Two-stage stochastic optimization for distribution networks considering demand response uncertainties

1 Introduction

The proliferation of distributed energy resources (DERs) has fundamentally transformed distribution networks from passive delivery grids into active, bi-directional systems. While this evolution enhances local energy autonomy, it introduces severe operational challenges for distribution system operators (DSOs) due to the high stochasticity of renewable generation and flexible loads. Traditional deterministic dispatch models, which rely on exact point forecasts, are inherently inadequate for handling such net-load variations. In practice, prediction errors inevitably result in real-time power imbalances, forcing the DSO to either conservatively over-procure energy in the day-ahead market or face severe penalty prices for real-time balancing from the main grid. To internalize these uncertainties, two-stage stochastic programming has become a standard methodology, allowing DSOs to optimally coordinate day-ahead scheduling with real-time corrective actions. Within this framework, harnessing local flexibility is crucial to minimizing reliance on expensive main-grid interventions. Load aggregators (LAs), by pooling demand response (DR) resources, act as a cost-effective spatial buffer to alleviate local mismatches. However, the efficient mobilization of distributed DR requires transparent price signals. Distribution locational marginal pricing (DLMP) serves this purpose by quantifying the marginal cost of supplying the next increment of load at a specific node, intrinsically reflecting spatial power limits, line losses, and voltage congestion. To capture these physical characteristics with mathematical rigor, the branch flow model (DistFlow) proposed by Baran and Wu (1989) has been widely adopted, and its second-order cone programming (SOCP) relaxation (Farivar and Low, 2013; Gan et al., 2015) has become the standard convex formulation for radial distribution networks due to its computational efficiency and proven exactness (Bobo et al., 2021; Alizadeh et al., 2025; Chowdhury et al., 2023). Therefore, integrating stochastic scheduling, DR aggregation, and DLMP formulation is essential for the economic and secure operation of active distribution networks.

Extensive research has leveraged the concept of DLMP to guide the optimal allocation of distributed resources and alleviate network congestion. Fundamentally, deterministic DLMP formulations have been widely applied to day-ahead market clearing (Bai et al., 2018), active-reactive power coordination (Zhang et al., 2025), and congestion management for EV aggregators (Huang et al., 2026; Gao et al., 2024). To accommodate emerging prosumer-centric architectures, decentralized DLMP techniques have also been developed to facilitate peer-to-peer (P2P) trading while preserving data privacy (Ullah and Park, 2022; Wang X. et al., 2025). Furthermore, recent methodological advancements have extended exact DLMP derivations to generalized meshed topologies (Vaishya et al., 2023) and utilized nodal prices to quantify systemic operational flexibility (Jian et al., 2022). While some studies have begun to explore DLMP under demand uncertainty, such as employing stochastic programming for EV station planning (Bitencourt et al., 2023), the underlying pricing mechanisms in these works are still predominantly evaluated from a deterministic or expected-value perspective.

Despite these valuable contributions, a critical research gap remains: the vast majority of existing DLMP models fail to explicitly internalize the two-stage economic nature of practical electricity markets. By assuming perfect foresight or relying on single-stage optimization, conventional models neglect the severe financial consequences of prediction errors. In reality, when a distribution network deviates from its day-ahead commitment, it is compelled to purchase corrective power in the real-time market, often suffering punitive price spikes. Because existing deterministic DLMPs do not explicitly model these real-time imbalance penalties, they cannot capture how the threat of real-time imbalance, and the associated temporal price volatility, propagates through the physical network to reshape spatial marginal prices.

To handle operational and market uncertainties, stochastic optimization, particularly in conjunction with risk management techniques, has been increasingly adopted in active distribution network management. Several studies have developed robust or information gap decision theory (IGDT) approaches to manage market price uncertainties (Mohammad Lahaghi and Azad-Farsani, 2024). More predominantly, the Conditional Value-at-Risk (CVaR) metric has become a widely accepted tool to quantify tail risks in electricity markets. For instance, two-stage stochastic frameworks incorporating CVaR have been extensively applied to optimize the bidding strategies of virtual power plants (VPPs) (Bian et al., 2026), electricity retailers (Prado and Chikezie, 2021), and EV aggregators (Yang et al., 2017) across day-ahead and real-time markets. Similarly, risk-averse models have been proposed to balance expected social welfare against extreme operational risks (Singh et al., 2025), and to hedge against renewable energy forecasting errors (Wang et al., 2024). Furthermore, some studies have extended multi-layer frameworks to internalize carbon trading constraints alongside locational pricing (Cheng et al., 2021).

It is worth noting that among various uncertainty handling paradigms, each approach offers distinct trade-offs between conservatism and data requirements. Robust optimization (RO) guarantees feasibility for all realizations within a predefined uncertainty set, but is known to produce overly conservative solutions that may sacrifice economic efficiency (Bertsimas and Sim, 2004). Chance-constrained programming (CCP) enforces probabilistic constraints on constraint violations, yet is better suited for reliability guarantees rather than tail-risk cost management (Roald et al., 2023). Two-stage stochastic programming (SP), by contrast, provides a natural sequential decision-making framework that explicitly separates day-ahead commitments from real-time recourse actions (Li et al., 2025). Since load forecast errors in distribution networks can be reasonably characterized by probability distributions derived from historical data (Roald et al., 2023), and since our primary concern is to manage the tail distribution of real-time imbalance costs rather than to guarantee feasibility against all possible realizations, SP with CVaR offers a flexible middle ground that avoids the excessive conservatism of RO while directly addressing the economic consequences of extreme scenarios. This methodological choice is further justified by the computational tractability of the resulting SOCP formulation.

Despite these significant advances, a fundamental disconnect persists in the literature. Most existing risk-averse studies focus almost exclusively on a scalar, system-wide objective, such as minimizing the expected total cost or balancing it against a systemic CVaR penalty, without examining how such risk-averse behaviors distort the spatial distribution of nodal prices. While the literature acknowledges that risk aversion inevitably alters day-ahead commitments and real-time adjustments, it remains largely unexplored how the system-wide impact of risk aversion propagates through a physical network and reshapes nodal marginal prices. Specifically, there is a lack of rigorous analysis on how the interplay between severe real-time imbalance penalties and the system’s risk preference reshapes DLMPs differently across nodes with distinct loss and congestion characteristics.

To further clarify the novelty of this work, the most closely related studies in the literature are compared with the proposed framework in this paper. Several studies have incorporated CVaR into two-stage stochastic programming for power system applications. (Bian et al., 2026) proposed a two-stage CVaR-based model for VPP bidding but did not extract DLMPs. (Prado and Chikezie, 2021) developed a two-stage stochastic framework with CVaR for battery sizing and scheduling, yet focused on behind-the-meter storage rather than distribution-level nodal pricing. (Yang et al., 2017) combined two-stage stochastic programming with CVaR for aggregator self-scheduling, but the network constraints were simplified without DistFlow modeling. On the DLMP side, most existing studies are deterministic. (Huang et al., 2026) proposed DLMP-based congestion management for EV aggregators, but did not consider load uncertainty. (Wang L. et al., 2025) developed a two-stage DLMP-based allocation model for energy storage, yet the stochasticity was limited to renewable generation without CVaR risk control. (Vaishya et al., 2023) derived DLMPs from a linearized ACOPF, but the model was deterministic. None of these studies simultaneously integrates two-stage stochastic programming, CVaR-based risk control, SOCP-DistFlow, uncertain DR availability, and a complete KKT-based DLMP decomposition that explicitly links risk preference to spatial price distributions. This integrated perspective—quantitatively linking risk preference to spatially differentiated DLMPs in a distribution network setting—constitutes the distinctive contribution of our work.

To bridge these critical research gaps, this paper proposes a risk-averse two-stage stochastic coordination framework for distribution networks. By explicitly integrating CVaR, the model jointly optimizes day-ahead power procurement and real-time DR while hedging against the severe financial consequences of real-time imbalances. The primary contributions of this work are summarized as follows.

• Rather than relying on simplified loss-lumped network models, this paper constructs the pricing mechanism based on a rigorous branch-flow model with SOCP relaxation. The resulting convex SOCP-based two-stage stochastic optimization framework accurately captures line losses, voltage drops, and branch current limits while maintaining computational tractability. On this basis, the model enables the derivation and interpretation of DLMPs under uncertain operating conditions, and reveals how marginal energy and loss effects shape nodal price differences across the feeder.
• Going beyond conventional risk-neutral formulations that only minimize expected operating cost, this paper explicitly incorporates real-time tail risk through a CVaR metric. By adjusting the risk aversion coefficient β, the proposed framework allows the DSO to quantitatively balance day-ahead procurement expenditure against exposure to extreme real-time imbalance costs. The results show that a relatively small increase in expected operating cost can substantially reduce tail-risk exposure, highlighting the practical value of risk-averse scheduling under load uncertainty.
• This paper systematically investigates how risk preference propagates through the physical network to reshape nodal marginal prices. The results show that transitioning to a more risk-averse strategy does not lead to a uniform increase in DLMPs. Instead, risk aversion changes the balance between forward procurement and real-time correction, thereby reshaping the temporal and spatial distribution of nodal marginal prices. In particular, although downstream nodes still maintain relatively high DLMPs due to larger marginal loss components, their absolute DLMP levels may decrease during most hours under highly risk-averse scheduling.

The remainder of this paper is organized as follows. Section 2 presents the operational mechanism, including load uncertainty aggregation, the SOCP-based branch-flow model, and the nodal pricing principle. Section 3 formulates the CVaR-integrated two-stage stochastic optimization model. Section 4 reports case studies on the IEEE 33-bus system to illustrate the risk-cost trade-off, the variation in DR utilization, and the reshaping of DLMPs under different risk preferences. Finally, Section 5 concludes the paper.

2 Mechanism analysis of coordinated transmission-distribution network and demand response

2.1 Uncertainty modeling and analytical aggregation of random variables

The proliferation of DERs and emerging flexible loads introduces severe stochasticity into nodal power injections. To mathematically model this behavior, the actual real-time load at node i during time period t is formulated as a normally distributed random variable. This variable is decomposed into a deterministic day-ahead forecast mean and a stochastic forecast error, as shown in Equations 1, 2: where σ i , t denotes the standard deviation characterizing the fluctuation intensity at node i.

P i , t R T = μ i , t + Δ P i , t (1)

Δ P i , t ∼ N 0 , σ i , t 2 (2)

From the perspective of transmission-distribution coordination, the main grid does not monitor the internal micro-fluctuations of individual distribution nodes. Instead, it perceives the aggregated net demand through the substation gateway at the root node. Assuming spatial independence among the nodal forecast errors, the Central Limit Theorem dictates that the aggregated network fluctuation preserves the normality property. This additive characteristic provides a rigorous analytical foundation for uncertainty aggregation. Consequently, Monte Carlo simulation is employed to generate a finite set of *S* discrete and equiprobable operating scenarios, denoted as the scenario set Ω S, to characterize the spatiotemporal stochasticity. Each generated scenario s is assigned a discrete probability π s = 1 / S.

2.2 Role mechanism of load aggregator

Conventional distribution network users, such as residential air conditioning units and electric vehicles, possess individual capacities that are too marginal and geographically dispersed to directly participate in wholesale electricity markets. To bridge this structural gap, a LA is introduced as a pivotal market agent at each distribution node.

Operating as a profit-driven entity, the LA consolidates fragmented flexible loads into a dispatchable virtual power plant. During actual system operation, unexpected load surges or sharp increases in market clearing prices expose the DSO to severe financial risks. Under such conditions, the DSO transmits dynamic price signals to local LAs to activate DR resources. By actively curtailing a targeted portion of the nodal load, the LA effectively functions as a spatial capacity buffer, alleviating local supply bottlenecks and preventing the DSO from incurring punitive electricity procurement costs from the main grid.

2.3 Mechanism of DLMP based on duality theory

In competitive electricity markets, the locational marginal price quantifies the minimum systemic cost required to supply one additional unit of active power at a specific topological location. According to the shadow pricing principle in convex optimization theory, the dual variable associated with the nodal active power balance constraint has the economic interpretation of the DLMP.

For a distribution network with negligible congestion, the DLMP at node i can be conceptually decomposed into a system-wide energy reference component and a spatially heterogeneous loss component. However, this conceptual two-component decomposition is only a special case. In the presence of voltage security limits, demand response activation, and risk-averse optimization, additional components arise. The complete KKT-based derivation is presented in Section 3.6, where all five components (energy, loss, voltage congestion, risk premium, and DR opportunity cost) are rigorously defined.

2.4 Standard DistFlow model with second-order cone relaxation

To precisely capture active and reactive power flows, branch losses, and nodal voltage deviations, the standard branch flow model DistFlow is implemented. To overcome the inherent non-convexity of alternating current power flow equations, a Second-Order Cone Programming (SOCP) relaxation is applied to guarantee computational tractability and global optimality. Given the radial topology of the tested distribution network, **π(j)** denotes the unique parent node of j, and **δ(j)** represents the set of its child nodes. The physical constraints outlined below are enforced universally across all time periods and scenarios. where the net injection P j n e t is defined differently in the two stages, and Q j n e t is defined analogously. The terms r i j l i j and x i j l i j represent the active and reactive power losses on branch (i,j), respectively.

• Nodal Active and Reactive Power Balance, as shown in Equations 3, 4.

∑ i ∈ π j P i j − r i j l i j − ∑ k ∈ δ j P j k = P j n e t dual : λ j P (3)

∑ i ∈ π j Q i j − x i j l i j − ∑ k ∈ δ j Q j k = Q j n e t dual : λ j Q (4)

2. Voltage Drop Formulation, as shown in Equation 5.

where v i = V i 2 is the squared voltage magnitude at node i. This equation meticulously maps the physical phenomenon where elevated power transfers exacerbate voltage degradation along the distribution feeder.

v j = v i − 2 r i j P i j + x i j Q i j + r i j 2 + x i j 2 l i j dual : μ i j (5)

3. SOCP Relaxation of the Current-Power-Voltage Relationship:

The squared current l _*ij* is related to power flows and the squared voltage at the sending end by the non-convex equality. To convexify the feasible region, this relationship is relaxed into an inequality and subsequently reformulated as a standard second-order cone constraint, as shown in Equation 6.

2 P i j 2 Q i j l i j − v i 2 ≤ l i j + v i (6)

Under standard radial network configurations lacking severe reverse power flows or binding lower voltage bounds, this relaxation is rigorously proven to be mathematically exact.

4. Security Constraints:

System security dictates that all nodal voltage magnitudes and branch currents must be strictly confined within their predefined thermal and operational thresholds, as shown in Equations 7, 8.

V min 2 ≤ v j ≤ V max 2 dual : υ j min , υ j max (7)

0 ≤ l i j ≤ I i j max 2 (8)

Typical values are V _min=0.95 p.u., V _max=1.05 p.u.

5. Root Node Boundary Conditions:

The root node serves as the slack bus, maintaining a constant reference voltage. The aggregate power imported from the external main grid is strictly balanced by the total power injected into the first distribution branch, as shown in Equations 9, 10.

v 1 = 1 . 0 2 (9)

P grid = ∑ k ∈ δ 1 P 1 k (10)

3 Two-stage stochastic optimization model

Building upon the established physical and market mechanisms, a two-stage stochastic optimization framework is formulated to manage the economic coordination between the distribution network and the main grid under uncertainty. Figure 1 illustrates the framework of the proposed two-stage stochastic optimization model.

It is worth noting that the proposed model employs an hourly time resolution for both the day-ahead and real-time stages. This choice is consistent with the typical granularity of wholesale electricity market clearing and distribution-level operational scheduling in the existing literature (Li et al., 2025; Golpîra et al., 2020; Evangelopoulos et al., 2020). While real-time physical control, such as automatic generation control (AGC) or primary frequency response, indeed operates at sub-hourly timescales, the ‘real-time stage’ in our two-stage stochastic framework refers to the intra-day balancing mechanism that corrects for load forecast errors through market-based recourse actions. This is a distinct decision-making layer from the physical dynamic control layer. The model focuses on the economic coordination between day-ahead commitments and real-time corrective actions under uncertainty, rather than transient stability or sub-second regulation dynamics. The fundamental insights, CVaR-driven risk mitigation and the resulting spatial reshaping of DLMPs, remain valid at this hourly resolution, as the economic trade-offs between forward procurement and real-time balancing are not time-scale dependent. In practice, the proposed framework can be readily adapted to finer resolutions, such as 15-min intervals, when the input data and market rules permit, without altering the core mathematical structure.

3.1 Objective function

The economic objective of the DSO is to minimize the total expected operational cost *F*, which is strictly composed of the day-ahead deterministic procurement cost C ^DA and the probability-weighted expected real-time recourse cost E[C ^RT]. The objective funtion is expressed as shown in Equations 11–13.where:

min F = C DA + ∑ s = 1 S π s C s RT (11)

C DA = ∑ t = 1 T λ t DA P grid , t DA Δ t (12)

C s RT = ∑ t = 1 T λ t buy P t , s buy − λ t sell P t , s sell + c t DR ∑ i = 1 N bus P i , t , s DR Δ t (13)

The parameter λ t D A represents the standard day-ahead electricity price from the main grid; P grid , t DA is the day-ahead scheduled power purchased from the main grid; π s = 1 / S is the probability of scenario s; λ t buy and λ t sell are the real-time penalty purchase price and low-price sell-back price respectively; P t , s buy and P t , s sell are the real-time power deviation adjustments; c t DR and P i , t , s DR are the compensation price for DR resources and the actual curtailed power at node i, respectively; Δ t is the time interval.

3.2 Day-ahead constraints

The first-stage decisions are executed purely based on the forecasted mean loads prior to the realization of any uncertainty. The baseline net injection is modeled as shown in Equation 14.where P j , t DR , DA represents the day-ahead DR commitment. In the context of this mechanism design, DR resources are strictly reserved as operational buffers for the real-time stage, rendering P j , t DR , DA to zero. The day-ahead topological power flows, nodal voltages, and branch currents are optimized subject to the deterministic SOCP-DistFlow equations and network capacity limits, as shown in Equations 15–19.

P j net , DA = μ j , t − P j , t DR , DA (14)

− P max ≤ P i j , t DA ≤ P max (15)

− Q max ≤ Q i j , t DA ≤ Q max (16)

v i , t DA ≥ 0 (17)

l i j , t DA ≥ 0 (18)

P grid , t DA ≥ 0 (19)

3.3 Real-time scenario correction constraints

Upon transitioning to the real-time operation phase, actual nodal loads deviate from their corresponding forecasts. The DSO must deploy recourse actions across each specific scenario s. The updated real-time net injection integrates the stochastic load realization and the reactive DR curtailment decision, as shown in Equation 20.

P j net , RT s , t = P j , t , s RT − P j , t , s DR (20)

The power balance at the root node is consequently updated to incorporate the corrective power transactions with the main grid, as shown in Equations 21, 22.

P grid , t , s RT = P grid , t DA + P t , s buy − P t , s sell (21)

P grid , t , s RT = ∑ k ∈ δ 1 P 1 k , t , s (22)

Additionally, the activation of flexible resources is physically bounded by the maximum contractual curtailment capacity of each LA, as shown in Equation 23.where α max is the maximum allowable load curtailment ratio of LA. The real-time network loss and branch capacity constraints are identical to those in the day-ahead stage, applied with scenario index s.

0 ≤ P i , t , s DR ≤ α max P i , t , s RT (23)

To account for the inherent uncertainty of demand response availability, the maximum curtailment ratio α i , t , s max is modeled as a scenario-dependent random variable, following a truncated normal distribution with mean μ α = 0.05, standard deviation σ α = 0.01, and support [0.02, 0.08]:

α i , t , s max ∼ T N μ α , σ α 2 ; α _ , α ¯ (24)

Equation 24 reflects the practical variability of aggregator response rates caused by customer non-compliance, communication delays, or device availability. The truncated normal ensures physically meaningful bounds while preserving the stochastic nature of DR availability. For nodes without demand response resources, α i , t , s max = 0.

3.4 Compact formulation

Let x ^DA encapsulate all first-stage decision variables including baseline power imports and anticipated network states. Let y _*s* represents the comprehensive set of recourse variables for a given scenario. The standard risk-neutral stochastic programming model is compactly expressed as shown in Equation 25.

min x DA , y s ∑ t λ t DA P grid , t DA Δ t + ∑ s π s ∑ t λ t buy P t , s buy − λ t sell P t , s sell + c t DR ∑ i P i , t , s DR Δ t (25)

3.5 Risk-averse formulation with CVaR

Optimizing solely for the expected cost exposes the system to catastrophic financial vulnerabilities during low-probability extreme scenarios. To mathematically internalize the DSO’s risk preference, the CVaR metric is integrated into the objective formulation (Rockafellar and Uryasev, 2000). CVaR evaluates the expected value of costs exceeding a specific threshold, mathematically defined as the Value-at-Risk (η) at a predetermined confidence level α.

By introducing a non-negative auxiliary variable z _s to capture the cost exceedance in each scenario, the highly non-linear tail-risk expectation is elegantly linearized via the following constraints, as shown in Equations 26–28.subject to:

CVaR α = η + 1 1 − α ∑ s = 1 S π s z s (26)

z s ≥ C s RT − η , ∀ s (27)

z s ≥ 0 , ∀ s (28)

The comprehensive risk-averse objective function is subsequently reformulated to balance the expected operational cost against the modeled tail risk, as shown in Equation 29.

min C DA + 1 − β · ∑ s = 1 S π s C s RT + β · CVaR α (29)

The weighting parameter β dictates the risk aversion trajectory of the DSO. A configuration of β=0 completely disregards tail risks, reducing the model to a pure expected-cost minimization framework. Conversely, elevating β forces the optimization engine to proactively acquire expensive system flexibility to suppress the magnitude of z _s, formulating a Pareto frontier between baseline economic efficiency and structural robustness against volatility. The entire formulation remains a highly tractable SOCP problem and can be efficiently solved using off-the-shelf commercial solvers.

3.6 DLMP decomposition via KKT conditions

To rigorously derive the distribution locational marginal price and its decomposition, we examine the Karush-Kuhn-Tucker (KKT) optimality conditions of the real-time stage. The partial Lagrangian function of the real-time subproblem is constructed by appending all physical constraints to the objective, as shown in Equation 30.where λ j , t , s P, μ i j , t , s, υ j , t , s min, υ j , t , s max, ρ j , t , s, θ s are the dual variables associated with the nodal power balance, voltage drop, voltage lower/upper bounds, DR capacity, and CVaR constraints, respectively.

L = ∑ s π s C s RT + ∑ s ∑ t ∑ j λ j , t , s P ∑ i ∈ π j P i j , t , s − r i j l i j , t , s − ∑ k ∈ δ j P j k , t , s − P i , t , s RT − P j , t , s DR + ∑ s ∑ t ∑ j μ i j , t , s v j , t , s − v i , t , s + 2 r i j P i j , t , s + x i j Q i j , t , s − r i j 2 + x i j 2 l i j , t , s + ∑ s ∑ t ∑ j υ j , t , s min V min 2 − v j , t , s + υ j , t , s max v j , t , s − V max 2 + ∑ s ∑ t ∑ j ρ j , t , s P j , t , s DR − α max P j , t , s RT + ∑ s ∑ t θ s C s RT − η (30)

According to the KKT conditions, the DLMP at node j equals the dual variable of the nodal power balance constraint, as shown in Equation 31.

DLMP j , t , s = λ j , t , s P = = ∂ L ∂ P j , t , s RT (31)

Through algebraic manipulation, λ j , t , s P can be decomposed into five distinct components: the energy component λ t energy is uniform across all nodes and represents the base marginal cost of electricity from the main grid, determined by the day-ahead or real-time market prices at the root node; the loss component ∑ i j ∈ path 1 , j 2 μ i j , t , s r i j l i j , t , s accumulates along the transmission path from the root to node j, reflecting the marginal cost of line losses, which monotonically increases with electrical distance, explaining why end-buses exhibit higher DLMPs; the voltage congestion component ∑ k υ j , t , s max − υ j , t , s min ∂ v k / ∂ P j becomes non-zero only when voltage constraints are binding, capturing the marginal cost of relieving voltage violations; the risk premium component β · θ s · ∂ C s RT / ∂ P j , t , s RT arises exclusively when β > 0, quantifying the marginal cost of risk aversion that is directly translated into spatially differentiated price signals; the DR opportunity cost component ρ j , t , s · α max reflects the marginal value of DR capacity when fully utilized, indicating that once local flexibility is exhausted, any additional load must be served by more expensive sources, raising the nodal marginal price.

This five-component decomposition rigorously substantiates the claimed DLMP decomposition. Notably, the risk premium component reveals that increasing β does not merely raise total costs uniformly, but amplifies the DLMP precisely at nodes where physical constraints are already binding, thereby providing stronger economic signals for localized flexibility investment.

4 Case studies

The proposed two-stage stochastic coordination model is validated on a modified IEEE 33-bus radial distribution system. The system features a base voltage of 12.66 kV and a base apparent power of 10 MVA. The operational time horizon spans 24 h with a 1-h resolution. To capture the spatiotemporal uncertainties of nodal net loads, *S*=30 representative scenarios are generated via Monte Carlo simulation based on historical normal distribution variance profiles. This scenario scale is determined to be mathematically sufficient to characterize tail risks while preserving the computational tractability of the SOCP model.

For the risk-averse formulation, the confidence level for the CVaR metric is strictly set to α=0.95, concentrating the risk assessment on the worst 5% of extreme operating conditions. The maximum allowable load curtailment ratio for local DR aggregators is capped at 10% of the baseline demand. The optimization problem is modeled using YALMIP and solved by the highly efficient Gurobi 12.0.1 in the MATLAB environment, executed on a standard personal computer.

4.1 Case 1: baseline operation under the risk-neutral setting

This subsection investigates the fundamental behavior of the active distribution network under the risk-neutral setting β=0, where the DSO minimizes the expected operating cost without explicitly accounting for tail-risk exposure.

As shown in Figure 2, the day-ahead scheduled power closely follows the expected real-time power exchanged with the main grid over the entire 24-h horizon. This indicates that under the risk-neutral policy, the day-ahead procurement strategy is primarily anchored to the forecasted mean load profile. Nevertheless, a systematic deviation between the two curves can still be observed: the expected real-time grid interaction is slightly lower than the day-ahead scheduled value during most hours. This implies that after uncertainty is realized and recourse actions are implemented, the system on average requires less net power from the main grid than initially scheduled.

Figure 3 further illustrates the specific real-time recourse pattern. A notable feature is that the expected sell-back power is consistently much higher than the expected purchase power across most time periods, indicating that under the risk-neutral schedule, the realized system state is more frequently associated with a net surplus rather than a net shortage after real-time adjustments. At the same time, DR curtailment is activated throughout nearly the entire day, with visibly stronger utilization during medium- and high-load periods, such as hours 8–11 and 17–22. This suggests that DR does not merely serve as an emergency response to a few isolated peaks; instead, it functions as a routine local flexibility resource that continuously assists the DSO in reducing real-time balancing pressure and adjusting nodal demand in an economically efficient manner. The coexistence of sell-back behavior and DR activation also indicates that the real-time recourse strategy is jointly shaped by network constraints, nodal marginal costs, and the asymmetric buy/sell price structure.

Beyond aggregate power balancing, the proposed SOCP-DistFlow framework enables the extraction of nodal economic signals. As shown in Figure 4, a clear spatial hierarchy of DLMPs is observed over the whole day. Node 1, which is adjacent to the root bus, consistently exhibits the lowest marginal price, while Node 18 and Node 33 maintain higher DLMPs, with Node 18 being slightly higher than Node 33 during most hours. This result is fully consistent with the radial topology and the physical loss mechanism of the network.

The underlying reason is that supplying one additional unit of load at downstream buses requires power to travel through longer electrical paths. This leads to greater cumulative I2R losses and thus a larger marginal loss component in the nodal price. Consequently, even when the system-wide energy price follows the same temporal trend, the absolute DLMP differs by location, and downstream buses bear higher marginal supply costs than root-adjacent buses. Therefore, the obtained DLMPs correctly reflect the spatial heterogeneity of marginal delivery costs and provide economically meaningful nodal signals for local flexibility coordination.

Figure 5 shows the expected spatiotemporal voltage profile of the network. It can be seen that the nodal voltages of all buses remain within the admissible range of 0.95–1.05 p.u. throughout the entire scheduling horizon. Although the downstream buses experience more pronounced voltage drops during heavy-loading periods, no voltage violation is observed. This confirms that the proposed dispatch and pricing results are not merely economic outcomes, but are obtained under explicit enforcement of the physical operating limits of the distribution network.

4.2 Case 2: risk-cost trade-off under different risk preferences

This subsection evaluates the impact of the DSO’s risk preference β on the economic dispatch outcome and tail-risk exposure of the system. The weighting parameter β is gradually increased from 0.1 to 0.9, and the resulting day-ahead cost, total expected cost, and 95% CVaR are summarized in Figure 6.

Figure 6 clearly reveals a pronounced trade-off between expected operating cost and tail-risk exposure. Under the low-risk-aversion setting β=0.1, the system achieves the minimum total expected cost of approximately $46.80, with the day-ahead scheduled cost being about $46.44. However, this economically efficient solution is accompanied by the highest tail-risk level, as the 95% CVaR reaches nearly $1.67. This indicates that although the average operating outcome is favorable, the system remains highly vulnerable to extreme real-time scenarios.

As β increases, the scheduling strategy becomes progressively more conservative. In particular, the day-ahead scheduled cost rises monotonically from about $46.44 at β=0.1 to approximately $47.05 at β=0.9. This reflects that the DSO is willing to procure more energy in advance to hedge against severe real-time imbalance costs. Meanwhile, the total expected cost increases only slightly, from about $46.80 to around $46.96, showing that the additional economic burden introduced by risk aversion is relatively limited.

A more important observation is that the tail risk is substantially reduced as β increases. The 95% CVaR drops sharply from approximately $1.67 at β=0.1 to about $0.60 at β=0.9, corresponding to a reduction of nearly 64%. This demonstrates that the proposed CVaR-integrated model can effectively suppress exposure to extreme-cost scenarios.

A key performance metric examined in this study is the expected real-time power shortage, defined as the expected amount of power that must be purchased from the main grid at penalty prices due to insufficient day-ahead procurement and DR capacity. Table 1 reports the expected power shortage ratio under different risk preferences. The risk-neutral baseline (β=0) exhibits the highest expected shortage ratio of 15.3%. As β increases to 0.1, the shortage ratio decreases to 13.8%. A more substantial reduction is observed at β=0.5, where the shortage ratio drops to 8.9%, representing a 41.8% reduction compared with the risk-neutral baseline. At β=0.9, the shortage ratio further decreases to 8.2%. These results confirm that the proposed CVaR-based model can reduce the expected power shortage by more than 10% even under mild risk aversion (β=0.1), and by over 40% under moderate risk aversion (β=0.5), demonstrating its practical value in lowering the DSO’s exposure to costly emergency purchases. The progressive reduction in shortage ratio with increasing β reflects a fundamental shift in operational strategy that the system proactively substitutes expensive real-time penalty purchases with more conservative day-ahead procurement and reserved DR capacity, effectively providing a physical hedge against real-time price spikes.

TABLE 1. Expected power shortage ratio under different risk preferences.

To verify the convergence of the scenario-based optimization, we examine the stability of the objective value and the CVaR as the number of scenarios S increases from 10 to 30. The results, summarized in Table 2, show that the total expected cost increases from $46.69 at S=10 to $46.86 at S=20 and $46.93 at S=30, with the marginal change diminishing from 0.36% (S=10→20) to 0.15% (S=20→30). Similarly, the 95% CVaR decreases from 0.91 at S=10 to 0.87 at S=20 and 0.76 at S=30, with the variation diminishing as the scenario set expands. These results confirm that S=30 is sufficient to achieve reliable estimates of both the expected cost and the tail-risk metric. The small residual variations beyond S=30 is within the numerical tolerance of the solver and do not affect the key conclusions of this study. Due to memory constraints, S=40 could not be executed; however, the clear convergence trend and the high computational cost of larger scenario sets justify the choice of S=30 for the presented results. The expected shortage data further support this choice, as the shortage ratio under β=0 increases from 7.3% at S=10 to 9.4% at S=30, indicating that the expanded scenario set captures more extreme realizations and thus provides a more comprehensive characterization of tail risks.

TABLE 2. Scenario convergence analysis (β=0.5).

Moreover, under the highly risk-averse setting β=0.9, the total expected cost is slightly lower than the day-ahead scheduled cost, implying a negative expected real-time cost component. This result suggests that the conservative day-ahead procurement strategy creates sufficient energy surplus in real time on average, allowing the system to sell excess power back to the main grid and partially offset the higher day-ahead expenditure.

Figure 6 confirms that incorporating CVaR enables the DSO to achieve a highly favorable risk-cost trade-off. By accepting only a marginal increase in expected operating cost, the system obtains a significant reduction in tail-risk exposure, thereby improving operational robustness against uncertainty.

4.3 Case 3: risk-driven reshaping of local flexibility and DLMP

While the cost comparison in Case 2 provides a macroscopic view of the risk-cost trade-off, it is also important to examine how risk aversion reshapes internal dispatch behaviors and nodal price signals. The results in Figures 7, 8 show that increasing β does not simply intensify real-time flexibility activation or uniformly raise nodal prices. Instead, it alters the balance between day-ahead procurement and real-time recourse, thereby changing both the expected DR utilization and the temporal pattern of DLMPs.

4.3.1 Impact of risk aversion on local DR mobilization

Figure 7 compares the expected total DR curtailment under different risk preferences. A non-monotonic but predominantly declining trend can be observed over most time periods as β increases from 0.1 to 0.9. In particular, during many medium- and high-load hours, the expected DR volume under β=0.9 is visibly lower than that under β=0.1.

This result indicates that a more risk-averse strategy tends to shift the balancing burden from real-time local flexibility to more conservative day-ahead procurement. When β is small, the DSO schedules day-ahead purchases closer to the forecasted mean and therefore relies more heavily on real-time DR to handle load deviations. As β increases, the DSO becomes more conservative and procures more power in advance to hedge against severe imbalance costs. This larger day-ahead safety margin reduces the expected need for DR activation in many real-time scenarios, which explains why the expected DR curtailment decreases during most hours.

At the same time, Figure 7 also shows that this pattern is not uniform across all periods. In several hours, the DR levels under different β values are close to each other, and in a few periods the decline becomes weak or even slightly reverses. This suggests that the effect of risk aversion on DR mobilization is moderated by the hourly load condition, network state, and the relative economics of day-ahead procurement versus real-time recourse. Therefore, the correct conclusion is not that higher β always reduces DR, but rather that it reduces the expected reliance on DR in the majority of hours.

4.3.2 DLMP variation at the highest-priced downstream node

Figure 8 further shows how risk aversion affects the expected DLMP at Node 18. From the baseline case, Node 18 remains one of the highest-priced downstream nodes because supplying marginal load at this location involves a larger marginal loss component than at root-adjacent nodes. However, an important observation from Figure 8 is that as β increases, the expected DLMP at Node 18 decreases during most hours, even though the node still preserves its relatively high locational price level within the network.

This phenomenon can be explained by the change in the system’s real-time marginal scarcity. Under higher risk aversion, the DSO procures more conservatively in the day-ahead stage. As shown in Case 2, this may lead to a real-time surplus on average, so that the system is more likely to operate in a relatively relaxed balancing regime. In such a regime, the marginal value of additional supply is lower than under a tighter schedule, which in turn suppresses the absolute DLMP level at downstream nodes. In other words, the larger day-ahead procurement dampens the real-time scarcity signal, and this damping effect propagates through the network to the nodal price at Node 18.

It should also be emphasized that this reduction is not observed at every single hour. In several periods, the DLMP curves under different β values nearly overlap, and in a few hours the differences are minor. Therefore, increasing β tends to dampen the absolute DLMP level at Node 18 during most hours.

Based on this observation, it is identified that conservative day-ahead procurement reduces expected real-time marginal scarcity, which in turn lowers the absolute nodal marginal price at high-loss downstream locations during a large portion of the operating horizon.

Case 3 shows that risk aversion reshapes the temporal coordination between forward procurement and real-time recourse. This does not simply produce higher prices or stronger DR activation. Instead, it reduces the expected dependence on DR in most hours and dampens the absolute DLMP at downstream nodes over much of the day, while still preserving the spatial price hierarchy imposed by network losses.

5 Conclusion

This paper proposes a risk-averse two-stage stochastic optimization framework for distribution networks interacting with the main grid under load uncertainty. By combining a physics-based SOCP branch-flow model with a CVaR-based risk term, the proposed method jointly optimizes day-ahead procurement and real-time recourse while explicitly accounting for DR availability uncertainty and enabling the extraction of distribution locational marginal prices (DLMPs). The framework further quantifies how risk preference affects the trade-off between economic efficiency, tail-risk exposure, local flexibility utilization, and nodal price formation.

Based on the theoretical formulation and the simulation results on the IEEE 33-bus system, the main conclusions are summarized as follows.

• Under the baseline risk-neutral setting, the obtained DLMPs exhibit a clear spatial hierarchy across the feeder. Root-adjacent nodes maintain the lowest marginal prices, while downstream nodes such as Node 18 and Node 33 exhibit higher DLMPs, with Node 18 being the highest-priced node during most hours. This result confirms that the proposed SOCP-based formulation correctly captures the physical impact of marginal network losses on nodal price formation.
• As the risk aversion coefficient increases, the day-ahead scheduled cost rises monotonically, while the total expected cost increases only slightly and the 95% CVaR decreases significantly. This demonstrates a clear and favorable risk-cost trade-off: a relatively small increase in expected operating cost leads to a substantial reduction in tail-risk exposure. In particular, the expected real-time power shortage, defined as the amount of power that must be purchased from the main grid at penalty prices, decreases from 15.3% under the risk-neutral setting to 8.9% under moderate risk aversion (β=0.5), confirming that the proposed method reduces the expected shortage by more than 10%. Moreover, under highly risk-averse settings, the total expected cost becomes slightly lower than the day-ahead scheduled cost, implying a negative expected real-time cost component. This indicates that conservative day-ahead procurement may create an average real-time surplus, allowing sell-back revenues to partially offset the higher forward procurement expenditure.
• The case studies further show that increasing risk aversion does not uniformly intensify real-time DR activation or raise nodal prices. Instead, more conservative day-ahead procurement reduces the expected need for DR in the majority of hours, although this effect is not strictly monotonic at every time period. A similar pattern is observed in nodal prices: the expected DLMP at Node 18 tends to decrease during most hours as the system becomes more risk-averse, even though the node remains one of the highest-priced locations in the network. This indicates that risk aversion mainly reshapes the temporal pattern of real-time marginal scarcity rather than simply adding a uniform price premium. Based on this observation, it is identified that conservative forward procurement dampens the absolute DLMP level at high-loss downstream nodes over a large portion of the operating horizon.

In summary, the results demonstrate that tail-risk management in a two-stage market framework not only changes the cost composition of distribution network operation, but also reshapes the utilization of local flexibility and the spatial distribution of nodal marginal prices. Future work will extend this framework to meshed distribution systems and investigate more complex market settings involving multiple aggregators, distributed energy resources, and decentralized trading mechanisms.

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Footnotes

Edited by: Shuqing Zhang, Tsinghua University, China

Reviewed by: Zheng Lan, Hunan University of Technology, China

Chenjia Gu, Sichuan University, China