Flexible resource aggregation two-stage scheduling optimization model considering multiple uncertainties in the market environment

1 Introduction

1.1 Motivation

Driven by the carbon-peaking and carbon-neutrality goals, China’s installed capacity of renewable energy has experienced sustained and rapid growth (Li et al., 2026). By the end of 2025, the cumulative installed capacity of wind and solar power nationwide had exceeded 1,700 GW, accounting for over 45% of total generation capacity (Xie et al., 2026). In several provinces, the instantaneous penetration rate of renewable generation during certain hours has repeatedly reached new highs. The strong randomness and intermittency inherent in wind and solar power output have, with such high penetration, profoundly transformed the power system from a “generation-follows-load” paradigm to a “source-grid-load-storage interaction” mode, significantly increasing both the dimensions and magnitude of operational uncertainty (Li J. et al., 2025). Flexibility has therefore become a core attribute for evaluating a power system’s ability to cope with uncertainty. Nevertheless, current power systems generally suffer from insufficient overall flexible-resource capacity and a lack of resource diversity, making it difficult to effectively accommodate rapidly growing renewable energy (Zhao et al., 2026; Pinheiro et al., 2026). Against this backdrop, the coordinated capacity configuration of multiple types of new flexibility resources—such as distributed generation, electrochemical energy storage, and hydrogen energy storage systems—is of great theoretical and practical significance for ensuring secure and stable system operation in an uncertain environment and for achieving the efficient integration of a high proportion of renewable energy.

1.2 Literature review

Rational planning and configuration of flexibility resources provide a fundamental basis for constructing a new-type power system and ensuring its safe and stable operation. Many scholars have conducted research on the uncertainty characteristics of flexibility resources participating in the new-type power system and on multi-objective collaborative optimization.

Regarding the uncertainty characteristics of flexibility resources participating in the power system, existing research mainly employs methods such as stochastic programming and robust optimization to address the intermittency of wind and solar power output on the source side and load-forecast deviations on the demand side. Meng et al. (2026) pointed out that the large-scale integration of renewable energy sources (RESs) and electric vehicles (EVs), especially under flexible load regulation of EVs, brings significant challenges related to uncertainty, system coordination, and dispatch complexity. Li et al. (2026) selected an optimal time-varying Copula function to characterize the joint uncertainty of wind and solar power and generated sample data to construct a Wasserstein-distance-based ambiguity set. Subsequently, by incorporating linearly adjustable strategies and flexible-resource management, they developed a model and reformulated it into a tractable form using duality theory and convex optimization. Simulation results demonstrated that the CTSDRO algorithm achieves a favorable balance among economic performance, robustness, and computational efficiency. Xie et al. (2026) proposed a hierarchical robust-bound estimation framework for power flexibility and uncertainty within distribution networks and microgrids. Moarref et al. (2025) introduced a three-level hybrid IGDT-robust framework covering wind-power and load uncertainty. Wang H. et al. (2026) presented a distributionally robust model-predictive-control framework for uncertainty from renewable energy sources and electric-vehicle charging. Li Y. et al. (2025) configured soft open points based on PV-load scenarios generated using WGAN-GP and k-means and then optimized the planning of PV, EV charging stations, energy storage, and active-management resources.

Regarding collaborative optimization of multiple flexibility resources, existing studies have established multi-objective frameworks that coordinate economic performance, reliability, and low-carbon characteristics. Ding et al. (2025) proposed a flexibility-oriented bi-level multi-objective model that coordinates day-ahead resource allocation with intraday operation strategies. Wang et al. (2025) optimized the configuration of pumped-storage power plants and other flexible resources in the electricity market. Liu et al. (2025) constructed a two-layer energy-dispatch and collaborative-optimization model for regional integrated energy systems. LiJ. et al. (2026) proposed a dispatch strategy that considers the active-power regulation capabilities of different flexibility resources and coordinated transmission-distribution optimization. Kong et al. (2025) designed an operating-reserve optimization framework for distribution networks based on a master-slave game and heuristic algorithms.

In summary, most existing studies focus only on a single type or a subset of flexibility resources, and few have attempted to integrate distributed generation, electrochemical energy storage, and hydrogen energy storage systems into a unified robust capacity-configuration framework. Furthermore, current methods often fail to fully integrate the quantitative characterization of flexibility demand with two-stage robust capacity configuration, leaving room for improvement in the coordination between economic performance and operational flexibility.

1.3 Our contributions

This study addresses the gap between flexibility-requirement assessment and capacity planning by coordinating distributed generation, battery storage, and hydrogen storage in one robust framework. Its contributions are:

• It derives hourly upward and downward flexibility requirements from feasible net-load intervals rather than from point forecasts alone.
• It embeds these requirements in a two-stage robust capacity-planning model that links first-stage installed capacities to second-stage operating limits.
• It develops a C&CG solution procedure with a KKT-based reformulation of the linear recourse subproblem.
• It evaluates deterministic, budgeted-robust, and full-box robust planning using anonymized provincial-grid data and quantifies the trade-off between cost and uncertainty protection.

1.4 Organization of this paper

The remainder of the paper is organized as follows. Section 2 quantifies upward and downward flexibility requirements from net-load intervals. Section 3 formulates the capacity-planning model for distributed generation, battery storage, and hydrogen storage. Section 4 presents the C&CG solution procedure and the KKT-based subproblem reformulation. Section 5 reports the case study and benchmark comparisons. Section 6 summarizes the findings and limitations.

2 Methodology

System flexibility is affected by both forecast uncertainty and time-varying net load. Controllable distributed generators and storage resources can provide upward or downward adjustments to meet these requirements (Pan et al., 2025). This section uses a common interval-based formulation for flexibility requirements (demand side) and flexibility capabilities (supply side).

2.1 Interval-based characterization of flexibility requirements

Let W t , S t , D t denote the forecasted wind power, solar PV power, and load demand at time t, respectively. Their relative uncertainty bounds are represented by coefficients α , β , γ ϵ [ 0 , 1, calibrated from historical forecast errors. Here, t indexes hourly time intervals, while α , β , γ denote the wind, PV, and load uncertainty radii, respectively. The possible ranges are calculated using the Equations 1–3:

W t max = 1 + α W t ; W t min = 1 − α W t (1)

S t max = 1 + β S t ; S t min = 1 − β S t (2)

D t max = 1 + γ D t ; D t min = 1 − γ D t (3)

The net load N t = D t − W t − S t is therefore uncertain. Its upper and lower bounds are obtained by combining worst-case deviations, as shown the Equations 4, 5:

N ¯ t = D t max − W t min − S t min (4)

N _ t = D t min − W t max − S t max (5)

Conventional methods compute flexibility requirements simply as the difference between consecutive point forecasts, ignoring forecast errors. Instead, we compare the previous realized net load N t − 1 r e a l with the current feasible interval. Given N _ t , N ¯ t, the upward and downward requirements at time t are Equations 6, 7:

R t u p = max N ¯ t − N t − 1 r e a l , 0 (6)

R t d o w n = max N t − 1 r e a l − N _ t , 0 (7)

Three distinct regimes emerge:

• Upward only: N t − 1 r e a l ≤ N _ t → R t d o w n = 0 , R t u p > 0
• Both directions: N _ t ≤ N t − 1 r e a l ≤ N ¯ t → R t u p > 0 , R t d o w n > 0
• Downward only: N t − 1 r e a l ≥ N ¯ t → R t u p = 0 , R t d o w n > 0

This interval-based formulation provides a rigorous, uncertainty-aware quantification of directional flexibility demands.

2.2 Capabilities of new-type flexibility resources

New-type flexibility resources—including controllable distributed generators and various energy storage systems—exhibit fast response and significant adjustment potential. Their ability to provide upward/downward flexibility is modeled below in a form consistent with the requirements defined above.

2.2.1 Controllable distributed generators

Controllable units can adjust their active power output to track net load variations, typically on a 15–60 min scale. For unit i at time t, the upward and downward flexibility provision capacities are limited by the distance to output limits and by ramping constraints over the dispatch interval T, where i indexes controllable distributed generators (Equations 8, 9): where P i , t represents the current output; P i max , P i min represents output limits; ρ i + , ρ i − are ramping rates; and T represents the scheduling interval.

Δ P i , t + = min P i max − P i , t , ρ i + T (8)

Δ P i , t − = min P i , t − P i min , ρ i − T (9)

2.2.2 Energy storage systems

Energy-storage technologies considered in the flexibility analysis include electrochemical batteries, hydrogen energy storage systems, superconducting magnetic energy storage, flywheels, and compressed-air energy storage (Zhang et al., 2024). Hydrogen storage can support both short- and long-duration flexibility, whereas batteries provide high charge/discharge rates for short-duration regulation. Because batteries are constrained by energy duration and cycling limits, hybrid battery-hydrogen configurations can provide complementary flexibility.

For storage unit j at time t, the available upward and downward flexibility is limited by rated charge/discharge power, ramping capability, and the available stored-energy margin. Let P j , t denote storage power, with positive values representing discharge and negative values representing charge. The corresponding power-flexibility limits are Equations 10, 11: where D j max and C j max are the maximum discharge and charge powers; σ j + and σ j − are the upward and downward ramping rates; Q j , t is stored energy; Q j min and Q j max are the energy limits; η j o u t and η j i n are the discharge and charge efficiencies; and Δ t is the dispatch interval. Because the energy margins are divided by Δ t, every term in Equations 10, 11 is a power quantity in MW. The storage flexibility variables are therefore denoted by Δ P j , t + and Δ P j , t − rather than Δ E.

Δ P j , t + = min D j max − P j , t , σ j + Δ t , Q j , t − Q j min η j o u t Δ t − P j , t (10)

Δ P j , t − = min C j max + P j , t , σ j − Δ t , Q j max − Q j , t η j i n Δ t + P j , t (11)

2.2.3 Matching requirements and provisions

Because both flexibility requirements and flexibility provisions are expressed in MW over the same dispatch interval, the planner can directly verify the upward and downward flexibility supply-demand balance. If the available provision is insufficient, additional capacity or external market support is required.

3 Capacity configuration model for new-type flexibility resources under uncertainty

3.1 Two-stage robust optimization framework

To address the impact of forecast uncertainties in wind power, photovoltaic generation, and load demand on capacity planning, this paper adopts a two-stage robust optimization approach (Al-Khafaji et al., 2025). The core idea is that first-stage investment decisions are made before uncertainties are realized, while second-stage operational decisions are made after the actual uncertain parameters are observed. The framework minimizes the comprehensive cost under the worst-case uncertainty scenario, thereby yielding a risk-averse planning solution, as shown in Figure 1. Accordingly, the overall model consistently follows a min–max–min structure: first-stage investment is minimized, the worst-case uncertainty realization is selected, and second-stage operating cost is minimized under that realization.

3.2 Two-stage robust model for flexibility resource capacity configuration

3.2.1 First stage: investment decisions

The first stage determines the rated capacities of three flexibility resource types: controllable distributed generators (DGs), battery storage, and hydrogen energy storage. Unless otherwise specified, power, energy, time, and cost quantities in the model are expressed in MW, MWh, h, and CNY, respectively. The decision vector is Equation 12:where y denotes the first-stage decision vector; P D G rated represents rated power of distributed generators (MW); ε B a t r a t e d represents rated energy capacity of battery storage (MWh); ε H y d r a t e d represents rated storage capacity of hydrogen energy storage system (MWh).

y = P D G r a t e d , ε B a t r a t e d , ε H y d r a t e d Τ (12)

The total annualized investment cost is Equation 13:where r represents annual discount rate (e.g., 5%–8%); L represents project economic lifetime (years); κ D G represents unit investment cost of distributed generators (CNY/MW); κ B a r represents unit investment cost of battery storage (CNY/MWh); κ H y d represents unit investment cost of hydrogen energy storage system (CNY/MWh).

Φ i n v = r 1 + r L 1 + r L − 1 κ D G P D G r a t e d + κ B a r ε B a r r a t e d + κ H y d ε H y d r a t e d (13)

The capacity-bound constraints are Equations 14–16: where P D G min and P D G max denote the lower and upper capacity bounds of controllable distributed generation, while ε B a r min , ε B a r max and ε H y d min , ε H y d max denote the corresponding storage-capacity bounds. A 1-h battery duration is assumed; therefore, when Δ t = 1 h, rated battery power (MW) and rated energy (MWh) are numerically equal in Equations 24, 25. Throughout the model, superscripts min and max denote lower and upper bounds, and rated denotes installed capacity. The operating limits of each aggregate resource are linked explicitly to its first-stage rated capacity.

P D G min ≤ P D G r a t e d ≤ P D G max (14)

ε B a r min ≤ ε B a r r a t e d ≤ ε B a r max (15)

ε H y d min ≤ ε H y d r a t e d ≤ ε H y d max (16)

3.2.2 Second stage: operational decisions

Given a capacity configuration y and a particular uncertainty realization u ∈ U, the system must determine the hourly dispatch of all units to satisfy load balance, device operating constraints, and flexibility supply–demand balance (Wang X. et al., 2026). The operating cost includes fuel costs, power purchase costs, and costs associated with grid interaction.

For each typical-day scenario w, the operational decision vector is Equation 17:where P i , t D G represents output of distributed generation i at time t; P j , t c h , P j , t d i s represents charging and discharging powers of battery i; P h , t e l , P h , t f c represents electrolyzer input and fuel-cell output of the hydrogen storage system; P t b u y , P t s e l l represents power purchased from/sold to the external grid; P t t h represents output of thermal generator; e j , t , e h , t represents stored energy in battery and hydrogen energy storage system. Superscripts ch and dis denote charging and discharging, el and fc denote electrolyzer and fuel-cell operation, and buy and sell denote electricity purchased from and sold to the external grid, respectively.

x ω = P i , t D G P j , t c h , P j , t d i s P h , t e l , P h , t f c P t b u y , P t s e l l P t t h e j , t , e h , t (17)

Operating cost function is Equation 18:where α i D G represents generation cost of distributed generator i (CNY/MWh); α t h represents generation cost of thermal generator (CNY/MWh); α b u y , α s e l l represents electricity purchase and sale prices (CNY/MWh), typically with α b u y > α s e l l; Δ t represents time interval (taken as 1 h); T represents dispatch horizon (24 h).

Φ o p x ω = ∑ t = 1 T ∑ i α i D G P i , t D G + α t h P t t h + α b u y P t b u y − α s e l l P t s e l l · Δ t (18)

When multiple typical days are considered, the total operating cost is the weighted expectation is Equation 19:where π ω represents the weight of typical day ω.

Φ o p t o t a l = ∑ ω π ω Φ o p x ω (19)

Constraint conditions:

3.2.2.1 Power balance constraint

At each time step, total generation must equal net load, as is Equation 20:where L ∼ i n e t represents the actual net load at time t.

∑ i P i , t D G + ∑ j P j , t d i s − P j , t c h + ∑ h P h , t f c − P h , t e l + P t b u y − P t s e l l + P t t h = L ∼ i n e t , ∀ t (20)

3.2.2.2 Grid interaction constraints

Grid interaction constraints as Equations 21-23. where P max b u y , P max s e l l denote the maximum purchase and sale powers (MW), respectively, and β t b u y , β t s e l l denote continuous mode indicators preventing simultaneous purchase and sale. The purchase and sale mode factors are continuous capacity-allocation variables. Under the non-negative tariffs, with the purchase price higher than the sale price, simultaneous buying and selling is economically dominated and does not occur at optimality; therefore, the second-stage recourse remains linear.

0 ≤ P t b u y ≤ β t b u y · P max b u y (21)

0 ≤ P t s e l l ≤ β t s e l l · P max s e l l (22)

β t b u y + β t s e l l ≤ 1 , β t b u y , β t s e l l ∈ 0 , 1 (23)

3.2.2.3 Battery storage constraints

Charge/discharge power limits are Equations 24, 25:

0 ≤ P j , t c h ≤ ε b a t r a t e d (24)

0 ≤ P j , t d i s ≤ ε b a t r a t e d (25)

State-of-charge limits is Equation 26:

E b a t min ≤ e j , t ≤ E b a t max (26)

Energy dynamics is Equation 27:

e j , t = e j , t − 1 + η b a t c h P j , t c h Δ t − P j , t d i s Δ t η b a t d i s (27)

Charge/discharge mutual exclusivity is Equation 28:where E b a t min , E b a t max represents minimum/maximum allowed stored energy (MWh), typically 10%–90% of rated capacity; η b a t c h , η b a t d i s represents charge and discharge efficiencies (0.92/0.91); χ j , t c h , χ j , t d i s denote continuous mode indicators for charging/discharging. For each typical day, the lower and upper battery-energy limits are set to 10% and 90% of rated capacity, respectively, and the terminal stored energy is constrained to equal its initial value. Because round-trip efficiency is below unity and operating costs are non-negative, simultaneous charging and discharging is economically dominated.

χ j , t c h + χ j , t d i s ≤ 1 , χ j , t c h , χ j , t d i s ∈ 0 , 1 (28)

3.2.2.4 Hydrogen energy storage system constraints

A hydrogen energy storage system consists of an electrolyzer, a hydrogen tank, and a fuel cell. Simultaneous electrolyzer and fuel-cell operation is excluded.

Electrolyzer power limit is Equation 29:

0 ≤ P h , t e l ≤ ε h y d r a t e d · ξ e l (29)

Fuel cell power limit is Equation 30:

0 ≤ P h , t f c ≤ ε h y d r a t e d · ξ f c (30)

Energy (equivalent hydrogen) dynamics is Equation 31:

e h , t = e h , t − 1 + η e l P h , t − 1 e l Δ t − P h , t f c Δ t η f c (31)

Storage capacity limits is Equation 32:where ξ e l , ξ f c represents power-to-capacity ratios for electrolyzer and fuel cell; η e l , η f c represents electrolyzer hydrogen production efficiency and fuel cell generation efficiency. Here, h indexes hydrogen-storage units. For each typical day, the terminal hydrogen-storage energy is constrained to equal its initial value. Because the combined electrolyzer and fuel-cell efficiency is below unity and operating costs are non-negative, simultaneous electrolyzer and fuel-cell operation is economically dominated.

E h y d min ≤ e h , t ε h y d r a t e d (32)

3.2.2.5 Distributed generator constraints

Power limits is Equation 33:

P i min ≤ P i , t D G ≤ P i max (33)

Ramping constraints is Equation 34:where denote the downward and upward ramping margins, respectively.

− Δ P i , t − ≤ P i , t D G − P i , t − 1 D G ≤ Δ P i , t + (34)

3.2.2.6 Thermal generator constraints

Power limits is Equation 35:

P t h min ≤ P t t h ≤ P t h max (35)

Ramping constraints is Equation 36:where R t h represents the thermal generator ramping rate (MW/h).

− R t h Δ t ≤ P t t h − P t − 1 t h ≤ R t h Δ t (36)

3.2.2.7 Flexibility supply–demand balance constraints

Upward flexibility is Equation 37:

∑ i Δ P i , t D G , + + ∑ j Δ P j , t B a t , + + ∑ h Δ P h , t H y d , + + Δ P t t h , + ≥ R t u p , ∀ t (37)

Downward flexibility is Equation 38:where R t u p and R t d o w n are the upward and downward flexibility requirements obtained from the interval method. The terms Δ P D G , Δ P B a t , Δ P H y d , Δ P t h denote power-flexibility contributions from distributed generation, battery storage, hydrogen storage, and thermal generation, respectively. Superscripts + and - indicate upward and downward flexibility. Thus, every term in Equations 37, 38 is consistently expressed in MW.

∑ i Δ P i , t D G , − + ∑ j Δ P j , t B a t , − + ∑ h Δ P h , t H y d , − + Δ P t t h , − ≥ R t d o w n , ∀ t (38)

3.2.3 Uncertainty modeling

A budgeted uncertainty set is used to represent net-load forecast deviations. For resource r ∈ W , P V , D, the normalized forecast error is defined as e r , t = x r , t a c t − x r , t f o r max x r , t f o r , ε r. To avoid inflated relative errors near zero output, ε W and ε P V are set to 1% of the corresponding rated renewable capacity, and ε D is set to 1% of peak load.

The 8,760 wind, PV, and load forecast errors are aligned by timestamp before the net-load error series is constructed as Δ N t = Δ D t − Δ W t − Δ S t. This preserves temporal alignment and directional compensation in the calibration samples. No parametric cross-variable correlation structure is imposed in the budgeted uncertainty set. The empirical 95th percentile of e r , t is rounded upward for engineering implementation, yielding α = 15 % for wind, β = 15 % for PV, and γ = 10 % for load. These radii determine deviation magnitude, whereas the robustness budget Γ limits the number of periods that may simultaneously approach adverse bounds.

The actual net load L ∼ i n e t at time t fluctuates around its forecasted value L ^ i n e t is Equation 39.:where N t f o r is the point forecast of net load (MW), Δ L t max is the maximum deviation magnitude, and z t + and z t − are the positive and negative deviation indicators. Positive and negative deviations cannot occur simultaneously in the same period.

L ∼ i n e t = L ^ i n e t + δ t + Δ L t max − δ t − Δ L t max (39)

Total deviation budget constraint is Equation 40:where Γ is the robustness budget. Γ = 0 gives deterministic planning, Γ = T gives the conventional full-box robust case, and 0 < Γ < T protects against adverse deviations in only a subset of the T periods. The percentage radii control deviation magnitude, while Γ controls temporal simultaneity.

∑ t = 1 T δ t + + δ t − ≤ Γ (40)

The uncertainty set U is formally defined as Equation 41:

U = u = δ t + , δ t − δ t + , δ t − ∈ 0 , 1 , δ t + + δ t − ≤ 1 , ∑ t = 1 T δ t + + δ ≤ Γ (41)

3.2.4 Two-stage robust optimization model

Integrating the above components, the complete two-stage robust capacity configuration model is Equation 42:

min y Φ i n v y + max u ∈ U min x ∈ χ y , u ∑ ω π ω Φ o p x ω (42)

4 Solution algorithm

The two-stage robust capacity-planning model is solved by the Column-and-Constraint Generation (C&CG) algorithm (Hu et al., 2025). The method alternates between a master problem and a subproblem, adding cuts generated from newly identified worst-case scenarios until the upper and lower bounds satisfy the stopping tolerance.

4.1 Master problem: investment decisions under given uncertainties

The master problem makes first-stage capacity decisions while assuming that the uncertainty parameters take a set of known worst-case scenarios. Its objective is to minimize the sum of investment cost and an auxiliary variable representing the future worst-case operating cost, subject to all operational constraints.

The compact formulation of the master problem is Equation 43:subject to Equations 44–49: where y represents first-stage capacity variables; η represents auxiliary variable approximating the worst-case expected operating cost in the second stage; l represents the current iteration index, L represents the number of cuts added so far; x ω l represents second-stage operational variables for typical day ω generated at iteration l ; u l represents worst-case uncertainty scenario returned by the subproblem at iteration l ; Φ i n v y represents annualized investment cost; Φ o p x ω l represents operating cost function; H , J , f , K , A o p , d o p , K o p , k o p , ∏ u n c represents coefficient matrices/vectors of appropriate dimensions, corresponding to power balance, device limits, energy dynamics, uncertainty coupling, etc.

min y , η Φ i n v y + η (43)

η ≥ ∑ ω π ω Φ o p x ω l , ∀ l ≤ L (44)

H x ω l + J y ≥ f − K u l , ∀ l ≤ L (45)

A o p x ω l ≥ d o p (46)

K o p x ω l ≥ k o p (47)

Π u n c x ω l = u l (48)

y ∈ Y , x ω l ∈ X , η ∈ R (49)

4.2 Subproblem: identifying the worst-case uncertainty scenario

Given a first-stage decision y * obtained from the master problem, the subproblem seeks the uncertainty scenario u ∈ U that maximizes the expected operating cost. Its mathematical form is Equation 50:

ψ y * = max u ∈ U min x ∈ X y * , u ∑ ω π ω Φ o p x ω (50)

The subproblem has a bilevel max-min structure: the outer problem selects an adverse uncertainty realization, and the inner problem minimizes operating cost under that realization. The inner problem is replaced by its Karush-Kuhn-Tucker (KKT) optimality conditions to obtain a single-level formulation.

Transformation steps:

• Inner-problem linearity: for fixed first-stage capacities and a fixed uncertainty realization, the recourse problem is linear. The mode variables in Equations 23, 28 are continuous on [0,1]. Because the purchase tariff exceeds the sale tariff and round-trip efficiencies are below one, simultaneous buying/selling and charging/discharging are economically dominated at optimality. This supports a KKT-based (equivalently, primal-dual) reformulation.
• Dual variables: Let γ , λ , π be the dual variables associated with the inequality constraints H x + J y ≥ f − K u , A o p x ≥ d o p, and the equality constraints K o p x = k o p, respectively.
• KKT conditions: These include primal feasibility, dual feasibility, complementary slackness, and stationarity.

After introducing the KKT conditions and big-M linearization of the complementary constraints, the subproblem is written as the following single-level mixed-integer linear program is Equation 51:

max u ∈ U , x , γ , λ , π ∑ ω π ω Φ o p x ω (51)

4.3 Solution process

Based on the master problem and subproblem defined above, the complete C&CG solution process is shown in Figure 2.

5 Case study

5.1 Basic data

The case study uses one anonymized operating year with 8,760 synchronized hourly forecast and realized observations from a provincial grid in Northwest China. K-means clustering is applied to the hourly net-load and renewable profiles to obtain three typical days representing spring/autumn, summer, and winter, as shown in Figure 3. Forecast errors are paired by timestamp, normalized using the denominator floors defined in Section 3.2.3, and calibrated from empirical 95th percentiles. The resulting engineering radii are 15% for wind, 15% for PV, and 10% for load, as shown in Table 1. Timestamp alignment is retained when the net-load error is constructed, but the budgeted uncertainty set does not represent a full joint probability distribution. The radii control deviation magnitude, while Γ controls the number of adverse periods in Equations 40, 41.

TABLE 1. Statistical basis and calibration of uncertainty radii.

In terms of system economic parameters, the economic lifetime of the project is set to 20 years, and the annual discount rate is set to 7%. The main parameters of the distributed generator, electrochemical storage, and hydrogen energy storage system are shown in Figure 4.

5.2 Algorithm validation and convergence analysis

Typical Day 3 is used as the convergence test case with Γ = 12 and fluctuation coefficient δ = 0.10. The stopping tolerance is 1%. At iteration k, the candidate upper bound is generated by the newly identified adverse scenario, whereas the incumbent upper bound is the best upper bound found through iteration k. The convergence gap is calculated from the incumbent upper bound as U B i n c − L B max 1 , U B i n c.

Figure 5 shows that the C&CG algorithm terminates after three iterations. The candidate upper bounds are 2.235, 2.234, and 2.243 million CNY/a; the incumbent upper bounds are 2.235, 2.234, and 2.234 million CNY/a; and the lower bounds are 1.890, 2.210, and 2.230 million CNY/a. Although the third candidate upper bound increases slightly, the incumbent upper bound remains 2.234 million CNY/a. The incumbent relative gaps are 15.44%, 1.07%, and 0.18%, so the final gap is below the 1% tolerance. The model was implemented in MATLAB R2022b with YALMIP R20230622 and Gurobi Optimizer 10.0.3 on a 64-bit Windows 11 workstation with an Intel Core i7-12700H CPU (2.30 GHz) and 16 GB RAM. Total wall-clock time was approximately 20 s as shown in Table 2.

TABLE 2. C&CG convergence history (million CNY/a).

5.3 Comparison with benchmark models

5.3.1 Comparison with deterministic and conventional robust planning

Figure 6 shows the dispatch schedule obtained from the budgeted-robust solution (Γ = 12, δ = 0.10). The capacity and cost comparison with deterministic and full-box robust planning is reported in Table 3.

TABLE 3. Comparison of deterministic, budgeted-robust, and full-box planning.

Relative to deterministic planning, the budgeted-robust solution at Γ = 12 increases distributed-generation and battery-storage capacities by 13.8% and 11.2%, respectively, while hydrogen-storage capacity changes only slightly. Planning cost increases by 7.91%, and comprehensive cost increases by approximately 16.4%. This difference is the cost of protection against adverse net-load deviations.

A conventional full-box robust benchmark is obtained by setting Γ = 24, so all hourly deviations may simultaneously reach their bounds. For a controlled comparison, the deterministic, proposed budgeted-robust, and full-box robust models use identical resource models, costs, and operating constraints; only the uncertainty treatment differs. At δ = 0.10, their comprehensive costs are approximately 1.925 × 10⁶, 2.240 × 10⁶, and 2.325 × 10⁶ CNY/a, respectively. The proposed solution is approximately 3.7% less costly than the full-box solution while protecting against adverse deviations in up to 12 of the 24 periods. This result demonstrates reduced conservatism relative to conventional box robustness; it does not imply universal superiority over stochastic or distributionally robust models, which rely on different probabilistic information.

5.3.2 Operational characteristics under typical scenarios

Battery and hydrogen storage provide different forms of flexibility. Battery storage follows a valley-charging and peak-discharging pattern, absorbing surplus renewable power around 03:00–04:00 and 21:00. Its energy-duration, cycling, and depth-of-discharge limits make hydrogen storage useful during sustained or extreme imbalances.

At night, thermal units remain above their minimum technical output, so both storage technologies charge to absorb surplus renewable generation and reduce curtailment. During high-net-load daytime periods, they discharge to relieve peak supply pressure.

Furthermore, constrained by both current cost levels and energy conversion efficiency, the installed capacity of hydrogen energy storage is relatively limited at this stage. Nevertheless, with continuous breakthroughs and maturation in technologies such as water electrolysis for hydrogen production, hydrogen fuel cell power generation, and related materials, the economic viability and technical feasibility of hydrogen energy storage are expected to steadily improve, and its future application prospects in power systems will become even broader.

5.4 Robustness analysis

5.4.1 Impact of robustness budget Γ

Under the baseline uncertainty radii of 15% for wind, 15% for PV, and 10% for load, Γ = 6 and Γ = 12 are compared at δ = 0.10. The corresponding capacities, planning costs, and comprehensive costs are shown in Figure 7.

As shown in Figure 7, increasing Γ from 6 to 12 expands the number of adverse net-load periods covered by the model. Distributed-generation and battery-storage capacities increase to provide additional upward flexibility, whereas hydrogen-storage capacity changes more moderately because it primarily supports sustained and extreme events.

5.4.2 Joint impact of Γ and δ

Figure 8 presents the capacity-configuration results for different combinations of Γ and fluctuation coefficient δ.

To quantify the economic effect of the robustness budget, Figure 9 presents comprehensive cost as Γ varies from 0 to 24 at δ = 0.10.

As shown in Figure 9, comprehensive cost increases monotonically with Γ, while the marginal increase declines as the solution approaches the fully conservative full-box case.

Taken together, Figures 8, 9 demonstrate the trade-off between robustness and economy: larger Γ and δ protect against more severe uncertainty realizations but require greater flexibility-resource investment.

5.5 Worst-case scenario dispatch analysis

5.5.1 Net load deviations and system dispatch results

Under the worst-case uncertainty realization, Figure 10 shows the lower bound, upper bound, and selected net-load realization, while Figure 11 shows the corresponding dispatch of generation, storage, and grid-exchange resources.

Figures 10, 11 show that positive net-load deviations occur mainly during the daytime peaks of 06:00–12:00 and 16:00–19:00, when the selected realization reaches the upper uncertainty bound. At 24:00, the selected realization falls below the forecast and produces a negative deviation.

The two daytime intervals require the most upward flexibility. Because the thermal generator ramps more slowly, distributed generation supplies most of the upward adjustment in these periods.

5.5.2 Complementary operation of battery and hydrogen storage

During the 12:00–14:00 high-price interval, battery discharge is used before additional grid purchases. By about 17:00, the battery reaches its minimum state of charge, after which hydrogen storage supplies the remaining deficit.

This sequence illustrates the complementary roles of the two storage technologies: the battery responds to short-duration high-price deficits, and hydrogen storage provides support after the battery energy limit becomes binding.

6 Conclusion

This study developed a two-stage robust capacity-planning model for distributed generation, battery storage, and hydrogen storage under wind, PV, and load uncertainty. The main findings are:

• Net-load intervals provide explicit hourly upward and downward flexibility requirements. Including these requirements in the planning model links capacity selection to operational flexibility.
• The planning model follows a min-max-min structure: first-stage decisions minimize annualized investment cost, the uncertainty set selects an adverse realization, and second-stage recourse minimizes operating cost. At Γ = 12, the optimal capacities are 2,472.30 MW for distributed generation, 852.40 MW for battery storage, and 446.20 MW for hydrogen energy storage.
• The dispatch results distinguish the roles of the two storage technologies. Battery storage responds to short-duration imbalances and high-price periods, whereas hydrogen storage provides support during longer deficits or after the battery reaches its energy limit.
• Relative to deterministic planning, the budgeted-robust solution increases distributed-generation capacity by 13.8% and battery-storage capacity by 11.2%, while comprehensive cost increases by approximately 16.4%. It is approximately 3.7% less costly than the full-box robust solution, indicating lower conservatism for the tested uncertainty set.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Zipeng Liang, Hong Kong Polytechnic University, SAR China

Reviewed by: Wei Wang, Ministry of Water Resources, China

Jinfang Zhang, North China Electric Power University, China