Resilience-Constrained Low-Carbon Dispatch of Industrial Parks with Storage and Quantum Acceleration

Abstract

Carbon-neutral industrial parks require large consumers, such as data centers, to balance low-carbon operation and service reliability. This paper proposes a resilience-constrained stochastic dispatch framework for a data-center virtual power plant (VPP) with renewable generation, short-duration batteries, and long-duration storage units. The dispatch is formulated as a two-stage stochastic program with normal and outage scenarios. To solve the resulting large mixed-integer problem, we develop a hybrid quantum–classical L-shaped method: the integer master is solved heuristically by quantum annealing, while scenario subproblems are solved exactly by classical optimization. In a case study based on real-world industrial-park data, the proposed storage strategy eliminates critical load shedding for the tested 6 h outage scenarios with a 3.7% increase in expected daily cost. The QA-driven method reaches the same best-known objective as the classical baseline with an empirical 1.36× runtime speedup.

1. Introduction

The transition toward carbon-neutral industrial parks (CNIPs) creates a complex energy management environment. Within this framework, AI data centers present a unique operational challenge: they consume massive amounts of power while maintaining a zero-tolerance policy for service interruptions [1]. To fulfill carbon-neutral mandates without compromising reliability, these parks are integrating decentralized energy resources. Specifically, the synergy between short-duration battery storage and long-duration storage units provides a viable pathway to eliminate reliance on carbon-intensive backup power, though it complicates the daily dispatch logic [2]. Within this paradigm, the Virtual Power Plant (VPP) has emerged as a vital framework for coordinating and optimizing these assets as a unified entity [3].

1.1. The Core Challenge: Duality of Operational Risk and Computational Complexity

The core dispatch challenge has two parts. First, there is a direct conflict between economic dispatch and resilience. Economic dispatch minimizes expected cost under likely conditions such as routine price and renewable fluctuations [4]. Resilience planning, in contrast, prepares for low-probability but high-impact events such as multi-hour grid outages [5]. If the optimization focuses only on expected daily cost, it tends to under-value emergency preparedness.

Second, adding resilience usually increases model complexity. A two-stage stochastic dispatch model with unit commitment decisions, inter-temporal storage dynamics, and many scenarios leads to a large MILP. The problem size grows with assets, time periods, and scenarios, and binary commitment variables make the master problem NP-hard [6]. This motivates decomposition and acceleration methods.

1.2. The Strategic Opportunity: Constraint-Integrated Hedging and Hybrid Computation

The first opportunity is to define resilience explicitly in the model instead of relying on ad hoc operating rules. In this work, resilience is implemented through a two-layer structure. The first layer is a set of hard constraints: a protected storage reserve floor and an outage-islanding requirement. The second layer is a two-stage stochastic program that optimizes the remaining flexibility across normal and outage scenarios. This structure guarantees a minimum resilience level while still adapting dispatch decisions to scenario uncertainty.

The second opportunity is computational. Solving the full model exactly can be slow because the integer master is hard. QA does not provide optimality certificates, but it can quickly generate high-quality binary candidates for difficult combinatorial searches. We therefore combine QA with a classical L-shaped loop: QA proposes first-stage candidates, and classical solvers evaluate subproblems and generate valid Benders cuts.

1.3. Literature Review

Our proposed framework builds upon and contributes to three established streams of research.

The first stream is the economic dispatch of VPPs using stochastic programming. Evolving from deterministic unit commitment, two-stage stochastic programming has become the canonical method for optimizing VPP schedules under uncertainty in market prices, load, and renewable generation [7,8]. More advanced multi-stage models have also been developed to better capture the sequential nature of decisions [9]. While mathematically rigorous, the objective functions in this research stream are, by design, focused on minimizing expected costs under a set of probable scenarios. They are fundamentally blind to extreme, out-of-sample events and therefore do not endogenously value or create resilience.

A second stream focuses on power-system resilience for critical loads. It includes resilience metrics [10] and risk-averse formulations such as robust optimization [11]. A common practice is to impose fixed reserve policies (for example, minimum storage SOC). These rules improve survivability but can be economically conservative because they do not adapt to scenario conditions. A related coordination problem appears in other fields. For example, the RIS-aided joint SWIPT-MEC framework studies coordinated use of heterogeneous storage under QoS constraints [12]. This cross-domain similarity supports our focus on jointly handling hard service constraints and adaptive optimization. The remaining gap is a dispatch model that combines explicit resilience constraints with scenario-aware economic optimization in one unified formulation.

The third stream concerns the computational methods for solving large-scale stochastic MILPs. For the problem structure at hand, Benders decomposition (the L-shaped method) is the standard solution technique, as it efficiently decomposes the problem into a master problem and parallelizable scenario subproblems [13]. However, the method’s performance is often limited by the slow convergence of the integer-constrained master problem, a well-documented bottleneck [14]. Recently, quantum computing techniques have been explored as alternative computational paradigms for challenging power system optimization and decision-making problems [15,16]. Among quantum computing approaches, QA has emerged as a promising heuristic for solving combinatorial problems that can be cast in a Quadratic Unconstrained Binary Optimization (QUBO) form [17,18,19]. Its application to power systems is nascent but advancing along several fronts. In operations, QA formulations of unit commitment demonstrate that annealers can rapidly sample high-quality binary schedules and mitigate slack-variable growth through qubit-efficient encodings [20,21,22]. At the network layer, annealing approaches have been proposed for power-flow and optimal-power-flow variants via QUBO reformulations, with partitioned schemes illustrating scalability on benchmark systems [23,24]. In distribution systems, QA-based reconfiguration and minimum-loss formulations yield feasible solutions on realistic feeders [25], and planning problems such as transmission expansion have also been explored in an annealing framework. The current gap in this domain is the absence of a framework that uses QA not as a standalone solver for an entire simplified problem, but as a surgical tool to accelerate the specific combinatorial bottleneck within a proven [26], powerful classical decomposition method.

1.4. Our Approach and Contributions

This paper presents an integrated framework that tackles the dual challenge of operational risk and computational complexity in VPP scheduling. The dispatch model combines explicit top-down resilience constraints (a protected reserve floor, an outage-islanding requirement, and a load-shedding cap) with two-stage stochastic optimization across normal and outage scenarios, so resilience is guaranteed by design, while the remaining flexibility is optimized adaptively. A hybrid quantum–classical L-shaped method handles the computational challenge. The quantum annealer heuristically solves the NP-hard integer master problem to provide high-quality schedules, while the classical L-shaped framework performs exact subproblem evaluation and cut generation for stable iterative improvement.

Compared with existing work, the main contributions are:

(1)

Constraint-Integrated Resilient Dispatch Model: We formulate a two-stage stochastic dispatch model with explicit resilience constraints (protected storage reserve floor, outage-islanding requirement, and outage load-shedding cap). Within these hard limits, dispatch is optimized across normal and outage scenarios.

(2)

A Hybrid Quantum-Classical L-shaped Decomposition Framework: We design a novel hybrid algorithm that integrates QA into the L-shaped method. In this approach, QA is used to solve the integer master problem, while all second-stage subproblems are solved exactly using classical optimization solvers. This accelerates the discovery of high-quality first-stage solutions and the generation of informative optimality cuts within the L-shaped framework.

(3)

A Systematic QUBO Reformulation for the Stochastic UC Master Problem: We present a detailed methodology for transforming the complete L-shaped master problem, including unit commitment logic, continuous recourse variables, and dynamically generated Benders cuts, into a QUBO form suitable for quantum annealers.

The remainder of this paper is organized as follows. Section 2 presents the detailed mathematical formulation. Section 3 describes the proposed hybrid solution methodology. Section 4 presents the case studies and discusses the results. Finally, Section 5 concludes the paper.

2. Problem Formulation

This section details the mathematical framework for the resilience-constrained VPP dispatch problem. The problem is formulated as a two-stage stochastic MILP to co-optimize the VPP’s day-ahead and real-time operations under uncertainties in renewable generation and market prices.

2.1. First-Stage Problem

The first-stage problem determines the optimal day-ahead unit commitment (UC) schedule [27,28], which is fixed across all scenarios. The mathematical formulation of this problem is presented as

$$\begin{array}{cc}\hfill \underset{u,v,w,\theta ,\eta }{min}& \sum _{t\in T}\sum _{g\in G}\left(C_{\mathrm{su},g}{v}_{g,t}+C_{\mathrm{sd},g}{w}_{g,t}\right)+\theta \hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill \mathit{s}.\mathit{t}.& v_{g,t}-w_{g,t}=u_{g,t}-u_{g,t-1},\hfill & \hfill & \forall g\in G,t\in T,\hfill \end{array}$$

(2)

$$\begin{array}{cccc}\hfill & v_{g,t}+w_{g,t}\le 1,\hfill & \hfill & \forall g\in G,t\in T,\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill & \theta \ge \sum _{s\in S}{\pi }_s{\eta }_s,\hfill & \hfill & \hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill & u_{g,t},v_{g,t},w_{g,t}\in \{0,1\},\hfill & \hfill & \forall g\in G,t\in T,\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill & \theta ,{\eta }_s\ge 0,\hfill & \hfill & \forall s\in S,\hfill \end{array}$$

(6)

where the sets, parameters, and variables are defined in

Table 1. Nomenclature for the first-stage problem.

Symbol	Definition
Sets
T	Set of time periods in the planning horizon, indexed by t.
G	Set of dispatchable thermal generators, indexed by g.
S *	Set of uncertainty scenarios, indexed by s.
Parameters
$C_{\mathrm{su},g},C_{\mathrm{sd},g}$	Start-up and shut-down costs of generator (g).
${\pi }_s$	Probability of occurrence for scenario s.
$u_{g,0}$	Initial on/off status of generator g at $t=0$.
Decision Variables
$u_{g,t}$	Binary variable for the operational status of generator g.
$v_{g,t}$	Binary variable for the start-up action of generator g.
$w_{g,t}$	Binary variable for the shut-down action of generator g.
$\theta$	Continuous variable for the expected total recourse cost.
${\eta }_s$	Continuous variable for the recourse cost of scenario s.

* The set S is partitioned into two disjoint subsets: $S=S_{\mathrm{norm}}\cup S_{\mathrm{outage}}$, where $S_{\mathrm{norm}}$ is the subset for normal grid-connected conditions, and $S_{\mathrm{outage}}$ is the subset for grid outage events.

.

The objective in (1) minimizes first-stage commitment costs plus expected recourse cost. Constraints (2) and (3) enforce standard unit-commitment logic. Constraint (4) links the master to scenario recourse costs. The first-stage solution is denoted by $x^{\ast }=\{u^{\ast },v^{\ast },w^{\ast }\}$.

2.2. Second-Stage Problem

Given the optimal first-stage decision $x^{\ast }$, the second-stage problem determines the optimal real-time dispatch for each scenario $s\in S$. The VPP coordinates a variety of assets, where the set B represents the hybrid energy storage system. In this study, B includes both short-duration electrochemical batteries and long-duration storage units, which jointly provide the energy buffer necessary for carbon-neutral operation and campus resilience. The optimization problem is

$$\begin{array}{cc}\hfill Q(s,x^{\ast })=\underset{y_s}{min}& \sum _{t\in T}(\sum _{g\in G}{C}_{\mathrm{op},g}{p}_{g,t,s}+C_{\mathrm{DR}}{\mathrm{flex}}_{t,s}+{\Lambda }_{\mathrm{shed}}{\mathrm{shed}}_{t,s}\hfill \\ \hfill & +{\Lambda }_{\mathrm{RT},s,t}{p}_{t,s}^{\mathrm{RT}}+M_{\mathrm{feas}}(z_{\mathrm{pos},t,s}+z_{\mathrm{neg},t,s}))\hfill \\ \hfill \mathit{s}.\mathit{t}.& \sum _{g\in G}{p}_{g,t,s}+\sum _{b\in B}(p_{\mathrm{dis},b,t,s}-p_{\mathrm{ch},b,t,s})+P{V}_{s,t}+p_{t,s}^{\mathrm{RT}}+z_{\mathrm{pos},t,s}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & =(P_{\mathrm{DC},t}^{\mathrm{base}}+P_{\mathrm{DC},t}^{\mathrm{flex}})-{\mathrm{flex}}_{t,s}-{\mathrm{shed}}_{t,s}+z_{\mathrm{neg},t,s},\forall t\in T,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & p_{t,s}^{\mathrm{RT}}=0,\forall t\in T,s\in S_{\mathrm{outage}},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \sum _{t\in T}{\mathrm{shed}}_{t,s}\le {\Gamma }_s^{\mathrm{res}},\forall s\in S_{\mathrm{outage}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & P_{\min ,g}{u}_{g,t}^{\ast }\le p_{g,t,s}\le P_{\max ,g}{u}_{g,t}^{\ast },\forall g\in G,t\in T,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & p_{g,t,s}-p_{g,t-1,s}\le {\mathrm{RU}}_g,\forall g\in G,t\in T,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & p_{g,t-1,s}-p_{g,t,s}\le {\mathrm{RD}}_g,\forall g\in G,t\in T,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & 0\le p_{\mathrm{ch},b,t,s}\le P_{\mathrm{ch},b}^{\max },\forall b\in B,t\in T,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & 0\le p_{\mathrm{dis},b,t,s}\le P_{\mathrm{dis},b}^{\max },\forall b\in B,t\in T,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\mathrm{soc}}_{b,t,s}={\mathrm{soc}}_{b,t-1,s}+({\eta }_{\mathrm{ch},b}{p}_{\mathrm{ch},b,t,s}-p_{\mathrm{dis},b,t,s}/{\eta }_{\mathrm{dis},b})\Delta t,\forall b\in B,t\in T,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & E_{\min ,b}\le {\mathrm{soc}}_{b,t,s}\le E_{\max ,b},\forall b\in B,t\in T,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & 0\le {\mathrm{flex}}_{t,s}\le P_{\mathrm{DC},t}^{\mathrm{flex}},\forall t\in T,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & 0\le {\mathrm{shed}}_{t,s}\le P_{\mathrm{DC},t}^{\mathrm{base}}+P_{\mathrm{DC},t}^{\mathrm{flex}},\forall t\in T,\hfill \end{array}$$

(19)

where the parameters and decision variables are listed in

Table 2. Nomenclature for the second-stage problem.

Symbol	Definition
Parameters
$C_{\mathrm{op},g}$	Variable operational cost of generator g.
$C_{\mathrm{DR}}$	Unit cost for activating demand response.
${\Lambda }_{\mathrm{shed}}$	Penalty cost for load shedding.
$P_{\max ,g},P_{\min ,g}$	Maximum/minimum power output of generator g.
${\mathrm{RU}}_g,{\mathrm{RD}}_g$	Ramp-up and ramp-down rate limits of generator g.
$P_{\mathrm{ch},b}^{\max },P_{\mathrm{dis},b}^{\max }$	Maximum charging/discharging power of BESS unit b.
$E_{\max ,b},E_{\min ,b}$	Maximum/minimum state of energy of unit b.
${\eta }_{\mathrm{ch},b},{\eta }_{\mathrm{dis},b}$	Charging/discharging efficiencies of unit b.
$E_{b,0}$	Initial state of energy of unit b.
$P_{\mathrm{DC},t}^{\mathrm{base}},P_{\mathrm{DC},t}^{\mathrm{flex}}$	Base and flexible components of the data center load.
$P{V}_{s,t}$	Power from photovoltaic sources.
${\Lambda }_{\mathrm{RT},s,t}$	Real-time market price.
$M_{\mathrm{feas}}$	Penalty factor for feasibility slack variables.
${\Gamma }_s^{\mathrm{res}}$	Maximum allowable total load shedding in outage scenario s (MWh).
Second-Stage Decision Variables ($y_s$)
$p_{g,t,s}$	Power dispatch of generator g.
$p_{\mathrm{ch},b,t,s},p_{\mathrm{dis},b,t,s}$	Charging/discharging power of unit b.
${\mathrm{soc}}_{b,t,s}$	State of charge of unit b.
${\mathrm{flex}}_{t,s}$	Activated demand response.
${\mathrm{shed}}_{t,s}$	Amount of load shedding.
$p_{t,s}^{\mathrm{RT}}$	Net power purchased from the real-time market.
$z_{\mathrm{pos},t,s},z_{\mathrm{neg},t,s}$	Slack variables for power balance feasibility.

. The objective (7) minimizes the scenario operating cost. Constraint (8) enforces hourly power balance. Resilience is represented by two constraints. Constraint (9) enforces islanded operation in outage scenarios by setting real-time market transactions to zero. Constraint (10) limits total outage load shedding to ${\Gamma }_s^{\mathrm{res}}$ MWh for each outage scenario. If ${\Gamma }_s^{\mathrm{res}}=0$, no shedding is allowed. If ${\Gamma }_s^{\mathrm{res}}>0$, limited shedding is allowed by design. Generator output and ramp limits are enforced by (11)–(13). Storage dynamics are enforced by (14)–(17). In addition, the protected reserve policy is implemented through the lower energy bound $E_{min,b}$ for the resilience-oriented storage unit during normal operation. During outages, this reserve can be used to maintain service continuity. The storage equations are technology-agnostic and parameterized; they represent generic long-duration storage rather than a hydrogen-physics model. Constraints (18) and (19) bound demand response and emergency shedding.

3. Solution Methodology

This section extends the classical multi-cut L-shaped framework by using quantum annealing (QA) as the master solver throughout the entire Benders iteration. Concretely, at each iteration, we reformulate the current master (including all accumulated Benders’ optimality cuts) into a QUBO and solve it on a quantum annealer to obtain a first-stage candidate $x_k$. We then solve the second-stage scenario subproblems exactly to evaluate the true recourse costs and to generate new valid Benders cuts, which are appended to the cut pool and incorporated into the next master QUBO. This loop is repeated until a prescribed stopping criterion is met.

3.1. Classical L-Shaped Method

To solve the computationally intractable two-stage stochastic MILP, we employ the multi-cut L-shaped method, an iterative decomposition algorithm. The algorithm converges by tightening a Lower Bound (LB), derived from an optimistic master problem, and an Upper Bound (UB), representing the true cost of a candidate first-stage decision $x_k$. The UB is calculated at each iteration k by summing the first-stage costs and the expected value of the true recourse costs, $Q(s,x_k)$, obtained from the subproblems:

$$U{B}_k=\sum _{t\in T}\sum _{g\in G}\left(C_{\mathrm{su},g}{v}_{g,t,k}+C_{\mathrm{sd},g}{w}_{g,t,k}\right)+\sum _{s\in S}{\pi }_{s}Q(s,x_k).$$

(20)

Convergence is driven by adding Benders optimality cuts to the master problem. These cuts are linear constraints derived from the subproblem dual variables ($\gamma$) that progressively refine the master’s approximation of the recourse function. A unique cut is generated for each scenario s, providing a lower bound on the cost variable ${\eta }_s$:

$${\eta }_s\ge Q(s,x_k)+\sum _{g\in G}\sum _{t\in T}\left({\displaystyle \frac{\partial Q(s,x_k)}{\partial u_{g,t}}}\right)(u_{g,t}-u_{g,t,k}),$$

(21)

where the subgradient components are constructed using the dual variables ($\gamma$) from the generator capacity constraints:

$${\displaystyle \frac{\partial Q(s,x_k)}{\partial u_{g,t}}}=P_{\max ,g}{\gamma }_{\max ,g,t,s,k}-P_{\min ,g}{\gamma }_{\min ,g,t,s,k}.$$

(22)

Adding these cuts guarantees the monotonic convergence of the LB to the UB. This iterative procedure is formally outlined in Algorithm 1.

Algorithm 1: The L-shaped method for stochastic VPP dispatch

3.2. Quantum Annealing for the Master Problem

QA addresses binary quadratic problems by approximately minimizing the energy of a transverse-field Ising Hamiltonian. The master problem, once reformulated into QUBO form:

$$\underset{x\in {\{0,1\}}^n}{min}f\left(x\right)=x^{\top }Qx+c^{\top }x,$$

(23)

is mapped to an Ising model through the affine change of variables:

$$x_i=\frac{1+s_i}{2},s_i\in \{-1,+1\},i=1,\dots ,n,$$

(24)

where $x\in {\{0,1\}}^n$ are binary decision variables of the QUBO, and $s=(s_1,\dots ,s_n)$ are the equivalent spin variables. Substituting yields the Ising energy function;

$$E\left(s\right)=\sum _{i=1}^n{h}_i{s}_i+\sum _{1\le i<j\le n}{J}_{ij}{s}_i{s}_j+\mathrm{const},$$

(25)

where $h_i\in \mathbb{R}$ are local biases, and $J_{ij}\in \mathbb{R}$ are pairwise couplings between spins i and j. Minimizing (25) is equivalent to solving the original QUBO (23).

A QA device realizes a time-dependent Hamiltonian:

$$H\left(\tau \right)=A\left(\tau \right)H_D+B\left(\tau \right)H_P,\tau \in [0,1],$$

(26)

where $H_D=-{\sum }_i{\sigma }_i^x$ is the driver Hamiltonian, and $H_P={\sum }_i{h}_i{\sigma }_i^z+{\sum }_{i<j}{J}_{ij}{\sigma }_i^z{\sigma }_j^z$ encodes the problem instance, with ${\sigma }^x,{\sigma }^z$ denoting Pauli-X and Pauli-Z matrices. Here, $\tau$ is the normalized annealing time, i.e., the physical time rescaled by the total evolution duration so that $\tau =0$ corresponds to the start of the anneal and $\tau =1$ to the end. As illustrated in Figure 1, the schedule functions $A\left(\tau \right)$ and $B\left(\tau \right)$ interpolate between the two Hamiltonians, satisfying $A\left(0\right)\gg B\left(0\right)$ and $A\left(1\right)\ll B\left(1\right)$. By the adiabatic theorem, if the total evolution time $T_{\mathrm{anneal}}$ is large compared to the inverse squared minimum spectral gap,

$$T_{\mathrm{anneal}}\gg {\displaystyle \frac{\parallel \dot{H}\parallel }{{\Delta }_{min}^2}},$$

(27)

the system remains close to the instantaneous ground state of $H\left(\tau \right)$, and the final spin configuration approaches the global minimizer of (25) in the ideal adiabatic limit.

In practice, modern QA devices and physics-inspired hybrid samplers perform repeated finite-time anneals and return a distribution of spin configurations s according to an approximate Gibbs law,

$$\mathbb{P}\{S=s\}\propto exp\left(-\frac{E\left(s\right)}{T_{\mathrm{eff}}}\right),$$

(28)

with effective temperature $T_{\mathrm{eff}}>0$. Low-energy samples correspond to high-quality feasible candidates for the first-stage master problem.

In the L-shaped context, the master must be expressed as a QUBO. We therefore encode all master constraints (including Benders cuts) as quadratic penalties and discretize the first-stage non-negative continuous variables with fixed-point binary expansions, yielding a binary quadratic model suitable for QA.

3.3. QUBO Reformulation of the Master

Let $u_{g,t},v_{g,t},w_{g,t}\in \{0,1\}$ denote the unit status and start/shutdown binaries for $g\in G,t\in T$. The continuous non-negative variables $\theta$ and ${\eta }_s$ are represented via non-negative fixed-point binary expansions:

$$x={\Delta }_x\sum _{k=0}^{K_x-1}{2}^k{b}_{x,k},b_{x,k}\in \{0,1\},x\in \{\theta ,{\eta }_s\},$$

(29)

where ${\Delta }_x>0$ denotes the discretization step size (i.e., the resolution of the fixed-point grid), and $K_x$ is the number of binary digits. Hence, the representable domain is $[0,(2^{K_x}-1){\Delta }_x]$, and the encoded value of x lies on a uniform grid with spacing ${\Delta }_x$. Any slack variables introduced below are also non-negative and encoded analogously. Since binary squares satisfy $b^2=b$, all penalty terms constructed from equalities remain quadratic.

The fixed-point encoding induces a quantization error since continuous variables are restricted to a discrete grid. For each variable, the closest grid-representable value differs from the exact continuous value by, at most, ${\Delta }_x/2$. Consequently, the QUBO master constitutes a discretized approximation of the classical MILP master rather than an exact reformulation. In general, the optimal QUBO solution may differ from the classical MILP optimum due to this discretization, and the resulting objective deviation depends on both ${\Delta }_x$ and the objective coefficient magnitudes. Exact representability of the classical optimum is guaranteed only when the optimal value lies exactly on the fixed-point grid.

In practice, selecting a sufficiently fine discretization (larger $K_x$, smaller ${\Delta }_x$) renders the approximation error negligible relative to the prescribed optimality tolerance $ϵ$. In our implementation, each continuous variable is encoded with $K_x=8$ bits, and ${\Delta }_x$ is chosen such that the representable range covers the full feasible domain with a resolution of approximately $0.01\%$ of the domain width. A sensitivity study over $K_x\in \{6,8,10\}$ is reported in Section 4.4.4, confirming that this discretization level is sufficient for the considered problem scale.

The unit-commitment logic is enforced by penalizing the status/start/stop balance. Let $t_0:=minT$ and let $u_{g,0}$ be the initial status. For the first period,

$$v_{g,t_0}-w_{g,t_0}-u_{g,t_0}+u_{g,0}=0\Rightarrow P_{\mathrm{logic}}{\left(v_{g,t_0}-w_{g,t_0}-u_{g,t_0}+u_{g,0}\right)}^2,$$

(30)

and for $t>t_0$,

$$v_{g,t}-w_{g,t}-u_{g,t}+u_{g,t-1}=0\Rightarrow P_{\mathrm{logic}}{\left(v_{g,t}-w_{g,t}-u_{g,t}+u_{g,t-1}\right)}^2.$$

(31)

The mutual-exclusivity inequality $v_{g,t}+w_{g,t}\le 1$ is modeled by penalizing only the forbidden combination:

$$P_{\mathrm{excl}}{v}_{g,t}{w}_{g,t},$$

(32)

which assigns zero penalty to $(v,w)\in \left\{\right(0,0),(1,0),(0,1\left)\right\}$ and a positive penalty to $(1,1)$.

The recourse coupling $\theta \ge {\sum }_{s\in S}{\pi }_s{\eta }_s$ is written as an equality with a non-negative slack $z^{\theta }\ge 0$:

$$\theta -\sum _{s\in S}{\pi }_s{\eta }_s-z^{\theta }=0,P_{\theta }{\left(\theta -\sum _{s\in S}{\pi }_s{\eta }_s-z^{\theta }\right)}^2,$$

(33)

so that, at any feasible solution with sufficiently large $P_{\theta }$, the slack resolves to the shortfall $z^{\theta }=max\{0,{\sum }_s{\pi }_s{\eta }_s-\theta \}$, reproducing the original inequality.

For scenario s at iteration k, the standard optimality cut reads

$${\eta }_s\ge Q(s,x_k)+\sum _{g,t}\left({\displaystyle \frac{\partial Q(s,x_k)}{\partial u_{g,t}}}\right)\left(u_{g,t}-u_{g,t,k}\right),$$

(34)

where ${\displaystyle \frac{\partial Q(s,x_k)}{\partial u_{g,t}}}=P_{max,g}{\gamma }_{max,g,t,s,k}-P_{min,g}{\gamma }_{min,g,t,s,k}=:a_{s,k}^{g,t}.$ Rearranging (34) yields

$${\eta }_s\ge \underset{:={\mathrm{const}}_{s,k}}{\underbrace{Q(s,x_k)-\sum _{g,t}{a}_{s,k}^{g,t}{u}_{g,t,k}}}+\sum _{g,t}{a}_{s,k}^{g,t}{u}_{g,t}.$$

(35)

To fit QUBO, we convert the inequality into an equality with a non-negative slack $z_{s,k}^{\mathrm{cut}}\ge 0$:

$${\eta }_s-{\mathrm{const}}_{s,k}-\sum _{g,t}{a}_{s,k}^{g,t}{u}_{g,t}-z_{s,k}^{\mathrm{cut}}=0,$$

(36)

and add the quadratic penalty:

$$P_{\mathrm{cut}}{\left({\eta }_s-{\mathrm{const}}_{s,k}-\sum _{g,t}{a}_{s,k}^{g,t}{u}_{g,t}-z_{s,k}^{\mathrm{cut}}\right)}^2.$$

(37)

A distinct slack $z_{s,k}^{\mathrm{cut}}$ is used for each cut; sharing a slack across cuts can weaken individual cuts through compensation effects.

Combining all terms, the QUBO master objective is

$$\begin{array}{cc}\hfill min& \sum _{g,t}\left(C_{\mathrm{su},g}{v}_{g,t}+C_{\mathrm{sd},g}{w}_{g,t}\right)+\theta +\sum _{g,t}{P}_{\mathrm{excl}}{v}_{g,t}{w}_{g,t}\hfill \\ \hfill & +\sum _g\left[P_{\mathrm{logic}}{\left(v_{g,t_0}-w_{g,t_0}-u_{g,t_0}+u_{g,0}\right)}^2+\sum _{t>t_0}{P}_{\mathrm{logic}}{\left(v_{g,t}-w_{g,t}-u_{g,t}+u_{g,t-1}\right)}^2\right]\hfill \\ \hfill & +P_{\theta }{\left(\theta -\sum _s{\pi }_s{\eta }_s-z^{\theta }\right)}^2+\sum _s\sum _{k\in \mathcal{K}\left(s\right)}{P}_{\mathrm{cut}}{\left({\eta }_s-{\mathrm{const}}_{s,k}-\sum _{g,t}{a}_{s,k}^{g,t}{u}_{g,t}-z_{s,k}^{\mathrm{cut}}\right)}^2,\hfill \end{array}$$

(38)

where $\mathcal{K}\left(s\right)$ indexes the cuts accumulated for scenario s. All symbols in (38) are binary (natively or via fixed-point encodings). When all penalties are zero, (38) coincides with the original master objective, ensuring consistency.

3.4. Coefficient Scaling and Penalty Initialization

After assembling the binary quadratic model (BQM) of (38), we apply two preprocessing steps to ensure numerical stability and enforce feasibility.

(i)

Sampler-range normalization. Let $h_i$ and $J_{ij}$ denote the BQM linear and quadratic coefficients. To respect hardware or sampler bounds, we scale all coefficients by a common factor $\alpha >0$ such that

$$|\alpha h_i|\le h_{max},\left|\alpha J_{ij}\right|\le J_{max},$$

(39)

where $(h_{max},J_{max})$ are sampler-recommended limits (e.g., $h_{max}=2$, $J_{max}=1$). This linear scaling preserves the minimizers of the QUBO.

(ii)

Penalty initialization from objective scale. Let

$$M_{\mathrm{obj}}:=\sum _{g,t}\left(|C_{\mathrm{su},g}|+|C_{\mathrm{sd},g}|\right)+{\widehat{\theta }}_{max},$$

(40)

where ${\widehat{\theta }}_{max}$ is a conservative upper bound on $\theta$ (e.g., obtained from a feasibility penalty envelope or a prior upper bound). The penalty coefficients for all constraint families are initialized as

$$P_{\mathrm{logic}}=P_{\mathrm{excl}}=P_{\theta }=P_{\mathrm{cut}}={\kappa }_0{M}_{\mathrm{obj}},{\kappa }_0\in [5,15],$$

(41)

ensuring that any violation induces a cost increment that dominates the linear contribution of the objective. The initialization of the penalty factor ${\kappa }_0$ reflects a practical trade-off. If ${\kappa }_0$ is too small, the QUBO minimizer may return solutions that violate the original constraints, resulting in infeasible master candidates. In contrast, if ${\kappa }_0$ is too large, the objective landscape becomes overly dominated by the penalty terms, which compresses the effective dynamic range of the problem coefficients and may degrade performance due to hardware coefficient range limitations and reduced energy resolution. A sensitivity study over ${\kappa }_0\in \{5,10,15\}$ and $K_x\in \{6,8,10\}$ is reported in Section 4.4.4, demonstrating that ${\kappa }_0=10$ and $K_x=8$ provide stable and near-optimal performance for the present problem scale.

3.5. QA-Driven Master Strategy and Overall Algorithm

The proposed strategy integrates QA into the L-shaped method by solving the master problem via QA at every iteration. At iteration k, the master MILP with all accumulated Benders cuts is reformulated as a BQM using (38), including fixed-point encodings for non-negative continuous variables and quadratic penalties for all master constraints. After coefficient normalization to sampler bounds and penalty initialization, QA (or a compatible sampler) is invoked to approximately minimize the resulting QUBO. Decoding yields an integer first-stage candidate $x_k=\{u_k,v_k,w_k\}$. The QA-based master solve is a heuristic: it does not provide a proof of master optimality. The resulting $x_k$ is a candidate solution. Subproblems are still solved exactly, so $U{B}_k$ is evaluated from exact recourse costs and cuts remain valid. However, the method does not inherit the finite-step convergence proof of the classical integer L-shaped algorithm. We therefore treat the approach as heuristic-accelerated Benders decomposition.

Given $x_k$, each scenario subproblem is solved exactly to obtain the true recourse value $Q(s,x_k)$. The corresponding upper-bound candidate is computed as

$$U{B}_k=\sum _{g,t}\left(C_{\mathrm{su},g}{v}_{g,t,k}+C_{\mathrm{sd},g}{w}_{g,t,k}\right)+\sum _{s\in S}{\pi }_{s}Q(s,x_k).$$

(42)

In parallel, the accumulated Benders cuts define a piecewise-affine lower model of the recourse function. For each scenario s, the cut set ${\mathcal{C}}_s$ defines

$${\underline{Q}}_s\left(u\right)=\underset{i\in {\mathcal{C}}_s}{max}\left\{{\mathrm{const}}_{s,i}+\sum _{g,t}{a}_{s,i}^{g,t}{u}_{g,t}\right\}.$$

(43)

Evaluating this model at the sampled commitment $u_k$ yields

$$L{B}_k^{\mathrm{ind}}=\sum _{g,t}\left(C_{\mathrm{su},g}{v}_{g,t,k}+C_{\mathrm{sd},g}{w}_{g,t,k}\right)+\sum _{s\in S}{\pi }_s{\underline{Q}}_s\left(u_k\right).$$

(44)

For a classical exact master solve, this quantity can be interpreted as a lower bound. For the QA-driven branch, the master is solved heuristically, so $L{B}_k^{\mathrm{ind}}$ is used only as a cut-model progress indicator (not a certified global lower bound). We track $L{B}^{\mathrm{ind}}\leftarrow max(L{B}^{\mathrm{ind}},L{B}_k^{\mathrm{ind}})$.

From the optimal dual variables of each subproblem, a per-scenario Benders optimality cut of the form

$${\eta }_s\ge {\mathrm{const}}_{s,k}+\sum _{g,t}{a}_{s,k}^{g,t}{u}_{g,t}$$

(45)

is generated and appended to the cut pool ${\mathcal{C}}_s$. These cuts are incorporated into the master QUBO at the next iteration.

The algorithm terminates when the reported gap indicator

$$max\{0,UB-L{B}^{\mathrm{ind}}\}/max\{1,|UB\left|\right\}$$

falls below a prescribed tolerance or when a maximum iteration budget is reached. The complete procedure is summarized in Algorithm 2.

Algorithm 2: QA-driven L-shaped method

4. Case Study

To validate the proposed resilience-constrained dispatch model and evaluate the hybrid quantum-classical algorithm, we conduct a case study using real-world data. This section describes the test system, benchmark design, implementation details, and results.

4.1. System and Data Setup

The test system is modeled as a data center VPP, comprising a critical load, on-site generation, and energy storage. The optimization is performed over a 24 h scheduling horizon with an hourly time resolution. All specific asset capacities, load profiles, and cost parameters are based on industry-standard values and are detailed in

Table 3. Key parameters of the VPP test system.

Component	Parameter	Value
VPP Assets and Load
Data Center Load	Peak Load	10 MW
	Critical Base Load	8 MW
	Flexible Load	2 MW
Solar PV System	Installed Capacity	5 MWp
Dispatchable Generator	Type	Natural Gas
	Maximum Capacity	12 MW
	Minimum Output	2 MW
	Ramp Rate	6 MW/h
	Variable Cost	$60/MWh
	Start-up Cost	$500
	Shut-down Cost	$300
Battery Energy Storage	Energy Capacity	20 MWh
	Power Rating	4 MW
	Round-trip Efficiency	92%
	Initial State of Charge	50%
Long-Duration Storage Unit	Charging Power Rating	2 MW
	Discharging Power Rating	2 MW
	Energy Capacity	40 MWh
	Charging Efficiency (${\eta }_{\mathrm{ch}}$)	65%
	Discharging Efficiency (${\eta }_{\mathrm{dis}}$)	50%
	Initial State of Charge	50%
Stochastic Model Parameters
Scheduling Horizon		24 h
Time Resolution		1 h
Number of Scenarios	Total	50
	Normal Operation ($S_{\mathrm{norm}}$)	45
	Grid Outage ($S_{\mathrm{outage}}$)	5
Outage Duration		6 h

.

The baseline case uses one generator and two storage units (Table 3). We keep this setup compact to allow exact classical verification before scaling up. We agree that one calibrated case is not enough for full scalability validation. To provide preliminary guidance,

Table 4. Illustrative projections of QUBO size and wall-clock time for three system scales under explicit assumptions. The scale-up includes larger generation fleets, storage-asset sets, scenario counts, and horizon lengths. QUBO size is estimated from the $\mathcal{O}\left(\right|G\left|\right|T\left|K_x\right)$ relation ($K_x=8$); classical subproblem time is assumed to scale near-linearly with $\left|S\right|$; QA annealing time per iteration is treated as hardware-limited (≈3.8 s).

Scale	$\left|\mathit{G}\right|$	$\left|\mathit{B}\right|$	$\left|\mathit{T}\right|$	$\left|\mathit{S}\right|$	QUBO Vars	Classical Time (s)	QA Time (s)
Baseline	1	2	24	50	∼200	202	148
Medium	5	6	48	100	∼2000	∼890	∼430
Large	20	20	168	200	∼27,000	>5000	∼1200

reports assumption-based scale-up results for baseline ($\left|G\right|=1$, $\left|B\right|=2$, $\left|T\right|=24$, $\left|S\right|=50$), medium ($\left|G\right|=5$, $\left|B\right|=6$, $\left|T\right|=48$, $\left|S\right|=100$), and large ($\left|G\right|=20$, $\left|B\right|=20$, $\left|T\right|=168$, $\left|S\right|=200$) settings. QUBO size follows the $\mathcal{O}\left(\right|G\left|\right|T\left|K_x\right)$ relation implied by the fixed-point encoding in Section 3.3. Runtime projections assume near-constant QA time per iteration (≈3.8 s on D-Wave Advantage) and near-linear growth of the classical subproblem time with $\left|S\right|$. These are assumption-bounded estimates, not definitive performance guarantees. We also report robustness analyses for outage assumptions (Section 4.4.2) and QA hyperparameters $(K_x,{\kappa }_0)$ (Section 4.4.4). Larger benchmark validation remains future work.

We use 50 scenarios in total. For the 45 normal scenarios, PV output is sampled from a Beta distribution scaled by installed capacity, and real-time market price is sampled from a log-normal distribution. The hourly load profile is perturbed by Gaussian noise (coefficient of variation 5%) to represent intra-day demand fluctuations. For the 5 outage scenarios, the outage start time is sampled uniformly from hours 0–6, and outage duration is fixed at 6 h. Scenario probabilities are calibrated so that weighted means and variances of uncertain inputs match historical statistics; the total outage-scenario probability is set to 0.10 in this case study.

Fixing the outage duration at 6 h is a simplifying assumption. To test robustness, Section 4.4.2 evaluates 3 h, 6 h, and 12 h outage cases. We also note that the current load model uses one Gaussian-noise regime; richer regime-switching demand models (e.g., weekday/weekend and seasonal patterns) are left for future work.

4.2. Experimental Design and Benchmark Models

To evaluate our two primary contributions, we establish the following experimental design.

4.2.1. Model Performance Evaluation:

To demonstrate the value of our proposed modeling framework, we compare the performance of two distinct operational models. Both models are solved using a standard commercial MILP solver to ensure a fair comparison of the formulations themselves.

• Case 1: Baseline-VPP: This benchmark uses a standard two-stage stochastic dispatch model with only the 45 normal-operation scenarios. It does not model outage risk explicitly and therefore prioritizes economic performance in normal conditions.
• Case 2: Resil-VPP (Proposed Model): This model enforces functional differentiation through storage reserve sizing. A 70% portion is used for daily economic dispatch, while a 30% minimum reserve is protected in normal operation through the lower energy bound. During outages, this reserve can be used to maintain supply continuity. This 30% reserve partition is an explicit top-down resilience constraint; the model contribution is to combine this guarantee with scenario-aware stochastic optimization so the remaining flexibility is allocated adaptively.

4.2.2. Validation of the Hybrid Quantum–Classical Algorithm:

To demonstrate the computational advantages of our proposed solution method, we solve the more complex Resil-VPP model using two different algorithmic approaches.

• Method A: Classical L-shaped (Classical Benchmark): This is the standard integer L-shaped method. The master problem is solved at each iteration using the Gurobi commercial MILP solver.
• Method B: QA-Driven L-shaped Method (Proposed Algorithm): This is our proposed hybrid algorithm. The QUBO-formulated master problem is solved at each iteration using the D-Wave Advantage quantum annealer.
• Method C: Genetic Algorithm (GA Heuristic): A population of 50 binary chromosomes (24 h unit-commitment schedules) is evolved for 80 generations. Single-point crossover and bit-flip mutation rates are 0.80 and 0.02. To keep the runtime manageable, fitness is evaluated on 10 representative scenarios; the best chromosome is then re-evaluated on all 50 scenarios for final reporting.

We compare three methods. Method A (classical L-shaped) is the certified-optimum baseline for the full two-stage MILP. Method B (QA-driven L-shaped) uses the same decomposition structure and exact subproblem evaluation, but its master step is heuristic, so performance is reported empirically. Method C (GA) is a classical heuristic reference without optimality certificates.

4.3. Implementation Details

The optimization models were implemented in the Julia programming language using the JuMP modeling package. All classical MILP and LP problems, including the subproblems and the master problem for Method A, were solved using Gurobi v9.5 on a workstation with Intel Xeon Gold 6148 CPUs (20 cores). For the QA-Driven L-shaped algorithm, the QUBO master problem was formulated and solved using the D-Wave Ocean SDK. The problem instances were submitted to the D-Wave Advantage quantum annealer. Each continuous recourse variable was discretized into eight binary variables, and the penalty coefficients for all constraint families were initialized with ${\kappa }_0=10$, about one order of magnitude above the objective scale to strongly penalize master-constraint violations.

4.4. Results and Discussion

4.4.1. Performance of the Resilience-Constrained Dispatch Model

The primary goal of the Resil-VPP model is to guarantee power continuity during outages with minimal impact on economic performance during normal operation.

As shown in Figure 2, which plots the BESS dispatch for a representative normal scenario, the two models exhibit fundamentally different behaviors. The Baseline-VPP utilizes the full range of the BESS capacity to maximize profits from price arbitrage. In contrast, the Resil-VPP operates the BESS within the bounds of its ‘economic reserve’, carefully preserving the ‘resilience reserve’ at a high state-of-charge throughout the day.

Table 5. Resilience performance under grid outage scenarios.

Model	Avg. Load Shed (MWh)	Reserve Used (%)	Critical Load (%)	RI (%)
Baseline-VPP	5.2	100	93.5	89.2
Resil-VPP	0.0	30	100.0	100.0

summarizes the VPP’s performance across all outage scenarios. The results clearly demonstrate that the Baseline-VPP, having depleted its energy reserves for economic gain, is forced to shed 5.2 MWh of critical load on average during an outage. The Resil-VPP, however, successfully leverages its protected resilience reserve to island the data center and serve the entire critical load, resulting in zero load shedding across all outage scenarios.

This enhanced resilience comes at a modest and quantifiable cost. Our analysis shows that the Resil-VPP model’s expected daily operational cost is only 3.7% higher than that of the Baseline-VPP. This finding highlights the economic viability of the proposed functional differentiation strategy, which provides complete immunity to the tested outages for a small, predictable premium.

4.4.2. Sensitivity to Outage Duration and Load Fluctuation

To partially address the limitation of the fixed 6 h outage assumption, we examine how the key performance metrics vary when the outage duration is extended or contracted.

Table 6. Sensitivity of Resil-VPP performance to outage duration.

Outage Duration (h)	Avg. Load Shed (MWh)	Cost Premium (%)	Reserve Depleted (%)
3	0.0	2.1	15
6	0.0	3.7	30
12	1.4	5.9	100

reports the Resil-VPP’s average load shedding and expected cost premium for three outage durations: 3, 6, and 12 h, holding all other parameters constant. The results show that the Resil-VPP maintains zero load shedding for the 3 h and 6 h cases, with the cost premium rising modestly from 2.1% to 3.7%. Under the more severe 12 h outage, the 30% resilience reserve is insufficient to cover the full outage window, leading to a small residual load shed of 1.4 MWh in the final two hours; the expected cost premium rises to 5.9%. This analysis confirms that the proposed resilience reserve sizing is effective for moderate outage durations and highlights that for extended grid disruptions, a larger resilience reserve or additional long-duration storage capacity would be required.

For dynamic load fluctuations, the scenario construction in Section 4.1 already adds time-varying Gaussian noise to the hourly demand profile, so intra-day variability is represented. A remaining limitation is that this study uses a single noise regime; richer regime-switching demand models (e.g., weekday/weekend and seasonal regimes) are left for future work.

4.4.3. Computational Performance of the QA-Driven L-Shaped Algorithm

The second set of experiments validates the computational benefits of incorporating a quantum annealer to solve the integer master problem. Figure 3 shows the evolution of upper and lower indicators for the classical and QA-driven L-shaped methods. The classical variant improves the lower bound rapidly in early iterations, whereas the hybrid variant progresses gradually at first and then exhibits a sharp lower-indicator jump. After this jump, the hybrid method reduces the reported gap indicator quickly and approaches the best-known objective.

A quantitative comparison is provided in

Table 7. Wall-clock time and solution quality comparison among the GA heuristic, classical L-shaped, and hybrid QA–L-shaped methods.

Method	Iters †	Avg. Time (s)	Total Time (s)	Final Obj. ($)
GA Heuristic	80	2.4	195.0	97,218 ‡
Classical L-shaped	34	5.9	202.2	96,843.04
Hybrid QA–L-shaped	39	3.8	148.3	96,843.04

^† For the L-shaped methods, “Iters” denotes Benders decomposition iterations; for GA, it denotes the number of generations (population size 50). ^‡ The GA objective is re-evaluated on all 50 scenarios after termination; no optimality bound is available.

. Figure 4 shows the same comparison in terms of runtime and final objective gap. Using the reported clipped UB–LB indicator from Algorithm 2, the QA-driven L-shaped method reaches the 1% threshold in about 150 s with 39 iterations, while the classical L-shaped method reaches the same threshold in about 200 s with 34 iterations. Both methods reach the same best-known objective ($96,843.04) in this case study, which corresponds to an empirical runtime speedup of about 1.36× for the QA-driven variant. The GA heuristic completes 80 generations in 195.0 s and returns $97,218, about 0.39% above the best-known objective. Overall, the QA-driven method shows a positive but moderate speed gain for this tested instance. Because the QA master is heuristic, this result should be interpreted as an empirical performance rather than a formal global-optimality guarantee.

A practical interpretation of the hybrid trajectory is the following. Early in the run, the cut pool is sparse, so QA explores a broad set of first-stage candidates and the lower-model indicator improves slowly. As cuts accumulate, the surrogate landscape becomes more informative; candidate quality improves, and the reported gap contracts faster. By comparison, the classical branch-and-bound master tends to show a steadier trajectory due to deterministic tree search. This interpretation is empirical and specific to the tested instance.

From a scalability perspective, the master-side variable budget in our implementation is dominated by fixed-point encodings of first-stage decisions and scales approximately as $\mathcal{O}\left(\right|G\left|\right|T\left|K_{\mathrm{bits}}\right)$. Additional auxiliary/slack variables from accumulated Benders cuts grow with the number of retained cuts, which depends on cut-management choices and stopping tolerance rather than on $\left|S\right|$ alone. Accordingly, increases in $\left|G\right|$, $\left|T\right|$, and effective cut-pool size raise both QUBO embedding difficulty and classical integer–master complexity. On the recourse side, subproblem solves are scenario-decomposable and parallelizable, but the wall-clock time still depends on the scenario count and available computing resources. The trends reported here should therefore be read as instance-specific empirical evidence under the stated settings and assumptions, not as a universal guarantee of speedup for all scales or all hardware generations.

4.4.4. Sensitivity to Discretization Resolution and Penalty Coefficient

We tested sensitivity to two QA hyperparameters: discretization bits per continuous variable $K_x\in \{6,8,10\}$ and penalty multiplier ${\kappa }_0\in \{5,10,15\}$. All other settings are fixed.

Table 8. Sensitivity of the QA master to discretization bits $K_x$ and penalty multiplier ${\kappa }_0$. Best feasible objective ($) and constraint-violation rate (%) across 100 QA samples per configuration.

${\mathit{K}}_{\mathit{x}}$	${\mathit{\kappa }}_0$	Best Feasible Obj. ($)	Violation Rate (%)
6	5	98,610	26.8
6	10	97,105	3.9
6	15	97,060	1.6
8	5	97,892	16.4
8	10	96,843	0.3
8	15	96,843	0.7
10	5	97,180	12.5
10	10	96,843	0.6
10	15	96,855	0.9

reports full-factorial results over these settings, using 100 QA samples per configuration. We report the best feasible objective and the constraint-violation rate (fraction of samples that violate at least one master constraint).

Two trends are clear. First, $K_x=6$ is too coarse: even with larger penalties, the best feasible objective stays above the classical reference. Moving to $K_x=8$ removes most discretization loss, and the QA master recovers the best-known objective ($96,843) when ${\kappa }_0\ge 10$. Increasing to $K_x=10$ gives little additional benefit but increases BQM size.

Second, ${\kappa }_0$ mainly controls feasibility. With ${\kappa }_0=5$, violation rates are high (12–27%). With ${\kappa }_0\ge 10$, violation rates drop below 1% for all tested $K_x$. The setting $(K_x,{\kappa }_0)=(8,10)$ achieves the lowest violation rate (0.3%) while matching the best-known objective.

Based on these results, we use $K_x=8$ and ${\kappa }_0=10$ as the default setting for this problem scale. The QA master is still heuristic, so agreement with the classical objective should be interpreted as empirical rather than certified.

For reproducibility, we apply a simple tuning rule: choose the smallest $K_x$ whose best feasible objective is within 0.1% of the classical reference, then choose the smallest ${\kappa }_0$ that keeps the violation rate below 1% under that $K_x$. This rule selects $(K_x,{\kappa }_0)=(8,10)$.

5. Conclusions

This study presents a two-stage stochastic dispatch framework for carbon-neutral industrial parks that balances economic efficiency and operational resilience. The proposed storage strategy combines short-duration batteries with long-duration storage and, in the tested 6-h outage scenarios, eliminates critical load shedding with a 3.7% expected cost increase. To handle the resulting MILP complexity, we implement a hybrid quantum-classical L-shaped algorithm. On the tested case, the QA-driven method reaches the same best-known objective as the classical baseline with an empirical 1.36× runtime speedup. Because the QA master is heuristic, this should be interpreted as empirical near-optimal performance rather than a certified global optimum. A key limitation is that detailed dispatch benchmarking is still based on one calibrated configuration; extending validation to larger multi-asset benchmark sets is an important next step. This study provides assumption-bounded scalability guidance and states the scope limits explicitly. The same resilience-constrained architecture can also be adapted to other mission-critical energy parks (e.g., hospital or telecom campuses) after parameter re-calibration.