pricing intervals, and the calibration of demand response activation thresholds. This facilitates more efficient use of flexible resources, improves cost recovery, and enhances market transparency. From a policy perspective, the establishment of standardized methods for time-interval segmentation and dynamic cost measurement would further support the scalable implementation of such pricing mechanisms and contribute to a more transparent and efficient electricity market.

Limitations of the study

We recognize some limitations in our study. Fixed operational assumptions and the exclusion of ancillary service costs may underestimate total system costs, especially as flexibility needs grow. In particular, ancillary services such as frequency regulation, reserve procurement, and flexibility support tend to exhibit strong temporal concentration and become increasingly important during periods of high renewable penetration and tight system flexibility. Their omission is therefore more likely to understate peak and near-peak costs than to uniformly bias average cost levels. Moreover, the proposed framework is primarily intended as a cost measurement and signal extraction tool for scheduled medium- and long-term markets, rather than a fully integrated market clearing or dispatch optimization model. As a result, the interaction between time-varying cost signals and actual contract execution or scheduling decisions is not explicitly modeled. In addition, the proposed framework focuses on intra-provincial medium- and long-term markets and does not explicitly consider inter-provincial transmission constraints, such as available transfer capability (ATC), or their interactions with provincial market outcomes. As inter-regional electricity trading expands under China’s unified national market reform, transmission capacity constraints may increasingly influence supply-demand conditions and cost formation across provinces. Similarly, while renewable energy integration is reflected through its impact on net load and cost variability, issues such as renewable curtailment, ancillary service provision, and cross-market coordination are not modeled endogenously.

Future research should include more detailed system constraints and market interactions through scenario analysis. In particular, extending the framework to incorporate inter-provincial transmission limits and coordinated multi-area market settings would enhance its applicability in integrated market environments. Further extensions could explicitly integrate ancillary service cost formation with energy cost signals, enabling a more comprehensive representation of total system costs under high-renewable operating conditions. Embedding the proposed time-varying cost signals into actual contract settlement rules, time-of-use pricing schemes, and demand response activation mechanisms would further improve market transparency and operational efficiency. It should also explore how policy and market changes affect generation costs. Evaluating dynamic pricing using real-world market data is crucial. These efforts will help China’s power market become more flexible, efficient, and market-oriented.

Resource availability

Lead contact

Further information and requests for resources and data should be directed to and will be fulfilled by the lead contact, Dr. Shanshan Huang (22007040089@stu.csust.edu.cn).

Materials availability

This study did not generate new materials.

Data and code availability

• •Data: The data reported in this paper are available from the lead contact upon reasonable request. Due to confidentiality restrictions associated with provincial power system data in China, the datasets cannot be deposited in a public repository. Data sources and related resources are described in the key resources table.
• •Code: This study includes original code used to implement the scenario identification model, capacity cost allocation algorithm, and clustering analysis. The code is available from the lead contact upon reasonable request.
• •Additional Information: Any additional information required to reanalyse the data reported in this paper is available from the lead contact upon reasonable request.

Acknowledgments

This work was supported by the Science and Technology Program of the Hunan Provincial Department of Education (no. 23B1098) and the Postgraduate Scientific Research Innovation Project of Hunan Province (CX20251399).

Author contributions

S.S.H.: conceptualization, formal analysis, methodology, software, visualization, and writing – original draft. Y.C.M.: data curation, validation, investigation, and writing – review and editing. Z.Y.: supervision.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
Hourly system load data	Provincial power grid operator	Available upon reasonable request
Renewable generation profiles (wind, solar, hydro)	Provincial energy authority	Available upon reasonable request
Thermal unit technical parameters	Power plant statistical yearbooks	Available upon reasonable request
Software and algorithms
MATLAB	MathWorks (R2024a)	https://www.mathworks.com/products/new_products/release2024a.html
Peak-load responsibility method	This paper	Implemented in MATLAB
Scenario-based cost modeling framework	This paper	Implemented in MATLAB
Fuzzy C-means clustering algorithm	This paper	Implemented in MATLAB

Method details

This section describes the modeling framework used to quantify time-varying generation costs under different supply-demand conditions, rather than to simulate market clearing or dispatch optimization.

Unit output model

This study considers four generation technologies: wind power, photovoltaic power, hydropower, and thermal power. These technologies are modeled using differentiated approaches to reflect their distinct operational roles in the power system. Wind and photovoltaic generation exhibit strong intermittency, high temporal variability, and near-zero marginal operating costs. Their outputs are treated as exogenous inputs and represented by scenario-dependent 8,760-hour time series derived from historical resource availability. These renewable profiles are used to characterize system-level supply-demand conditions and determine the residual load, rather than for endogenous cost calculation.

Thermal power plants are modeled as dispatchable and cost-bearing units responsible for balancing the system. Their hourly output is endogenously adjusted to meet the residual load, subject to technical constraints, including minimum and maximum output limits and ramping constraints, as defined in Eqs. 1, 2, and 3.

$$P_{\text{th}}^{\min }\le P_{\text{th},i}\le P_{\text{th}}^{\max }$$ (Equation 1)

$$P_{\text{th},i}-P_{\text{th},i-1}\le \Delta P_{\text{th}}^U$$ (Equation 2)

$$P_{\text{th},i-1}-P_{\text{th},i}\le \Delta P_{\text{th}}^D$$ (Equation 3)

where P_th,i represents the output of thermal units at time i; $P_{\text{th}}^{\min }$ and $P_{\text{th}}^{\max }$ are respectively the minimum and maximum output of thermal units; $\Delta P_{\text{th}}^U$ and $\Delta P_{\text{th}}^D$ are respectively the ramp-up and ramp-down rates of thermal units.

Hydropower output is modeled based on water head, flow rate, and conversion efficiency (Eq. 4), and constrained by seasonal availability and capacity limits (Eq. 5). Wind power output follows a piecewise power curve as a function of wind speed (Eq. 6), with uncertainty represented using a Weibull distribution (Eqs. 7, 8, and 9). Photovoltaic output is calculated as a function of solar irradiance, conversion efficiency, and module area (Eq. 10), with irradiance uncertainty modeled using a Beta distribution (Eq. 11).

$$P_{\text{hy},i}=g{\eta }_{\text{hy}}{\phi }_i{\lambda }_i$$ (Equation 4)

$$P_{\text{hy}}^{\min }\le P_{\text{hy},i}\le P_{\text{hy}}^{\max }$$ (Equation 5)

$$P_{\text{wi},i}=\{\begin{array}{cc}0& 0\le v_i\le v_{\text{ci}},v\ge v_{\text{cu}}\\ P_{\text{wi}}^{\mathrm{r}}\frac{v_i-v_{\text{ci}}}{v_{\mathrm{r}}-v_{\text{ci}}}& v_{\text{ci}}\le v_i\le v_{\mathrm{r}}\\ P_{\text{wi}}^{\mathrm{r}}& v_{\mathrm{r}}\le v_i\le v_{\text{cu}}\end{array}$$ (Equation 6)

$$v_i=c{\left(-\ln \beta \right)}^{1/k}$$ (Equation 7)

$$c=\frac{E_{\text{wi}}}{\mathrm{\Gamma }\left(1+1/k\right)}$$ (Equation 8)

$$k={\left(\frac{{\sigma }_{\text{wi}}}{E_{\text{wi}}}\right)}^{-1.086}$$ (Equation 9)

$$P_{\text{pv},i}={\lambda }_{\text{pv}}{\gamma }_{\text{pv},i}{S}_{\text{pv}}$$ (Equation 10)

$$f\left({\gamma }_{\text{pv},i}\right)=\frac{\mathrm{\Gamma }\left(a+b\right)}{{\gamma }_{\max }\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\left(\frac{{\gamma }_{\text{pv},i}}{{\gamma }_{\max }}\right)}^{a-1}{\left(1-\frac{{\gamma }_{\text{pv},i}}{{\gamma }_{\max }}\right)}^{b-1}$$ (Equation 11)

Where P_hy,i denotes the output of the hydro unit at time i; g denotes the gravitational acceleration; η_hy denotes the efficiency of the hydro unit; φ_i denotes the net water head of power generation at time i; and λ_i denotes the water flow rate at time i; $P_{\text{hy}}^{\min }$ and $P_{\text{hy}}^{\max }$ are respectively the minimum and maximum output of the hydropower unit; P_wi,i represents the output of the wind unit at time i; $P_{\text{wi}}^{\mathrm{r}}$ represents the rated power of the wind unit; v_i represents the wind speed at time i; v_r, v_ci and v_cu are respectively the rated wind speed, the cut-in wind speed, and the cut-out wind speed of the wind unit; c represents the scale parameter; k represents the shape parameter; β represents a random number obeying a uniform distribution on 0∼1; E_wi represents the mean value of the wind speed; σ_wi represents the standard deviation of the wind speed; Γ() represents the gamma function; P_pv,i denotes the predicted output of PV at time i; λ_pv denotes the rated photovoltaic conversion efficiency; γ_pv,i denotes the radiation intensity of the irradiated PV module at time i; and S_pv denotes the total area of the photovoltaic module; γ_max represents the maximum value of the radiation intensity of the photovoltaic module; a and b are respectively the shape parameters of the Beta distribution.

Scenario identification model

Four supply-demand scenarios are identified by comparing the net system load with the feasible operating range of thermal power units. As the primary dispatchable and cost-bearing resources in China’s medium- and long-term electricity market, thermal units provide an economically and operationally meaningful basis for distinguishing system states with different cost formation mechanisms. Importantly, scenario boundaries are endogenously determined by dispatchable generation constraints and residual load conditions, rather than by empirical thresholds or exogenous parameter settings.

The net system load is defined as total system demand minus aggregated renewable generation, including wind, photovoltaic, and hydropower outputs (Eqs. 12 and 13).

$$P_{\mathrm{s},i}=P_{\text{th},i}+P_{\text{hy},i}+P_{\text{wi},i}+P_{\text{pv},i}$$ (Equation 12)

$$P_{\text{net},i}=P_{\mathrm{s},i}-P_{\text{hy},i}-P_{\text{wi},i}-P_{\text{pv},i}$$ (Equation 13)

where P_s,i represents the load of the system at time i; P_net,i represents the net load of the system at time i.

When the net load falls within the feasible dispatchable range of thermal units, the system is in a balanced supply-demand state, where generation costs are primarily driven by utilization levels and normal operating conditions (Eq. 14). When the net load exceeds the maximum available thermal output, dispatchable capacity becomes binding and the system enters a supply shortage state, characterized by scarcity and rapidly increasing marginal costs (Eq. 15). When the net load remains positive but falls below the minimum stable output of thermal units, downward regulation constraints lead to surplus conditions and over-generation costs (Eq. 16). Finally, when the net load becomes negative, renewable generation exceeds demand, resulting in extreme oversupply conditions typically associated with curtailment and negative residual load (Eq. 17).

$$P_{\text{th}}^{\min }\le P_{\text{net},\mathrm{i}}\le P_{\text{th}}^{\max }$$ (Equation 14)

$$P_{\text{net},i}>P_{\text{th}}^{\max }$$ (Equation 15)

$$0\le P_{\text{net},i}<P_{\text{th}}^{\min }$$ (Equation 16)

$$P_{\text{net},i}<0$$ (Equation 17)

Time-varying capacity cost allocation

Total capacity cost consists of capital investment, fixed operation and maintenance expenses, and other long-term fixed costs required to ensure system adequacy (Eq. 18). These costs are allocated across hours based on utilization rates using the peak-load responsibility method.

$$C_{\text{fx}}=\sum C_{\text{dep}}+\sum C_{\text{fix}}+\sum C_{\text{mat}}+\sum C_{\text{sal}}+\sum C_{\text{else}}$$ (Equation 18)

Where C_fx denotes the total units capacity cost; C_dep denotes the depreciation cost of fixed assets of the units; C_fix denotes the repair cost of the units; C_mat denotes the material cost of the units; C_sal denotes the employee salary and welfare expenditure of the units; C_else denotes the other costs of the generating units.

The annual sequential load curve is first transformed into a load duration curve by aggregating load levels with identical durations, as illustrated in Figure S1.

Capacity costs are then allocated horizontally across load levels and vertically across load durations, as formulated in Eqs. 19, 20, 21, 22, 23, 24, 25, 26, and 27.

$$\Delta C_i=\frac{C_{\text{fx}}\Delta P_i}{P_{\max }}$$ (Equation 19)

$$Q_i=\sum _{j=1}^i{Q}_{i,j}\left(1\le j\le i\right)$$ (Equation 20)

$$\Delta P_i=P_i-P_{i-1}$$ (Equation 21)

$$\Delta Q_i=\Delta P_i{t}_i$$ (Equation 22)

Where ΔQ_i represents the electricity from P_i-1 to P_i duration t_i; ΔC_i represents the capacity cost of electricity ΔQ_i; P_max represents the maximum load; ΔP_i represents the load increment from P_i-1 to P_i.

Apportion the cost of capacity to different periods by vertical stacking. Where ΔQ_i share capacity cost per unit of electricity ΔD_i is in Eq. 23.

$$\Delta D_i=\frac{\Delta C_i}{\Delta Q_i}=\frac{C_{\text{fx}}}{P_{\max }{t}_i}$$ (Equation 23)

The capacity costs of different attributions are calculated according to the compliance duration, and the capacity cost C_i,i of the electricity block Q_i,i is shown in Eq. 24.

$$C_{i,i}=\Delta D_i{Q}_{i,i}=\frac{C_{\text{fx}}\Delta P_i\Delta T}{P_{\max }{t}_i}$$ (Equation 24)

By stacking vertically and summing the capacity costs in the vertical direction, the capacity cost C_i,j on the power block Q_i,j is in Eq. 25.

$$C_{i,j}=\Delta D_i{Q}_{i,j}=\frac{C_{\text{fx}}\Delta P_j\Delta T}{P_{\max }{t}_j}\left(1\le j\le i\right)$$ (Equation 25)

The sum of the capacity costs C_i apportioned to electricity with a load level of P_i is in Eq. 26.

$$C_i=\sum _{j=1}^i{C}_{i,j}\left(1\le j\le i\right)$$ (Equation 26)

Therefore, the capacity cost F_i for a unit of electricity with a load level of P_i is in Eq. 27:

$$F_i=\frac{C_i}{Q_i}=\frac{C_{\text{fx}}}{P_{\max }{P}_i}\sum _{j=1}^i\frac{\Delta P_j}{t_j}$$ (Equation 27)

This procedure allocates capacity costs according to continuous load levels and utilization rates derived from the load duration curve, rather than relying on ex ante time-period classifications such as peak, flat, or valley hours. Consequently, capacity costs reflect both the magnitude and persistence of system load, enabling a more economically consistent allocation of long-term fixed costs across hours.

Scenario-based generation cost calculation

Under supply-demand equilibrium conditions, generation cost is calculated as the sum of time-varying capacity cost and variable operating cost (Eqs. 28 and 29).

$$C_{\text{mc},i}=\left({\eta }_{\text{coal}}{p}_{\text{coal}}{P}_{\text{th},i}+p_{{\text{co}}_2}{\delta }_{{\text{co}}_2}{P}_{\text{th},i}\right)/P_{\mathrm{S},i}$$ (Equation 28)

$$C_{1,i}=\frac{C_{\text{fx}}}{P_{\max }{P}_i}\sum _{j=1}^i\frac{\Delta P_j}{t_j}+\frac{{\eta }_{\text{coal}}{p}_{\text{coal}}{P}_{\text{th},i}+p_{{\text{co}}_2}{\delta }_{{\text{co}}_2}{P}_{\text{th},i}}{P_{\mathrm{S},i}}$$ (Equation 29)

Where C_mc,i represents the marginal generation cost at time i; η_coal represents the standard coal consumption rate of thermal power; p_coal represents the unit price of standard coal at time i; $p_{{\text{co}}_2}$ represents the output of thermal power units in the interval of i; ${\delta }_{{\text{co}}_2}$ represents the standard coal consumption rate of thermal power; C_1,i denotes the time-varying cost for the market supply-demand equilibrium scenario.

In supply shortage scenarios, the customer-side Value of Lost Load (VOLL) is used to represent the economic cost of unmet demand, capturing the opportunity cost of supply interruptions (Eq. 30).

$$C_{2,i}=VOL{L}_{user,i}=GVA/G$$ (Equation 30)

Where C_2,i represents the time-varying cost for the market oversupply scenario; GVA represents the total value added to the GVA of the customer; G represents the electricity consumption of the customer; VOLL_user,i represents the value of customer loss loads.

In oversupply scenarios, generation cost is measured using marginal cost, reflecting short-run variable operating expenses and the reduced system value of electricity (Eq. 31).

$$C_{3,i}=\frac{{\eta }_{\text{coal}}{p}_{\text{coal}}{P}_{\text{th},i}+p_{{\text{co}}_2}{\delta }_{{\text{co}}_2}{P}_{\text{th},i}}{P_{\mathrm{S},i}}$$ (Equation 31)

Where C_3,i is the time-varying cost for the short-term oversupply scenarios.

In extreme oversupply scenarios, the generation-side VOLL is applied to represent the minimum acceptable revenue for generators to avoid shutdown, accounting for start-up and shutdown costs (Eq. 32).

$$C_{4,i}=C_i^{\text{QT}}/Q^{\mathrm{D}}$$ (Equation 32)

Where C_4,i is the time-varying cost for the extreme oversupply scenario; $C_i^{\text{QT}}$ is the startup and shutdown cost of the unit; Q^D is the amount of generation lost due to unit shutdown.

Solution flow

Figure 2 illustrates the solution process of the proposed scenario-based time-varying generation cost model. The procedure includes: (1) inputting system load, renewable generation, and cost parameters; (2) calculating net load; (3) identifying supply-demand scenarios by comparing net load with thermal unit constraints; (4) selecting the corresponding cost calculation model; and (5) computing hourly generation costs. All calculations and simulations are implemented using MATLAB R2024a.

Quantification and statistical analysis

Hourly generation cost results are statistically analyzed to characterize annual distributions, seasonal variations, and representative daily patterns. Descriptive statistics, including annual averages, extrema, standard deviations, and peak-valley spreads, are used to summarize the overall magnitude and volatility of time-varying generation costs.

To identify representative time-of-use periods, Fuzzy C-Means (FCM) clustering is applied to the annual hourly load profile, which reflects the underlying interaction between system demand conditions and generation flexibility requirements. Load is selected as the clustering variable because it directly characterizes system operating states that give rise to cost differentiation, rather than being an outcome of pricing mechanisms. FCM is adopted because it allows each hourly observation to belong to multiple clusters with different membership degrees, thereby capturing the smooth transitions between load states commonly observed in power system operations. The clustering process minimizes the weighted within-cluster squared distance between hourly load observations and cluster centers under probabilistic membership constraints.

The number of clusters is evaluated over a candidate range from two to six and determined based on a joint assessment of statistical validity and economic interpretability. Specifically, the Partition Coefficient (PC), Classification Entropy (CE), Xie-Beni (XB) index, and mean silhouette coefficient are employed to assess membership concentration, classification uncertainty, cluster separation, and structural stability. Among the tested configurations, the three-cluster solution achieves a favorable balance between clustering quality and interpretability. The corresponding evaluation results for alternative cluster numbers are reported in Table S1.

Accordingly, a three-period structure corresponding to valley, flat, and peak load conditions is adopted. The resulting classification is subsequently mapped to hourly generation cost outcomes to compute average costs for each period and to examine the consistency between load-based clustering and cost differentiation. This procedure ensures that the identified time-of-use periods are statistically robust and interpretable in the context of system operation, providing a consistent basis for subsequent cost analysis and time-differentiated evaluation.

Published: March 13, 2026