Stochastic Bidding for Hydro–Wind–Solar Systems in Cross-Provincial Forward–Spot Markets: A Dimensionality-Reduced and Transmission-Aware Framework

Abstract

Integrated hydro–wind–solar power generators (IPGs) in China face multi-timescale bidding challenges across provincial forward–spot markets, which are further compounded by hydropower nonconvexity and transmission constraints. This study proposes a stochastic optimization model addressing uncertainties from wind–solar generation and spot prices through scenario-based programming, integrating three innovations: average-day dimensionality reduction to harmonize monthly–hourly decisions, local generation function approximation to linearize hydropower operations, and transmission-aware coordination for cross-provincial allocation. Validation on a southwestern China cascade hydropower base transmitting power to eastern load centers shows that the model establishes hydropower-mediated complementarity with daily “solar–daytime, wind–nighttime” and seasonal “wind–winter, solar–summer” patterns. Furthermore, by optimizing cross-provincial power allocation, strategic spot market participation yields 46.4% revenue from 30% generation volume. Additionally, two transmission capacity thresholds are found to guide grid planning: 43.75% capacity marks the economic optimization inflection point, while 75% represents technical saturation. This framework ensures robustness and computational tractability while enabling IPGs to optimize profits and stability in multi-market environments.

1. Introduction

As the global demand for sustainable energy continues to rise, electricity has emerged as one of the key energy carriers for the future, playing a strategic role in energy transition pathways [1]. With the continued development of electricity infrastructure over the past few decades, market-based institutional mechanisms have played a pivotal role in enabling the integration of clean energy into power systems.

In China, where electricity market reforms have been implemented for a decade [2,3], remarkable progress has been achieved; by the end of 2024, the installed capacity of hydropower has increased from 319.37 GW to 435.95 GW, while wind and solar power capacity has surged from 170.58 GW to 1407 GW. Concurrently, the proportion of electricity traded through electricity market amounted to 63% of China’s annual generation [4]. These developments have been driven by continuously evolving market mechanisms, including multi-timescale market structures and inter-provincial power resource coordination. However, in such a complex market environment, coordinated bidding strategies remain a critical challenge for emerging integrated hydro–wind–solar power generators (IPGs) in China.

First, hydropower must serve as a flexible regulator to mitigate renewable intermittency. The inherent variability and intermittency of wind and solar power create significant risks of penalties for independent market participation. To address this, China’s dominant approach involves co-locating wind and solar farms with large-scale cascade hydropower systems [5]. This hybrid configuration leverages hydropower’s dispatchability to smooth volatile renewable output profiles, thereby enhancing bidding success rates and market compliance capabilities. Consequently, cascade hydropower is increasingly tasked with balancing intermittent renewables rather than solely maximizing its own generation.

Second, IPGs must navigate multi-timescale market dynamics spanning forward and spot markets. In forward markets, IPGs optimize monthly reservoir levels to secure long-term revenue, while in spot markets they must rapidly adapt to hourly price fluctuations and real-time renewable generation variability in order to minimize curtailment losses [6,7]. The temporal coupling of these decisions significantly amplifies the operational complexity for cascade hydropower systems, which must reconcile conflicting objectives across hourly and monthly horizons.

Third, IPGs face spatial coordination challenges due to uneven distribution of energy resources and electricity demand. For example, hydropower resources are predominantly concentrated in the southwestern region of China, while the major load centers are located in the eastern and southeastern coastal areas. As a result, hydropower generated in the southwest must be transmitted to the eastern provinces through the West-to-East Electricity Transmission Project [8]. This compels IPGs to simultaneously participate in multiple provincial electricity markets, each governed by distinct pricing mechanisms and regulatory constraints. The resulting cross-regional bidding requirements further complicate decision-making, as hydropower discharge decisions in one province directly affect generation capacity and market participation in others.

These multidimensional challenges—temporal flexibility, spatial coordination, and hybrid resource synergy—fundamentally disrupt traditional hydropower scheduling paradigms. Thus, developing coordinated bidding strategies for IPGs operating across heterogeneous timescales and provincial markets has emerged as a pressing engineering challenge for China’s major power utilities. This study addresses this critical gap by proposing a multi-timescale coordinated bidding model for integrated hydro–wind–solar systems in the context of cross-provincial forward–spot markets.

1.1. Literature Review

The operational coordination of IPGs in cross-provincial multi-timescale electricity markets constitutes a cross-disciplinary challenge intersecting two domains: (i) hydro–wind–solar complementary scheduling under meteorological uncertainties, and (ii) multi-market bidding strategies across temporal scales.

1.1.1. Coordinated Scheduling in Non-Market Contexts

Current research on IPG coordination bifurcates into two categories under non-market paradigms.

(1) Grid-Oriented Scheduling Optimization. The first category addresses technical constraints from large-scale renewable integration, particularly grid peaking challenges. For instance, Wang et al. [9] proposed a long-term complementary scheduling model for hydro–wind–solar systems under extreme drought conditions, focusing on guaranteed output and grid fluctuation mitigation. Similarly, Zhao et al. [10] developed a short-term optimization model to minimize residual peak–valley load differences through scenario-based renewable uncertainty management.

(2) Generation-Side Operational Frameworks. The second category designs scheduling algorithms and market-agnostic participation mechanisms. Cao et al. [11] introduced a rolling update model for hydro–solar systems using stochastic scenario trees, while Zhang et al. [12] enhanced system benefits via multivariate probabilistic forecasting of inflow and wind speed. Apostolopoulou et al. [13] improved short-term stability through hydro–solar joint robust scheduling with probabilistic solar forecasts.

1.1.2. Market-Driven Bidding Strategies

However, the above studies remain confined to non-market contexts. With electricity marketization accelerating, research has shifted toward IPGs’ bidding strategies. Early efforts by Cerejo et al. [14] and Ilak et al. [15] demonstrated hydropower’s role in mitigating wind uncertainty to enhance market revenues. Li et al. [16] advanced bilevel nested bidding models balancing immediate profits and long-term utility. European studies such as Riddervold et al. [17] and Angarita et al. [18,19] have validated hydro–wind complementarity in reducing imbalance costs. Jurasz et al. [20] developed day-ahead bidding strategies based on solar and hydro forecasts, but focused solely on single-market contexts.

However, these contributions are predominantly focused on daily or spot markets, neglecting the interplay between forward and spot markets. While Morstyn et al. [21] highlighted IPGs’ preference for forward contracts to secure long-term revenue, effective coordination across multi-timescale markets remains underexplored.

1.1.3. Forward-Spot Market Coordination

Recent attempts to bridge this gap exhibit limitations. For example, de la Nieta et al. [22] proposed stochastic programming for wind-pumped hydro coordination, but oversimplified long-term contract distributions. Yu et al. [23] revealed how bilateral contract decomposition impacts spot market revenues, yet their single-market scope restricts practical applicability. Cheng et al. [24] explored peak–valley allocation of long-term contracts, but confined their analysis to a single provincial market. Although Silva et al. [25] optimized multi-market bidding for wind–solar–storage systems, their assumption of fixed forward contracts overlooks dynamic adjustments.

1.1.4. Spatial Coordination Challenges

A further limitation lies in the neglect of spatial constraints. While Tang et al. [26] emphasized the necessity of linking forward and spot markets, their framework lacked mechanisms to handle interprovincial transmission bottlenecks. Gupta et al. [27] focused on single-region congestion management, while Mi et al. [28] correlated locational prices with line congestion. However, cross-provincial transmission complexities, which are critical in China’s west-to-east power transmission paradigm, remain unaddressed.

1.1.5. Nonconvexity of Hydropower Scheduling

The nonconvex nature of hydropower operations presents a fundamental challenge in solving hydro–wind–solar joint scheduling systems. To address this issue, scholars have explored evolutionary algorithms. For instance, Zhang et al. [29] implemented Genetic Algorithms (GA), while Li et al. [30] adopted the Cuckoo Search method, both of which circumvent hydropower’s complex characteristics through metaheuristic approaches. However, these methods suffer from prolonged computational times and struggle to ensure solution accuracy.

Alternative approaches employ linearization techniques such as SOS2 (Special Ordered Sets of Type 2) to reformulate nonlinear problems into Mixed-Integer Linear Programming (MILP) frameworks [31,32]. Nevertheless, these methods encounter challenges such as significant expansion of the problem scale and non-negligible accuracy degradation.

Gupta et al. [33] proposed a dual-strategy approach combining nonlinear optimization with logical optimization to handle nonconvexity. While nonlinear optimization exhibits slower convergence rates, logical optimization—despite its operational simplicity—excessively depends on empirical knowledge from production practices.

1.1.6. Summary of Related Literature

A comparative summary is provided in

Table 1. Comparison of this study with existing research.

Category	Representative Studies	Market Scope	Timescale Coordination	Spatial Scope	Hydropower Modeling	Innovation and Limitation
Non-market Scheduling	[9,10,11]	No market participation	Long-term or short-term only	Single region	Scenario-based or simplified	Focused on renewable smoothing and peak shaving; lack of market mechanisms
Single-Market Bidding	[14,15,16,20]	Spot or forward market only	Single timescale	Single region	Linearized or heuristic	Addressed strategic bidding; ignored inter-scale coupling and grid congestion
Forward-Spot Coordination	[21,22,23,24]	Both markets	Considered two timescales	Mostly single market	Fixed or pre-assumed contracts	Forward–spot linkage discussed; lacked spatial granularity and adaptability
Spatial Coordination/Transmission	[26,27,28]	Spot market focus	Short-term only	Inter-provincial	Simplified congestion models	Considered locational prices; failed to integrate with forward planning
Nonconvexity Handling	[29,30,31,32,33]	Mostly non-market models	–	–	Metaheuristics or MILP	Tackled nonconvexity; suffered from scale issues or limited accuracy
This Study	–	Coordinated forward and spot markets	monthly-daily scale coupling via average-day compression	Cross-provincial with grid constraints	Mixed-integer bilinear programming with reduced binary variables	Unified framework addressing temporal–spatial coupling and scalable computation

to clarify the novelty of this study against the existing literature.

The following contributions distinguish this work from the existing literature.

1.2. Contributions

This study develops a coordinated bidding framework for integrated hydro–wind–solar systems participating in cross-provincial forward and spot electricity markets. The main contributions are as follows:

• Joint forward–spot market coordination: A unified optimization model is established to integrate monthly forward contracts with daily spot market decisions. This enables consistent revenue-risk balancing across timescales, an aspect that is rarely addressed in existing studies.
• Average-day compression for temporal dimensionality reduction: A novel average-day abstraction is introduced to compress long-term forward bidding decisions into representative daily profiles. This approach effectively reduces the model dimensionality while preserving inter-temporal coupling accuracy.
• Spatially-aware market participation: The proposed model incorporates provincial-level transmission constraints and cross-regional power delivery requirements, addressing spatial coordination challenges overlooked in conventional single-region bidding models.
• Hydropower approximation for computational scalability: A local linear approximation method is proposed to represent hydropower generation functions, significantly reducing the number of binary variables while maintaining acceptable modeling precision.

1.3. Paper Organization

The remainder of this paper is structured as follows: Section 2 outlines the system configuration and market environment; Section 3 details the mathematical formulation; Section 4 validates the model using a case study; Section 5 discusses the model’s potential international applicability and variants; finally, Section 6 concludes the paper.

2. Methodology

Figure 1 illustrates the studied power system, challenges in modeling, and the solution proposed in this paper.

In this section, we first describe the overall system and market environment (Section 2.1), then analyze the key challenges of cross-temporal and cross-provincial bidding (Section 2.2), present our Local Generation Function Approximation Method (Section 2.3) and Average-Day Dimensionality Reduction Strategy (Section 2.4), and finally outline the core elements of the proposed bidding model (Section 2.5). Further elaboration of the proposed model is provided in Section 5.

2.1. System Description

As illustrated in Figure 1, the studied power system consists of three fundamental components: the Integrated Power Generator (IPG) under study, the load provinces, and their interconnecting transmission infrastructure.

2.1.1. Integrated Power Generator

The IPG coordinates large-scale cascade hydropower systems with adjacent wind and solar farms under unified management. This configuration utilizes hydropower’s fast ramping capabilities and storage capacity to mitigate renewable energy fluctuations during market participation, which is a typical arrangement in China’s southwestern regions. Electricity generated by the IPG is transmitted through dedicated networks to multiple load provinces. Given the substantial generation capacity of cross-provincial IPGs, simultaneous power delivery to several provinces has become standard practice.

2.1.2. Transmission Network

The transmission network primarily consists of DC lines from China’s West-to-East Power Transmission Project. The physical characteristics of DC transmission impose strict constraints on both power transfer capacities and ramping rates. For example, transmission levels between adjacent hours must maintain relative stability to ensure grid security.

2.1.3. Load Provinces and Market Environment

Eastern and southeastern coastal provinces form the primary load centers, where electricity demand consistently exceeds local supply. Provincial market mechanisms govern electricity transactions between IPGs and load provinces. China’s electricity markets operate at the provincial level, resulting in functionally independent market systems. Each load province maintains its own forward and spot markets with distinct operational rules.

(1)

Forward Market

Following international conventions, China’s forward markets enable price stabilization through advance contracts. IPGs participate in yearly and monthly markets, establishing bilateral agreements that specify monthly delivery quantities and prices to designated provinces. These contracts allow pre-delivery quantity adjustments to accommodate renewable generation uncertainties. However, IPGs’ large-scale generation capacity and competitive clean energy pricing create cross-provincial competition risks. To address this, the Chinese government mandates allocation ratios through centralized coordination, ensuring predetermined power distribution proportions among load provinces.

(2)

Spot Market

Spot markets manage supply–demand imbalances through three sequential stages: day-ahead, intraday, and real-time markets. Current provincial implementations primarily focus on day-ahead markets. These markets employ hourly timescales for auction-based clearing mechanisms. Although transaction volumes in spot markets are smaller compared to forward markets, they serve crucial roles in balancing short-term mismatches.

2.2. Modeling Challenges

Developing an interprovincial multi-timescale bidding model for IPGs presents two critical challenges, as outlined below.

(1)

Hydropower Nonconvexity

Forward market participation necessitates monthly contract adjustments based on updated forecasts. For IPGs dominated by cascade hydropower, these financial adjustments must be coordinated with physical hydropower operations. The nonlinear nature of hydropower generation functions introduces nonconvex optimization complexities that complicate coordinated scheduling.

(2)

Temporal Scale Conflict

Market mechanisms require IPGs to make hourly decisions in spot markets versus monthly decisions in forward markets, creating fundamental temporal conflicts. Hydropower scheduling typically follows annual cycles with hourly resolution, resulting in 24 × 365 = 8760 decision variables. When combined with nonlinear constraints and stochastic variables, this creates computationally intractable models. Conversely, monthly resolution strategies essentially nullify spot market functions.

To overcome these challenges, this paper proposes two methodological innovations: the Local Generation Function Approximation Method, and the Average-Day Dimensionality Reduction Method.

2.3. Local Generation Function Approximation Method

The nonconvex characteristics of hydropower originate from nonlinear generation functions. Conventional approaches employ MILP to transform these nonlinear constraints into piecewise linear approximations [31,32]. This process introduces integer decision variables that significantly increase computational efficiency compared to linear programming. More precise nonlinear approximations require more segments, further slowing computation speeds. With numerous decision intervals, as in this paper, conventional methods become computationally prohibitive.

To reduce segmentation while maintaining accuracy, this paper proposes the Local Generation Function Approximation Method. This approach constrains water level ranges using historical operational data, thereby minimizing integer variable requirements. As shown in Figure 2, traditional MILP modeling divides the entire water level range (from dead water level to normal high water level) into multiple segments. The proposed method narrows the feasible water level range based on historical operation patterns, enabling accurate two-segment approximations with single-integer variables.

2.4. Average-Day Dimensionality Reduction Method

To resolve temporal scale conflicts between forward and spot markets, this paper proposes a dimensionality reduction method based on the average day (see Figure 3). The concept of an “average day” constructs a representative daily profile that captures monthly average conditions: the hydropower inflow and discharge equal their monthly averages, while the renewable generation outputs represent monthly mean values. Decisions made for this hypothetical day approximate monthly forward market decisions through daily-to-monthly scaling factors (e.g., 30-day multipliers). This method reduces monthly decision variables from 8760 hourly intervals to 24 representative hours, achieving dimensional consistency between forward and spot markets while dramatically decreasing computational complexity.

Notably, using monthly average inflow and generation parameters is sufficient for hydropower scheduling and represents a common practice in the field. For wind and solar power, the average-day approach captures their long-term generation patterns, while stochastic scenarios in spot markets address short-term variability.

2.5. Model Elements

The proposed bidding model incorporates three key decision variables: reservoir levels at the end of each month, hourly hydropower generation schedules for average days, and hourly bidding quantities in both markets. Operational constraints include water balance equations connecting monthly inflows, discharges, and storage levels; hydropower plant capacity and ramping limits; and DC transmission capacity restrictions. The optimization objective, detailed in Section 3, maximizes expected profits while ensuring revenue stability across temporal and spatial dimensions.

3. The Proposed Bidding Model

A bidding model for IPGs operating in cross-provincial multi-timescale markets is established in this section. The model coordinates spot market bidding with forward contract adjustments and monthly reservoir level management. The proposed Average-Day Dimensionality Reduction Method couples decision variables between forward and spot markets. The proposed model also takes interprovincial transmission constraints into account. Given the stochastic nature of renewable generation (wind/solar) and spot prices, the proposed framework adopts stochastic optimization, in which uncertainties are captured through different scenarios. Each scenario contains renewable output and spot price realizations. Because forward market prices are predetermined, they introduce no additional uncertainty.

3.1. Objective

The IPG seeks to maximize total revenue across electricity markets, as formulated in Equation (1). In this equation, $\xi$ denotes the scenario representing uncertainty, while the multiplier ${\pi }_{\xi }$ indicates the probability of scenario $\xi$. The first term inside the brackets represents the expected revenue of the IPG in the forward market under scenario $\xi$, while the second term corresponds to the expected revenue in the spot market; thus, Equation (1) expresses the weighted expected revenue of the IPG across different scenarios. It should be noted that forward market prices are assumed to be known, while spot market prices are based on forecasts. The IPG is treated as a price-taker, implying that it cannot influence market prices and that demand in the load provinces is sufficiently high to fully absorb its supply.

$$maxR=\sum _{\xi \in \Xi }{\pi }_{\xi }\left[\sum _{m\in \mathbb{M}}\sum _{t\in \mathbb{T}}\sum _{k\in \mathbb{K}}{P}_{\xi ,k,m,t}^{\mathrm{fw}}\Delta t·{\lambda }_{k,m}^{\mathrm{fw}}+\left(P_{\xi ,k,m,t}^{\mathrm{sum}}-P_{\xi ,k,m,t}^{\mathrm{fw}}\right)\Delta t·{\lambda }_{\xi ,k,m,t}^{\mathrm{spot}}\right]$$

(1)

The total generation output $P_{\xi ,m,t}^{\mathrm{sum}}$ at hour t of the representative average day in month m under scenario $\xi$ aggregates contributions from all generation sources (Equation (2)), which equals the total power delivered to all provinces (Equation (3)).

$$P_{\xi ,m,t}^{\mathrm{sum}}=P_{\xi ,m,t}^{\mathrm{hydro}}+P_{\xi ,m,t}^{\mathrm{wind}}+P_{\xi ,m,t}^{\mathrm{solar}}$$

(2)

$$P_{\xi ,m,t}^{\mathrm{sum}}=\sum _{k\in \mathbb{K}}{P}_{\xi ,k,m,t}^{\mathrm{sum}}$$

(3)

It is important to note that in electricity markets, all participants—including both the generation and consumption sides—follow a unified standard for electricity measurement; for instance, in China the electricity delivered to the grid by power generators serves as the standardized commodity, represented by $P_{\xi ,m,t}^{\mathrm{sum}}$ in Equation (3). Under this mechanism, transmission losses (e.g., approximately 5% for 1000 kV lines [34]) are handled by the market operator during the settlement stage, and as such do not necessarily need to be considered by market participants during the bidding process.

The aggregated output of the hydropower cascade ($P_{\xi ,m,t}^{\mathrm{hydro}}$) in Equation (2) is defined by Equation (4).

$$P_{\xi ,m,t}^{\mathrm{hydro}}=\sum _{i\in \mathbb{I}}{P}_{\xi ,i,m,t}^{\mathrm{hydro}}$$

(4)

3.2. Hydropower Constraints

3.2.1. Water Balance Equation

The water balance relationship is governed by Equation (5), with generation discharge defined in Equation (6). The coefficient $T_m$, representing the number of days in month m, physically converts daily discharge operations (via average-day representation) to the monthly water storage calculations in Equation (5), as established in Section 2.4.

$$V_{i,m+1}=V_{i,m}+\left(I_{\xi ,i,m}+\sum _{u\in {\mathbb{U}}_i}{Q}_{\xi ,u,m}-Q_{\xi ,i,m}\right)\Delta t·T_m$$

(5)

$$Q_{\xi ,i,m}=G_{\xi ,i,m}+S_{\xi ,i,m}$$

(6)

3.2.2. Power Generation Function

The power generation function is formulated in Equations (7) and (8). As detailed in Section 2.3, the proposed Local Generation Function Approximation Method constrains water level ranges using historical operational data. This permits two-segment linearization with midpoint division, where $n_{i,m,t}$ indicates the active segment. The water discharge rate of hydropower plants differs across segments; Equation (7) specifies the calculation method of the generation function, while Equation (8) determines the water level segment (upper/lower relative to the division threshold) for decision variable $Z_{i,m,t}$.

$$P_{\xi ,i,m,t}·\left[n_{i,m,t}·{\epsilon }_{i,m}^{\mathrm{above}}+(1-n_{i,m,t})·{\epsilon }_{i,m}^{\mathrm{under}}\right]=G_{\xi ,i,m,t}$$

(7)

$$-M·(1-n_{i,m,t})\le Z_{i,m,t}-Z_{i,m}^{\mathrm{dl}}\le M·n_{i,m,t}$$

(8)

3.2.3. Water Storage

Equation (9) provides the piecewise model of the relationship between water level and storage.

$$V_{i,m,t}=\left[n_{i,m,t}·a_{i,m}^{\mathrm{above}}+(1-n_{i,m,t})·a_{i,m}^{\mathrm{under}}\right]Z_{i,m,t}+\left[n_{i,m,t}·b_{i,m}^{\mathrm{above}}+(1-n_{i,m,t})·b_{i,m}^{\mathrm{under}}\right]$$

(9)

3.2.4. Boundary Conditions and Constraints

Equations (10) and (11) respectively limit the ranges of outflows and generation flows. Equation (12) provides the feasible range of water levels, which varies monthly. Equations (13) and (14) define the preset initial and final water levels. Finally, Equations (15) and (16) represent the ramp-up and ramp-down constraints for hydropower in spot markets.

$$Q_i^{\min }\le Q_{\xi ,i,m}\le Q_i^{\max }$$

(10)

$$G_{i,m}^{\min }\le G_{\xi ,i,m}\le G_{i,m}^{\max }$$

(11)

$$Z_{i,m}^{\min }\le Z_{i,m,t}\le Z_{i,m}^{\max }$$

(12)

$$Z_{i,1}=Z_i^{\mathrm{initial}}\forall i\in \mathbb{I}$$

(13)

$$Z_{i,\left|\mathbb{M}\right|+1}=Z_i^{\mathrm{final}}\forall i\in \mathbb{I}$$

(14)

$$P_{\xi ,i,m,t}^{\mathrm{hydro}}-P_{\xi ,i,m,t-1}^{\mathrm{hydro}}\le r^{\mathrm{up}}$$

(15)

$$P_{\xi ,i,m,t-1}^{\mathrm{hydro}}-P_{\xi ,i,m,t}^{\mathrm{hydro}}\le r^{\mathrm{down}}$$

(16)

3.3. Forward–Spot Market Linkage

As per Section 2.4, monthly average days are used to bridge forward and spot market decisions through scaling factors (Equation (17)).

$$P_{\xi ,i,m}^{\mathrm{hydro}}·T_m=\sum _{t\in \mathbb{T}}{P}_{\xi ,i,m,t}^{\mathrm{hydro}}$$

(17)

3.4. Inter-Provincial Power Flow Constraints

To prevent interprovincial conflicts, pre-negotiated allocation ratios (${\eta }_k$) are enforced through Equation (18). Equation (19) ensures that forward contracts maintain a sufficient proportion (minimum ${\mu }_k$) of total generation for revenue stability (see Section 2.1.3).

$$P_{\xi ,k,m,t}^{\mathrm{sum}}={\eta }_k·P_{\xi ,m,t}^{\mathrm{sum}}$$

(18)

$$\sum _{t\in \mathbb{T}}{P}_{\xi ,k,m,t}^{\mathrm{fw}}\ge {\mu }_k·\sum _{t\in \mathbb{T}}{P}_{\xi ,k,m,t}^{\mathrm{sum}}$$

(19)

3.5. Transmission Network Constraints

Equation (20) enforces transmission capacity limits. For DC tie-lines, Equations (21)–(25) regulate the power adjustment amplitudes and frequency.

$$L_k^{\min }\le P_{\xi ,k,m,t}^{\mathrm{sum}}\le L_k^{\max }$$

(20)

$$P_{\xi ,k,m,t}^{\mathrm{sum}}-P_{\xi ,k,m,t-1}^{\mathrm{sum}}\le \Delta P_k^{\mathrm{up}}{x}_{\xi ,k,m,t}^{\mathrm{up}}$$

(21)

$$P_{\xi ,k,m,t-1}^{\mathrm{sum}}-P_{\xi ,k,m,t}^{\mathrm{sum}}\le \Delta P_k^{\mathrm{down}}{x}_{\xi ,k,m,t}^{\mathrm{down}}$$

(22)

$$x_{\xi ,k,m,t}^{\mathrm{up}}+x_{\xi ,k,m,t}^{\mathrm{down}}\le 1$$

(23)

$$\sum _{t\in \mathbb{T}}{x}_{\xi ,k,m,t}^{\mathrm{up}}\le x_k^{\mathrm{up},\max }$$

(24)

$$\sum _{t\in \mathbb{T}}{x}_{\xi ,k,m,t}^{\mathrm{down}}\le x_k^{\mathrm{down},\max }$$

(25)

3.6. Solution Methodology

The proposed model is solvable using commercial solvers such as Gurobi [35]. Because the constraints in Equations (7) and (9) are bilinear, Gurobi reformulates them into a Mixed-Integer Linear Programming (MILP) problem using McCormick envelope convexification, then applies spatial branch-and-bound.

4. Case Study

4.1. Case Settings

In Southwest China, several hydro–wind–solar power generation bases transmit electricity to eastern provinces through the West-to-East Power Transmission project. One of these bases is selected as the IPG in this study. It should be noted that this selection does not imply that the proposed model is applicable only to this specific IPG.

The chosen IPG comprises four large-scale cascade hydropower plants and their co-located wind/solar farms, which deliver electricity to two electricity-intensive load provinces in eastern China. The selected case utilizes the actual operational parameters of the hydropower plants, including historical inflow data from 2020 and turbine characteristics. The provincial electricity allocation ratio is equally distributed at 1:1 (i.e., ${\eta }_k=0.5$), with transmission capacity constrained to 75% of the IPG’s total generation capacity. Forward electricity contracts are required to maintain a minimum proportion of 70% (i.e., ${\mu }_k=0.7$) in the total contractual agreements.

Uncertainty characterization employs historical operational data from 2016 to 2020. For each uncertainty source—wind power output, solar power output, and spot market prices—100 distinct scenarios are generated. The Cartesian product of these scenarios creates a combinatorial space of 1,000,000 (100 × 100 × 100) potential scenarios. Through Latin hypercube sampling, 1000 representative scenarios are systematically extracted, which are further reduced to five characteristic scenarios by using k-means clustering to capture essential uncertainty patterns. To assess model robustness, the effect of varying the number of characteristic scenarios is further discussed in Section 4.5. Forward market prices are derived from historical data on the load provinces’ electricity markets.

Numerical experiments were conducted on a computer running Windows 11, equipped with an Intel Core i7 processor and 16 GB of RAM. The model was implemented in Python (version 3.10) using the Pyomo package (version 6.9) [36] and solved using Gurobi (version 11.0) [35]. Specifically, with NonConvex=2 enabled in Gurobi, global optimality was achieved through recursive domain partitioning and convex relaxation tightening. This solution framework ensures a balance between computational efficiency and mathematical rigor, making it well suited for the proposed model.

4.2. Bidding Result and Analysis

The effectiveness of the proposed model is validated through multi-timescale analysis of power generation and market participation behaviors.

4.2.1. Daily Complementarity Strategy

Figure 4 demonstrates the diurnal complementarity patterns among different energy sources. Solar power exhibits distinct day–night variation, reaching its maximum output of 5243.54 MW at noon during summer months while dropping to zero at night. Wind power shows an inverse pattern, with higher output during nighttime periods, for instance generating 3238.23 MW at 20:00 versus 2709.71 MW at noon on the average January day. Hydropower actively compensates for renewable variability through flexible adjustments, increasing output to 36,360.53 MW (31.7% of daily generation) during solar power troughs (21:00–24:00 in January) while reducing to 25,740.23 MW during solar peaks (13:00 in July). This operational strategy establishes a coordinated paradigm with solar dominant during daytime, supplemented by wind at nighttime and hydropower for flexibility.

4.2.2. Seasonal Complementarity Strategy

Comparative analysis of the seasonal patterns in Figure 4 reveals distinct operational characteristics. During winter months (particularly January and December), wind power contributes 15.1–26.1% of total generation, with an average output of 2914.59 MW in January, while solar power remains constrained by limited daylight availability (391.91 MW average). Hydropower compensates through evening peak regulation, reaching 36,360.53 MW at 21:00 in January. Summer operation demonstrates solar dominance, with maximum output reaching 5243.54 MW in June, while hydropower provides rapid ramping capacity during afternoon solar decline. Transitional seasons (April and October) exhibit alternating wind–solar dominance facilitated by hydropower’s flexible regulation, achieving daily output fluctuations up to 567.4% in April (see Equation (26)). This seasonal pattern establishes an operational paradigm with a wind-dominated winter and solar-driven summer, with hydropower maintaining continuous flexibility.

Complementarily, the daily output fluctuations rate is calculated as follows:

$$\mathrm{Fluctuation}={\displaystyle \frac{P_{\max }-P_{\min }}{P_{\min }}}\times 100\%$$

(26)

where $P_{\max }$ and $P_{\min }$ denote the maximum and minimum hourly hydropower outputs ($P^{\mathrm{hydro}}$) within a day.

4.2.3. Long-Term Hydropower Scheduling

Figure 5 presents annual cascade hydropower operations characterized by distinct long-term patterns. Total generation peaks at 28,391.81 MW during flood season while decreasing to 8672.56 MW in the December dry season. Despite these seasonal variations, monthly generation transitions remain smooth due to the IPG’s strong regulation capacity. Lower December generation results from dual factors, namely, reduced dry-season inflows and strategic reservoir level maintenance to satisfy annual storage requirements.

Figure 6 displays reservoir level trajectories within engineering-defined operational boundaries (shaded areas). The scheduling demonstrates conventional hydropower management strategies consisting of pre-flood drawdown and post-flood replenishment. This confirms that integrating moderate-scale renewables (wind/solar) with cascade hydropower forms an effective engineering practice, as traditional hydropower operations remain largely unaffected when renewable penetration remains within manageable levels.

The bidding strategy enables multi-level temporal complementarity, with hydropower’s rapid adjustment capability (demonstrated by 567.4% daily fluctuation) serving as the cornerstone for system stability across timescales.

4.3. Interprovincial Market Interactions

Figure 7 details provincial market participation under a characteristic scenario. While maintaining the ∼70% forward contract, the IPG strategically allocates ∼30% generation to spot markets during price peaks (19:00–21:00 in January; 10:00–11:00 and 19:00–21:00 in September). This optimized strategy yields 46.4% total revenue from spot markets despite quantity limitations, demonstrating spot market price premiums averaging twice the forward market rates. These results align with market intuition while providing precise quantitative verification, confirming the model’s capability for cross-provincial multi-timescale bidding.

4.4. Sensitivity Analysis

4.4.1. Market Share Constraints

Figure 8 examines the impact of spot market participation limits on total revenue. When the allowable spot market share increases from 0% to 100% in 10% increments, the results demonstrate near-linear revenue growth. This indicates that expanding spot market participation proportionally enhances profitability—each 10% increase in spot market share yields approximately 4.8% revenue growth. However, the analysis also cautions that despite these potential revenue gains, excessive spot market exposure without robust decision-support tools may introduce significant operational risks.

4.4.2. Transmission Capacity Impact

Figure 9 analyzes the relationship between transmission capacity and revenue generation through parametric simulations. The transmission capacity ratio varies from 25% to 100% of total generation capacity in 6.25% increments, simulating progressive infrastructure expansion. Three distinct operational phases are identified. The constrained phase (25–43.75% capacity ratio) exhibits 47.5% revenue growth (CNY 8.28–12.21B), indicating realization of suppressed transmission demand through capacity expansion. The transitional phase (43.75–75% capacity ratio) shows diminishing marginal returns with 0.38% revenue gain per 6.25% capacity increase, signaling that the system’s optimization limits are being approached. The saturated phase (>75% capacity ratio) demonstrates market demand saturation with stagnated growth (<0.12% per increment).

Two critical infrastructure thresholds emerge: 43.75% capacity marks the economic optimization inflection point, while 75% represents technical saturation. These findings recommend phased capacity expansion, prioritizing investments up to 43.75% capacity ratio prior to conducting cost–benefit analyses for further upgrades; beyond 75% capacity, the marginal 1.03% additional revenue (¥1.26B) cannot justify typical transmission infrastructure costs, emphasizing the importance of economic–technical balance in grid planning.

4.5. Robustness Evaluation

Because the proposed model is a scenario-based stochastic optimization model, its performance and properties are closely related to the number of scenarios used in the analysis. To evaluate the robustness of the model, outcomes under varying numbers of characteristic scenarios are compared. Specifically, the number of characteristic scenarios is set to several representative values ranging from 1 to 15. Among these, the case with only one scenario corresponds to a deterministic setting in which wind power output, solar power output, and spot market prices are treated as single-point forecasts. When the number of scenarios exceeds one, the model accounts for uncertainty within a stochastic framework. To ensure comparability, each setting was tested five times with independently regenerated scenario sets. The results are illustrated in Figure 10.

Figure 10 presents the total revenue of the IPG under different numbers of characteristic scenarios. Each green dot represents the result of a single experiment, while the blue squares and red error bars indicate the average and standard deviation across five experiments, respectively.

It is evident that in the deterministic case (scenario number = 1), the IPG’s revenue exhibits large variability, with significant increases or decreases across different trials. This is because the bidding and scheduling decisions are tailored to a single forecast and may perform poorly when actual conditions deviate from the predicted scenario, a deviation that is generally inevitable.

In contrast, the proposed stochastic optimization model demonstrates greater adaptability to uncertainty. As shown in Figure 10, when the number of characteristic scenarios is greater than or equal to 2, the variability across experiments is significantly reduced and the total revenue becomes more stable. This highlights the advantage of the stochastic approach. Moreover, as the number of scenarios increases, the total revenue remains relatively consistent and the standard deviation across trials gradually decreases. These results indicate that the proposed model is robust to future uncertainty. Considering that more scenarios lead to longer computation times, selecting a moderate number of scenarios offers a practical tradeoff between accuracy and efficiency.

5. More Discussion on the Model

While the core proposal of this paper has been presented in detail, we believe that there are two additional aspects of the proposed model that merit further discussion.

5.1. Model Applicability in Different Regions

Unlike conventional commodity markets, electricity markets are strongly influenced by local energy resources, socioeconomic conditions, infrastructure development, and policy environments; consequently, each regional electricity market exhibits distinct characteristics. Nevertheless, due to shared theoretical foundations and broadly similar energy technologies, electricity markets across different regions also demonstrate significant structural commonalities.

Although the proposed model has been developed in the context of China, it has strong potential for application in other regional markets. For example, the European electricity market—an integrated system composed of multiple regional markets—can be mapped to the framework presented here, where the “provinces” in the model correspond to “bidding zones.” The inter-provincial power flow constraints are analogous to the Net Transfer Capacities (NTCs) that limit exchanges between bidding zones in Europe. Moreover, the logic of multi-timescale bidding coordination—including temporal coupling and uncertainty modeling—remains applicable. The primary structural distinction lies in Europe’s reliance on day-ahead and intraday markets [37,38,39].

In Latin America—particularly in Brazil, where hydropower dominates electricity generation—electricity trading operates through two parallel systems: the regulated (captive) market and the free market. These market types differ in terms of risk, regulatory control, and operational flexibility, requiring coordination across market segments. Additionally, Brazil’s segmentation into four interconnected submarkets (Southeast, Northeast, South, and North) resembles the inter-regional structure of China’s power system, indicating that the proposed model is also adaptable to such market settings [40,41,42,43].

5.2. Modeling on Pricing

In the proposed model, the electricity market under study is assumed to be fully competitive; as such, the IPG is treated as a price-taker, meaning that it does not have the ability to influence market prices.

However, in other scenarios, such as during the early stages of market development or under inter-market transmission congestion, market participants may act as price-makers. In such cases, the bidding model should incorporate a price formation mechanism and treat bidding prices as decision variables [44]. When modeling price formation, the market demand curve is typically represented as a residual demand curve; simultaneously, the bidding (or pricing) model becomes more complex. A common approach is to adopt a bilevel optimization framework in which the lower-level problem describes the market clearing process [45].

6. Conclusions

This paper proposes a coordinated bidding model for integrated hydro-wind–solar systems operating in cross-provincial electricity markets. Empirical validation through a real-world case study in Southwest China yields three fundamental advancements with quantifiable impacts.

First, the proposed Average-Day Dimensionality Reduction Method effectively bridges forward and spot market operations, enabling hydropower to balance significant renewable intermittency. As demonstrated via operational profiles, this approach facilitates distinct complementarity patterns in which solar-dominated daytime generation coordinates with wind-supplemented nighttime output while hydropower provides critical flexibility, as evidenced by daily output fluctuations reaching 567.4% during transitional seasons. This temporal coordination generates substantial revenue enhancement, with 46.4% of total revenue originating from strategic spot market participation using only 30% of generation volumes during peak periods.

Second, our transmission-aware optimization approach resolves spatial conflicts through systematic integration of provincial constraints. Sensitivity analysis reveals critical infrastructure thresholds: capacity expansion below 43.75% yields 47.5% revenue growth, while beyond 75% capacity the marginal gains diminish to just 1.03% per additional investment. These quantifiable thresholds provide actionable benchmarks for grid planning, particularly demonstrating that transmission expansion exceeding 75% capacity becomes economically unjustifiable despite technical feasibility.

Third, our computational framework provides effective solutions to long-standing optimization challenges. The proposed Local Generation Function Approximation Method maintains solution accuracy while reducing computational complexity. When combined with the proposed Average-Day Dimensionality Reduction Method, these techniques enable efficient solving of large-scale problems within practical timeframes. The model also demonstrates robust performance across uncertainty scenarios, significantly reducing revenue volatility compared to deterministic approaches.

These advances provide power utilities, particularly IPG operators, with a validated decision-support tool for navigating complex electricity markets. In practice, the proposed model is typically embedded as the computational core of decision software deployed for daily operations within IPG dispatching departments and marketing divisions. Future work will extend this framework to incorporate detailed renewable generation characteristics as well as to conduct cross-market comparative analysis examining institutional impacts on decision-making behaviors.