Multi-Time-Scale Optimization and Control Method for High-Penetration Photovoltaic Electrolytic Aluminum Plants

Abstract

In response to the high energy consumption and carbon emission issues in the electrolytic aluminum industry, this paper proposes a multi-time-scale optimization and control method for electrolytic aluminum plants with high photovoltaic penetration. First, a plant architecture is established, which includes traditional power systems, renewable energy systems, and electrolytic aluminum loads. A mathematical model for flexible resources such as thermal power units, on-load tap-changing transformers, thyristor-controlled voltage regulators, saturable reactors, and electrolytic cells is developed. Based on this, a two-level optimization control strategy is designed, consisting of a day-ahead and real-time control layer: the day-ahead layer targets economic and low-carbon operation, while the real-time layer aims to stabilize the DC bus voltage. Using actual data from an electrolytic aluminum plant in Southwest China, simulations are conducted on the MATLAB 2021a platform, and the effectiveness of the strategy is verified through hardware-in-the-loop experiments. The results demonstrate that the proposed method can effectively increase the photovoltaic utilization rate, reduce thermal power output and operational costs, and decrease carbon emissions, providing a feasible solution for the green and low-carbon transformation of the electrolytic aluminum industry.

1. Introduction

In response to the high energy consumption and significant carbon emissions in the electrolytic aluminum industry in China, it is imperative to establish a new metallurgical system driven by renewable energy. The volatility and intermittency of renewable energy can cause fluctuations in the voltage and current of electrolytic cells, leading to disturbances in the magnetic and thermal fields of the electrolytic cells. This results in a reduction in current efficiency, decreased aluminum yield and quality, and increased energy and material consumption. Compared to residential and commercial loads, industrial load infrastructure is more complete, with greater response capacity and stronger feasibility for implementation. It is well-suited for high-precision flexible regulation within the multi-time-scale energy management and scheduling framework of “day-ahead, intra-day, and real-time”. The study of energy-saving optimization technologies for electrolytic cells under the integration of renewable green electricity is of great significance for the efficient utilization of renewable energy in the electrolytic industry [1,2,3].

Recent advancements in hierarchical energy management and distributed optimization frameworks have shown promise in coordinating microgrids and complex power systems. For example, Reference [4] proposed a multi-objective optimization method for a steam cracking microgrid, balancing economic and environmental goals by considering renewable energy volatility. Reference [5] introduced a two-stage hierarchical energy management system for microgrids, optimizing scheduling efficiency through Extreme Learning Machine (ELM) for better renewable energy forecasting. Reference [6] presented a stochastic energy management method for microgrids, integrating fuzzy multi-objective enhanced Grey Wolf Optimizer (MOEGWO) to address dynamic storage and demand response (DR) optimization. Reference [7] proposed a hierarchical and distributed energy management framework for AC/DC hybrid distribution systems, effectively handling multi-time-scale optimization tasks. Reference [8] developed a fully distributed parallel method for energy management in power distribution networks and discrete manufacturing systems, solving coordination challenges and ensuring computational efficiency. These studies demonstrate that hierarchical and distributed optimization methods are crucial for efficient energy management in energy-intensive industries like electrolytic aluminum plants.

Existing research has systematically explored the modeling, control, and planning of electrolytic aluminum. Reference [1] constructs an equivalent model of the “cell temperature—current efficiency—yield—power trajectory” and segmented regulation costs for day-ahead/intra-day optimization, showing that electrolytic aluminum can participate in consumption and peak shaving without compromising the process. However, the unified modeling of DC-side power quality and key device dynamics remains insufficient. Reference [9] proposes a layered cooperative frequency control strategy for ultra-high-voltage DC terminals, while Reference [10] offers a network-load interaction strategy based on process flow and regulation costs. Both studies validate that electrolytic aluminum can provide fast secondary frequency regulation via rectifiers, but the integration with economic dispatch and auxiliary service compensation still needs to be strengthened. Reference [11] addresses capacity and configuration optimization from an integrated “wind-solar-storage-aluminum” perspective, quantifying local consumption and economic improvement. However, how to implement these adjustable spaces according to plan at the operational level while simultaneously meeting real-time power quality indicators still requires supportive mechanisms.

On the methodological front, Reference [2] proposes a two-layer, multi-time-scale model predictive control (MPC) strategy, validated through hardware-in-the-loop (HIL) experiments, while Reference [3] presents a “day-ahead, intra-day, real-time” three-stage scheduling model incorporating demand response. In terms of larger-scale collaboration, References [12,13,14,15] provide solution approaches using scenario-based methods, robust optimization, and decomposition techniques for microgrids, integrated energy systems, and virtual power plants, which can be applied to electrolytic aluminum plants. On the policy side, Reference [16] reviews industrial demand response and low-carbon incentives, providing valuable insights for the design of pricing and compensation mechanisms.

Overall, these methods have improved rolling optimization and uncertainty handling abilities, but most fail to integrate bus voltage deviations/ripples and electrolytic cell state machines as hard constraints within the model.

Domestic research has also been advancing rapidly. Reference [17] proposes a control strategy for electrolytic aluminum’s participation in source-grid-load coordination for secondary frequency regulation, validating improvements in response speed and frequency quality. Reference [18] focuses on source-load coordination configuration at the plant side, showing good engineering feasibility. Reference [19] establishes a dual-layer rolling optimization model for pricing and dispatch, analyzing enterprise responses under different pricing strategies. Reference [20] constructs a layered optimization model in a “high wind power + thermal power deep peak shaving + high energy consumption load control” scenario, improving the economic efficiency of system peak shaving. Reference [21] provides an energy flow optimization evaluation method to assess adjustable capacity, offering an integrated “production-regulation” calculation approach. These studies lay the foundation for this paper while also revealing three common issues: (1) the lack of unified modeling for process constraints, device dynamics, and system operation; (2) unclear transfer between economic/low-carbon objectives and power quality/process safety objectives across the “day-ahead, intra-day, and real-time” layers; and (3) validation remains simulation-based, lacking “simulation + physical/hardware-in-the-loop” closed-loop evaluation.

Based on the above analysis, this paper presents the following innovative contributions:

(1). To address the issue of DC power quality disturbance (bus voltage deviation/ripple) caused by the fluctuations of wind and solar power and the insufficient coordination in the “source-network-load-storage-exports” system, a high photovoltaic penetration electrolytic aluminum plant system architecture is proposed. The architecture includes energy storage/flexible pathways at the plant side and establishes hard power quality constraints and fast control loops for devices at the system side.

(2). To address the common problem of electrolytic aluminum often being treated as a general load with uncoupled process and device constraints, a multi-type flexibility resource modeling approach is proposed for high photovoltaic penetration electrolytic aluminum plants. This includes unified coupling of power-temperature-yield with state machines and slope/hold boundaries, incorporating models for rectifiers, saturable reactors, and DC/DC converters.

(3). To address the issue of “day-ahead to real-time (including frequency stabilization)” target transmission and forecast deviations, a multi-time-scale optimization strategy for high photovoltaic penetration electrolytic aluminum plants is proposed. This strategy determines the economic/low-carbon optimal plan in the day-ahead stage, rolls forward corrections during operation based on the latest forecasts while leaving a margin, and sets bus voltage deviations/ripples as hard constraints in real-time. Fast adjustments via devices such as photovoltaic DC–DC converters and saturable reactors are used to achieve stable control, with integration into auxiliary services and carbon trading settlements.

(4). To verify the proposed scheme, simulation models and physical equipment are built. The effectiveness of the scheme is validated through MATLAB/Simulink + YALMIP/GUROBI solvers and semi-physical/hardware-in-the-loop testing, quantifying renewable energy consumption, operational costs, frequency, and yield improvements.

Section 2 introduces the system architecture for high photovoltaic penetration electrolytic aluminum plants; Section 3 presents the flexibility resource modeling approach for high photovoltaic penetration electrolytic aluminum plants; Section 4 discusses the multi-time-scale control strategy; Section 5 verifies the proposed multi-time-scale collaborative control strategy using actual data from an electrolytic aluminum plant in Southwest China, utilizing MATLAB/Simulink for simulation, and solving the optimization problem via MATLAB + YALMIP with GUROBI. Additionally, physical equipment is used for verification. Section 6 provides a conclusion and comparative analysis. Section 7 presents prospects. Simulation results validate the effectiveness of the proposed strategy.

2. System Architecture for High Photovoltaic Penetration Electrolytic Aluminum Plants

Figure 1 illustrates the system architecture of a high photovoltaic penetration electrolytic aluminum plant, which consists of three main components: the traditional power supply system, the renewable energy supply system, and the energy consumption system. In the traditional power supply system, thermal power units transmit electrical energy to the electrolytic aluminum plant’s AC bus through step-up transformers. The large power grid is connected to the AC bus via a step-up transformer. The AC bus is then connected to the DC bus through an on-load tap-changing transformer and rectifiers. In the renewable energy supply system, solar energy is converted into electricity by photovoltaic panels and further connected to the DC bus through a DC-DC converter. The energy consumption system mainly refers to the electrolytic aluminum industrial load, which requires long-term stable and safe operation to ensure product quality and has high power quality requirements.

Within this system, the flexible adjustable resources primarily include the thermal power units, on-load tap-changing transformers, rectifiers, and DC-DC converters. The following sections will focus on the modeling and optimization control of these four components.

3. Modeling of Flexibility Resources for High-Photovoltaic-Penetration Electrolytic Aluminum Plant

3.1. Detailed Modeling of Electrolytic Aluminum Load

The active-power–voltage external characteristic of the electrolytic aluminum load is modeled and analyzed. In the aluminum electrolysis process, all electrolytic cells are connected in series, whereas the rectifier units operate in parallel. Each electrolytic cell can be equivalently represented by a series resistance R and a counter-electromotive force E. The corresponding equivalent circuit is illustrated in Figure 2.

The rectifier in the figure consists of a thyristor phase-controlled section and a saturated reactor, while the equivalent electrolytic aluminum load comprises a voltage source E and an impedance R. Accordingly, the DC bus voltage balance equation of the electrolytic aluminum plant can be expressed as follows:

$$V_d=I_{d}R+E$$

(1)

In this equation, V_d denotes the DC bus voltage, and I_d represents the current of the equivalent electrolytic aluminum load. The equivalent resistance R and the equivalent counter-electromotive force E depend only on the electrolyte composition, cell temperature, and electrode spacing of the electrolytic cell, and can generally be considered constant for a given cell. The active power consumed by the electrolytic aluminum load P_ASL is expressed as:

$$P_{\mathrm{ASL}}=V_d{I}_d={\displaystyle \frac{V_d(V_d-E)}{R}}$$

(2)

As indicated by (2), the active power P_ASL of the high-energy-consumption electrolytic aluminum load is strongly coupled with its DC-side voltage V_d. In practical aluminum electrolysis operations, voltage regulation can be achieved through a combination of on-load tap changers (OLTCs), thyristor phase control, and saturated reactors, allowing the electrolytic aluminum load power to be adjusted continuously and smoothly over a wide range.

3.1.1. On-Load Tap-Changer (OLTC) Model

The OLTC adjusts the AC voltage supplied to the rectifier, maintaining the DC bus voltage within its allowable range. Its discrete voltage transformation can be expressed as:

$$U_{\mathrm{out},t}=U_{\mathrm{in},t}\left(1+\alpha n_t\right)$$

(3)

where $U_{\mathrm{in},t}$ and $U_{\mathrm{out},t}$ are the input and output voltages at time $t$; $\alpha$ is the step ratio of each tap change; and $n_t$ denotes the tap position (integer variable). Under steady operation, this can be approximated by a continuous mapping:

$$U_{\mathrm{out}}=U_{\mathrm{in}}(1+\alpha n)$$

(4)

where n represents the steady tap level. The range of tap movement satisfies:

$$n_{\min }\le n_t\le n_{\max }$$

(5)

where $n_{\min }$ and $n_{\max }$ are the mechanical limits. To limit mechanical wear and avoid voltage oscillations, the total number of tap changes per day is constrained by

$$\sum _{t\in \mathcal{T}}\left|\mathsf{\Delta }{n}_t\right|\le N_{\max }$$

(6)

where N_max denotes the daily switching limit.

In practical systems, deadband and dwell-time control logic are applied to suppress unnecessary switching when the voltage fluctuates slightly. This model thus preserves both the discrete nature of OLTC operation and its impact on system-level voltage regulation.

3.1.2. Thyristor-Controlled Rectifier Model

The thyristor-controlled rectifier (TCR) converts AC voltage into the DC voltage that supplies the electrolytic bus. The average DC output voltage of a six-pulse converter is given by

$$V_{dc}={\displaystyle \frac{3\sqrt{2}}{\pi }}{V}_{LL}\cos {\alpha }_f$$

(7)

where $V_{\mathrm{LL}}$ is the AC line-to-line RMS voltage, and ${\alpha }_f$ is the firing angle of the thyristor. To facilitate inclusion in linear optimization models, this nonlinear cosine function is linearized within each control interval as:

$$V_{dc}\approx a_i{V}_{LL}+b_i, {\alpha }_f\in \left[{\alpha }_i,{\alpha }_{i+1}\right]$$

(8)

where $a_{i,a_i}$ and $b_{i,b_i}$ are coefficients determined from piecewise-linear fitting. This simplified linear model is widely used for large-scale optimization while maintaining sufficient accuracy for control implementation. The rectifier’s ability to adjust ${\alpha }_f$ provides rapid voltage control, complementing the slower mechanical actions of the OLTC.

3.1.3. Saturable Reactor (SR) Model

The saturable reactor (SR) acts as a fast-response component in the DC circuit to suppress current ripple and transient fluctuations caused by load or supply disturbances. It provides a magnetic buffer between the rectifier and the electrolytic bus, improving the dynamic voltage stability of the DC link.

The instantaneous voltage-current relationship of the SR is expressed as:

$$V_{SR}=L_{eq}{\displaystyle \frac{d{I}_{SR}}{dt}}$$

(9)

where $V_{SR}$ and $I_{SR}$ are the instantaneous voltage and current of the reactor, and $L_{eq}$ is the equivalent inductance. The inductance depends on the magnetic properties of the core, which vary with the magnetic flux density. According to the electromagnetic theory of magnetic circuits:

$$L_{eq}={\displaystyle \frac{N^2{\mu }_0{\mu }_r{A}_c}{l_m}}$$

(10)

where NNN is the number of turns, $A_c$ is the effective cross-sectional area of the magnetic core, $l_m$ is the magnetic path length, and ${\mu }_r$ is the relative permeability of the core.

During normal operation, the permeability ${\mu }_r$ decreases as the core approaches magnetic saturation, reducing the effective inductance and allowing faster current variation. This behavior enables the SR to compensate rapid disturbances in the DC bus voltage within milliseconds, forming a fast inner control loop that complements the slower OLTC regulation.

3.1.4. Electrolytic Cell Flexibility Model

The electrolytic cell represents the dominant thermal–electrical coupling element in the aluminum production line. Its response to current and voltage changes determines the plant’s flexibility and efficiency. The energy balance of a single cell can be expressed as:

$$C_T{\displaystyle \frac{dT}{dt}}=I_{ASL}^{2}R-hA(T-T_{env})+P_{aux}$$

(11)

where $C_T$ is the thermal capacity of the cell, $hA(T-T_{\mathrm{env}})$ represents convective and radiative heat losses, and $P_{\mathrm{aux}}$ denotes auxiliary heating or cooling power.

Equation (11) captures the interaction between electrical power input and temperature evolution. A higher current increases ohmic losses and raises the bath temperature, while excessive cooling or heat loss can lower the temperature below the stable range. Maintaining $T$ within a narrow band is essential to ensure alumina dissolution and prevent solidification.

The feasible electrical power range is constrained as

$$P_{\min }\le P_{ASL}\le P_{\max }$$

(12)

where $P_{\min }$ and $P_{\max }$ correspond to the process-defined limits ensuring product quality and equipment safety. In real operations, the change rate of $P_{\mathrm{ASL}}$ is further restricted to avoid abrupt thermal shocks to the cell. These dynamic and steady-state limits define the “flexible operating window” of the electrolytic process, which is integrated into the optimization and dispatch models in Section 4.

3.2. Modeling of the Conventional Energy Supply System

Coal-fired units generally generate electricity through coal combustion and can participate in flexible regulation. Their active power $P_G$ and reactive power Q_G are subject to the following constraints:

$$\left\{\begin{array}{c}{P}_G^{m,\min }\le P_G^m\le P_G^{m,\max }\\ P_G^{m,R,d}\le P_G^m\left(t\right)-P_G^m\left(t-1\right)\le P_G^{m,R,u},t\ge 2\\ Q_G^{m,\min }\le Q_G^m\le Q_G^{m,\max }\end{array}\right.,m\in M$$

(13)

In Equation (23), M is the number of coal-fired units in the active distribution network, $P_G^m$ is the generation of the m unit, $P_G^{m,\min }$ and $P_G^{m,\max }$ are the active power limits, $Q_G^{m,\min }$ and $Q_G^{m,\max }$ are the reactive power limits, and $P_G^{m,R,d}$ and $P_G^{m,R,u}$ are the ramp-rate limits.

3.3. Modeling of the Renewable Energy Supply System

The photovoltaic generation power P_PV is closely related to the regional irradiance G and temperature T, and is expressed as follows:

$$P_{PV}=\left\{\begin{array}{ccc}0& ,& G<G_{\min }\\ {\eta }_{std}\left(\begin{array}{l}1-\beta \cdot \left({\displaystyle \frac{T-T_{std}}{T_{std}}}\right)\\ -\alpha \cdot \left({\displaystyle \frac{G}{G_{rated}}}-1\right)\end{array}\right)P_{PV}^{reted}& ,& \begin{array}{c}{G}_{\min }\le G\\ \le G_{reted}\end{array}\\ P_{PV}^{reted}& ,& G_{PV}^{reted}<G\end{array}\right.$$

(14)

$$0\le P_{PV}^r\le P_{PV}$$

(15)

Here, η_std and T_std are the efficiency and temperature under standard test conditions, G_rated is the rated irradiance, and $P_{\mathrm{PV}}^{\mathrm{reted}}$ is the rated PV power. α and β are the coefficients representing the effects of irradiance and temperature on PV efficiency. Since temperature variations can be neglected compared to irradiance changes, only the relationship between PV power and irradiance is considered, with T = T_std held constant. Moreover, allowing for a certain degree of curtailment, $P_{PV}^r$ is defined as the actual PV output power.

4. Multi-Time-Scale Control Strategy for High-Penetration PV Electrolytic Aluminum Parks

The day-ahead to real-time multi-time-scale optimization control flow is shown in Figure 3. First, input the allocation of various resources in the system (mainly flexibility resources), further input the power prediction curve of renewable energy generation and the load prediction curve to conduct multi-time scale optimization control, and finally output the real-time control instructions of the equipment, thereby guiding the system to optimize its operation. In the day-ahead optimization model, the controlled units include PV, coal-fired units, OLTC, thyristor-controlled circuits, saturated reactors, and electrolytic cells, with the optimization objectives of economic operation and low carbon emissions. In the real-time control model, the controlled units are the PV DC–DC converters and saturated reactors, with the control objective of maintaining DC bus voltage stability.

For day-ahead dispatching, its dispatching cycle is relatively long, with the unit dispatching cycle being at the hour level. The dispatching objective is the economic operation of the system, and the optimization result is the reference value of the equipment operating power. For real-time control, its control cycle is relatively short, with the unit control cycle at the second level, and the control objective is voltage stability. The control objects and control objectives of the two are different, and they cooperate with each other to achieve the overall optimized operation of the system.

The proposed day-ahead to real-time multi-time-scale control strategy effectively addresses the uncertainties inherent in solar irradiance and load forecasting through a hierarchical, coordinated framework. This approach leverages the complementary strengths of long-term economic optimization and short-term stability control. The day-ahead scheduling stage, operating on an hourly cycle, incorporates forecast data to generate an economically optimal and low-carbon reference plan for unit commitment and power dispatch, establishing a robust baseline that anticipates expected variability.

To mitigate the inevitable deviations between day-ahead forecasts and real-time conditions, the subsequent real-time control layer acts as a dynamic corrective mechanism. Operating on a second-level cycle, it continuously adjusts the power outputs of PV DC–DC converters and saturated reactors based on actual system measurements. This rapid response ensures DC bus voltage stability against sudden, unpredicted fluctuations in PV generation or load demand, which the slower day-ahead model cannot capture.

The key advantage lies in the seamless cooperation between these scales: the day-ahead model provides a cost-effective and stable set-point, while the real-time controller handles fast disturbances, together enhancing the system’s overall robustness, economic efficiency, and reliability against forecasting uncertainties.

4.1. Day-Ahead Optimization Model-Objective Function

The objective function covers OLTCs, thyristor phase control, saturated reactors, electrolytic cells, thermal power, and photovoltaics. Based on the flexibility-resource models, a high-penetration PV electrolytic aluminum park optimization model aimed at low-carbon operation is proposed, encompassing both economic and carbon-reduction objectives. It is expressed as follows:

$$\min f_1=C_{eco}+C_{C{O}_2}$$

(16)

$$C_{eco}=C_{re}+C_G+C_{grid}$$

(17)

In the equation, $C_{eco}$ and $C_{C{O}_2}$ denote equipment operating cost and carbon emission cost, respectively. $C_{re}$, $C_G$ and $C_{grid}$ denote PV curtailment cost, thermal-generation operating cost, and grid-interaction cost, respectively, as detailed below.

• Curtailment Cost

To improve the renewable energy utilization rate, the photovoltaic curtailment cost is calculated as follows:

$$C_{re}={\displaystyle \sum _{t=1}^T{c}_{re}}\left[P_{PV}\left(t\right)-P_{PV}^r\left(t\right)\right]$$

(18)

where $C_{re}$ denotes the penalty cost per unit of curtailed photovoltaic energy.

2.

Thermal Power Generation Cost

The thermal power generation cost mainly consists of equipment depreciation, fuel cost, and fixed operating cost, expressed as follows:

$$C_G={\displaystyle \sum _{m=1}^M\left[{\displaystyle \sum _{t=1}^T\left(a_m{\left(P_G^m\left(t\right)\right)}^2+b_m{P}_G^m\left(t\right)\right)+c_m}\right]}$$

(19)

where a_m, b_m, and c_m are the cost coefficients of different components in the operating cost of the m thermal power unit. Furthermore, since the value of a is relatively small compared with the unit’s capacity, the cost function can be approximately fitted by the following linear expression:

$$C_G={\displaystyle \sum _{m=1}^M\left[{\displaystyle \sum _{t=1}^T\left(\widehat{b_m}{P}_G^m\left(t\right)\right)+\widehat{c_m}}\right]}$$

(20)

where $\widehat{b_m}$ and $\widehat{c_m}$ are the coefficients obtained from the linear fit.

3.

Grid Power Transaction Cost

$$C_{grid}={\displaystyle \sum _{t=1}^T\left[R_{grid,b}(t)P_{grid,b}(t)-R_{grid,s}(t)P_{grid,s}(t)\right]}$$

(21)

where $P_{grid,b}$, $P_{grid,s}$, $R_{grid,b}$, and $R_{grid,s}$ denote the grid power purchase, grid power sale, purchase price, and sale price, respectively.

4.

Carbon Emission Cost

Carbon emissions in the electrolytic aluminum park are mainly generated by thermal power units and the grid. Over the total scheduling period, the park’s total carbon emissions E_G are expressed as follows:

$$E_G={\displaystyle \sum _{t=1}^T\left(c_{grid}{P}_{grid,b}(t)+{\displaystyle \sum _{m=1}^M{\eta }_m{P}_G^m\left(t\right)}\right)}$$

(22)

where $c_{grid}$ is the regional carbon emission factor, and η_m is the carbon emission per unit of power generated by the m-th unit. Furthermore, to effectively curb unregulated emissions from individual enterprises, a carbon emission cost model under a tiered carbon trading mechanism $C_{C{O}_2}$ for the electrolytic aluminum park is established as follows:

$$E_{ex}=E_G-c_q{\displaystyle \sum _{m=1}^M{Q}_G^m}$$

(23)

$$C_{C{O}_2}=\left\{\begin{array}{ccc}{c}_{C{O}_2}\lambda \gamma +\left(1+2\lambda \right)E_{ex}& ,& E_{ex}\le -\gamma \\ c_{C{O}_2}\left(1+\lambda \right)E_{ex}& ,& -\gamma <E_{ex}\le 0\\ c_{C{O}_2}{E}_{ex}& ,& 0<E_{ex}\le \gamma \\ c_{C{O}_2}\gamma +c_{C{O}_2}\left(1+\mu \right)\left(E_{ex}-\gamma \right)& ,& \gamma <E_{ex}\le 2\gamma \\ \begin{array}{c}{c}_{C{O}_2}\left(2+\mu \right)\gamma \\ +c_{C{O}_2}\left(1+2\mu \right)\left(E_{ex}-2\gamma \right)\end{array}& ,& 2\gamma <E_{ex}\le 3\gamma \\ \begin{array}{c}{c}_{C{O}_2}\left(3+3\mu \right)\gamma \\ +c_{C{O}_2}\left(1+3\mu \right)\left(E_{ex}-3\gamma \right)\end{array}& ,& 3\gamma <E_{ex}\le 4\gamma \\ \begin{array}{c}{c}_{C{O}_2}\left(4+6\mu \right)\gamma \\ +c_{C{O}_2}\left(1+4\mu \right)\left(E_{ex}-4\gamma \right)\end{array}& ,& 4\gamma <E_{ex}\end{array}\right.$$

(24)

where E_q is the carbon quota of the active distribution network, c_q is the quota coefficient,$Q_G^m$ is the capacity of the m-th thermal unit, E_ex denotes the emissions exceeding the quota, $c_{C{O}_2}$ is the trading price per unit of carbon emission, μ is the carbon-emission penalty coefficient, λ is the carbon-emission reward coefficient, and γ gamma is the unit interval. To linearize the piecewise function in the above expression, the piecewise function is first abstracted as follows:

$$f\left(x\right)=\left\{\begin{array}{ccc}{a}_{1}x+b_1& ,& x\le d_1\\ a_{2}x+b_2& ,& d_1<x\le d_2\\ ⋮& & ⋮\\ a_{n-1}x+b_{n-1}& ,& d_{n-2}<x\le d_{n-1}\\ a_{n}x+b_n& ,& d_{n-1}<x\end{array}\right.$$

(25)

Further, define a set of continuous variables w and binary variables z to assist linearization, with the following relations:

$$\left\{\begin{array}{c}x={\displaystyle \sum _{k=1}^{n-1}{w}_k{d}_k}\\ f\left(x\right)={\displaystyle \sum _{k=1}^{n-1}{w}_{k}f(d_k)}\end{array}\right.$$

(26)

The following constraints must be satisfied:

$$\left\{\begin{array}{c}{w}_1\le z_1+z_2,w_2\le z_2+z_3,\dots ,w_{n-1}\le z_{n-1}+z_n\\ {\displaystyle \sum _{k=1}^{n-1}{w}_k=1},{\displaystyle \sum _{k=1}^n{z}_k=1}\end{array}\right.$$

(27)

4.2. Day-Ahead Optimization Model-Constraints

• Power Balance Constraints

The photovoltaic units, thermal power units, aluminum-electrolysis load, and grid-exchange power must satisfy the supply–demand balance, expressed as:

$${\displaystyle \sum _{m=1}^M{P}_G^m\left(t\right)}+P_{PV}^r(t)+P_{grid,b}(t)-P_{grid,s}(t)=P_{ASL}(t)$$

(28)

2.

State Variable Constraints

For the production adjustment of the aluminum electrolysis load, three operating states are defined: power-holding, power-increase, and power-decrease. Three binary variables—δ_keep(t), δ_up(t), and δ_down(t)—are introduced to indicate whether the load at time t is in the power-holding, power-increase, or power-decrease state, respectively (1 if yes, 0 if no). At any time t, the following constraint ensures that the load is in only one state:

$${\delta }_{\mathrm{keep}}(t)+{\delta }_{\mathrm{up}}(t)+{\delta }_{\mathrm{down}}(t)=1$$

(29)

3.

Maximum Regulation Times Constraints

To prevent excessive adjustment of the aluminum electrolysis load from causing wear on the mechanical components, the maximum number of adjustments within a scheduling period is limited. The constraint is expressed as follows:

$${\displaystyle \sum _{t=1}^T({\delta }_{\mathrm{up}}(t)+{\delta }_{\mathrm{down}}(t))}\le N_{\mathrm{ASL}}$$

(30)

where NASL denotes the maximum allowable number of adjustments for the aluminum electrolysis load within one scheduling period T.

4.

Continuous Regulation Constraints

Continuous adjustments of aluminum load can cause frequent fluctuations in the electrolytic cell temperature. To maintain production stability, consecutive adjustments are prohibited, including consecutive upward, downward, or simultaneous up-and-down adjustments. Let T_keep denote the minimum duration that the load must remain at a set power after each adjustment; the constraint is expressed as follows:

$$({\delta }_{\mathrm{up}}(t)+{\delta }_{\mathrm{down}}(t))+({\delta }_{\mathrm{up}}(t-1)+{\delta }_{\mathrm{down}}(t-1))\le 1$$

(31)

$$({\delta }_{\mathrm{keep}}(t-1)-{\delta }_{\mathrm{keep}}(t))(T_{\mathrm{keep}}(t-1)-T_{\mathrm{keep}})=1$$

(32)

The equipment in the aluminum smelter park must comply with standard constraints such as power limits and capacity limits, which are not reiterated here.

4.3. Real-Time Control Model

The control targets are the PV DC–DC converters and the saturated reactors, with the overall power balance equation given by:

$$C_d{\displaystyle \frac{d{V}_d}{dt}}=I_{\mathrm{PV}}^r+I_R-I_{\mathrm{d}}$$

(33)

In the equation, $C_d$ is the equivalent capacitance of the DC bus, $I_{\mathrm{PV}}^r$ is the actual output current of the PV DC–DC converter, $I_R$ is the output current of the saturated reactor, and $I_{\mathrm{d}}$ is the equivalent current of the aluminum electrolysis load. The output current of the PV DC–DC (boost) converter is calculated as follows:

$$I_{PV}^r={\displaystyle \frac{{\left(V_{PV}^r\right)}^2{d}_{PV}\cdot (1-d_{PV})}{{\eta }_{PV}\cdot R\cdot V_d}}$$

(34)

$$V_d=V_{PV}^r{\displaystyle \frac{d_{PV}}{1-d_{PV}}}$$

(35)

In the equation, $V_{PV}^r$ is the actual voltage on the PV panel side, $d_{PV}$ is the converter duty cycle, and ${\eta }_{PV}$ is the converter efficiency. The output current of the saturated reactor is calculated as follows:

$$I_R={\displaystyle \frac{V_d{l}_c\cdot (1+k\cdot I_{\mathrm{bias}}^2)}{2\pi f\cdot {\mu }_0{\mu }_i{N}^2{A}_c}}$$

(36)

In the equation, $l_c$ is the magnetic path length, $A_c$ is the core cross-sectional area, ${\mu }_0$ is the initial permeability, ${\mu }_i$ is the relative permeability, and $N$ is the number of winding turns. Real-time control can be further implemented using a PID controller, which is not elaborated here.

5. Simulation Verification and Analysis

Based on the actual operation of an aluminum electrolysis plant in Southwest China, the proposed strategy was validated through simulations using MATLAB and the YALMIP modeling toolbox, with the IBM ILOG CPLEX solver employed to compute the optimal solution. The CPLEX solver was selected for its high-performance algorithms in handling mixed-integer linear programming (MILP) problems, utilizing its advanced simplex and branch-and-cut methods to ensure computational efficiency and robust convergence. The system demonstrated satisfactory convergence characteristics, achieving a near-optimal solution within acceptable tolerance levels in a finite number of iterations, which underscores the stability and practicality of the proposed multi-time-scale optimization framework. The system simulation parameters are detailed in

Table 1. System Parameters.

Variable	Unit	Value	Title 4
Simulation horizon (T)	hours	24	24 h scheduling period
Electrolysis load ($P_{ASL}$)	MW	100~750	Load peak 700 MW, valley 100 MW
PV installed capacity ($P_{pv}^{capacity}$)	MW	400	About 50% of the maximum load
PV output curve ($P_{PV}^r$)	MW	0~400	Reaches 100% output at noon (t = 10–12)
Network loss coefficient ($P_{loss}^{coef}$)	——	0.02	Network loss calculated as 2% of the load
Generation lower limit ($P_G^{\min }$)	MW	200	Minimum technical output
Generation upper limit ($P_G^{\max }$)	MW	800	Maximum technical output
Generation ramp rate ($P_G^{ramp}$)	MW/h	100	Maximum power change between adjacent periods
Tap position minimum ($Ta{p}^{\min }$)	tap	1	Minimum tap position
Tap position maximum ($Ta{p}^{\max }$)	tap	15	Maximum tap position
Max tap change rate ($\mathsf{\Delta }Ta{p}^{\max }$)	tap/h	3	Maximum tap change between adjacent periods
Reactor current minimum ($I_R^{\min }$)	kA	0	Minimum operating current
Reactor current maximum ($I_R^{\max }$)	kA	5	Maximum operating current
Reactor current change rate ($\mathsf{\Delta }{I}_R$)	kA/h	0.5	Maximum current change between adjacent periods
Grid exchange power limit ($P_{grid}^{\max }$)	MW	±300	Maximum exchange power with the main grid
Thermal generation cost coefficient (${\alpha }_G$)	104 CNY /MWh unit	0.1	——
OLTC tap-change cost coefficient (${\alpha }_{Tap}$)	104 CNY /tap unit	0.01	——
Reactor current cost coefficient (${\alpha }_R$)	104 CNY /kA unit	0.05	——

.

5.1. Day-Ahead Simulation Results

The algorithm demonstrated high computational efficiency, completing the optimization process in 1.3906 s of CPU time, which reflects the total processor time dedicated to the computation. This metric, measured under controlled conditions, serves as a key indicator of the method’s practical feasibility for real-time or large-scale applications.

The day-ahead simulation is used to evaluate the economic performance and renewable-energy utilization of the proposed multi-time-scale control strategy over a full-day operation. The results show that the strategy achieves coordinated operation among thermal generation, photovoltaic generation, and the power grid, meeting the aluminum electrolysis load while maximizing PV utilization, reducing total system cost, and ensuring secure grid interaction.

The simulation results demonstrate the effectiveness of the proposed multi-time-scale control strategy for the high-photovoltaic aluminum smelting park. Under the optimized operation, the total system cost was reduced to 9.5949 million CNY. The PV installation reached 400 MW, achieving a maximum penetration rate of 53.33% and an average utilization rate of 33.33%. Meanwhile, the average output of thermal units decreased to 398 MW, representing a 24.61% reduction, and the average grid exchange power was maintained at 7.14 MW. These results indicate that the proposed strategy successfully enhances renewable energy utilization, reduces conventional generation, and minimizes grid dependency, while maintaining system stability and operational feasibility.

To demonstrate the adaptability of the proposed model to photovoltaic power of different scales and loads of different scales, the initial photovoltaic capacity and load conditions were set at 100%, and the proportion of photovoltaic power and the scale of the load were respectively changed for simulation, thereby observing their impact on the results. The results of parameter sensitivity analysis are shown in

Table 2. Sensitivity analysis.

Proportion of Photovoltaic Capacity	Load Scale	Cost (million CNY)	Maximum Penetration Rate	Solution Time (s)
50%	100%	10.88	26.67%	1.2969
75%	100%	10.11	40%	0.79688
100%	100%	9.59	53.33%	1.3906
150%	100%	9.19	80%	1.0312
100%	50%	8.56	106.67%	1.1406
100%	75%	9.10	71.11%	1.2188
100%	100%	9.59	53.33%	1.3906
100%	150%	17.85	35.56%	0.9375

.

It can be seen in Table 2 that the proposed scheme demonstrates good adaptability: When the photovoltaic capacity was raised from 50% to 150% (with the load scale fixed at 100%), the system cost gradually decreased from 10.88 million CNY to 9.19 million CNY, and the maximum penetration rate of photovoltaic power significantly increased from 26.67% to 80%. This indicates that the scheme can effectively support a high proportion of photovoltaic access and achieve economic operation. In terms of load scale changes, when the photovoltaic capacity is fixed at 100%, the load increases from 50% to 100% while still maintaining good economic efficiency (the cost rises from 8.56 million CNY to 9.59 million CNY), and the penetration rate drops from 106.67% to 53.33%. However, when the load further increased to 150%, the cost rose sharply to 17.85 million CNY, and the penetration rate dropped to 35.56%, indicating that the applicability of the solution was limited when the load was significantly higher than the photovoltaic capacity. The solution time remains stable at around 1 s in all scenarios, demonstrating real-time control capabilities. In conclusion, this solution is particularly suitable for scenarios where the photovoltaic capacity is sufficient and the load scale is moderate.

As shown in Figure 4, the power balance between photovoltaic generation, thermal power, and grid supply is presented. It illustrates how renewable energy (PV) contributes to meeting energy demand, especially during peak solar hours, while thermal power and grid exchange supplement the system during low solar generation periods. The coordination of these power sources reduces reliance on thermal generation, optimizing overall system efficiency and reducing operational costs.

As shown in Figure 5, the status of control devices, such as on-load tap-changing transformers (OLTC), saturated reactors, and thyristor-controlled rectifiers, is presented during the multi-time-scale optimization process. It highlights how these devices adjust according to the day-ahead scheduling and real-time control strategies. The coordination between long-term economic objectives and short-term voltage stability ensures the optimal operation of all devices within their specified limits.

As shown in Figure 6, the composition of power sources used to meet the energy demand in the electrolytic aluminum park is presented. The figure illustrates the relative contributions of thermal power, photovoltaic generation, and grid power. During peak solar hours, photovoltaic energy significantly reduces the reliance on thermal generation. The integration of PV energy helps minimize operational costs, ensuring that the system operates efficiently throughout the day.

As shown in Figure 7, the power exchanged with the grid to maintain overall system balance is presented. It illustrates how the electrolytic aluminum park interacts with the grid, exchanging power as needed to meet energy demand while minimizing reliance on the grid, especially during periods of high renewable energy generation. This interaction ensures the system remains balanced and stable, even during fluctuating power generation.

Figure 8 presents the variation of the photovoltaic penetration rate throughout the day is presented. The penetration rate is calculated as the ratio of PV power output to total energy demand. It reflects the system’s ability to integrate PV energy into its operations, reducing reliance on thermal power during peak solar hours. The curve shows a sharp increase in PV penetration during midday, when solar generation is highest.

5.2. Real-Time Control Results

The real-time simulation is carried out on the RT-Lab platform to verify the dynamic response of the proposed multi-time-scale control strategy. The results indicate that, under sudden load changes and rapid PV output fluctuations, the DC bus voltage and key power indicators remain within the permissible range, and the system responds to external disturbances on a millisecond timescale. These findings confirm that real-time simulation effectively validates the stability and robustness of the control algorithm, providing a reliable foundation for subsequent engineering applications.

As shown in Figure 9, the output of thermal power units throughout the day is presented. The figure demonstrates how thermal units adjust their output based on the optimized schedule to complement renewable energy generation. The thermal units provide additional power during periods of low renewable energy output, ensuring that the electrolytic aluminum park meets its energy demands while maintaining system stability.

As shown in Figure 10, the variation of DC bus voltage over time is presented, confirming the stability of the system under the real-time control strategy. The figure highlights how the control strategy effectively maintains voltage stability even during fluctuations in renewable generation and load demand, ensuring that the DC bus voltage remains within acceptable limits for stable operation.

6. Conclusions and Comparative Analysis

This study presents an innovative multi-time-scale optimization and control framework for high-penetration photovoltaic (PV) integrated electrolytic aluminum parks. The proposed architecture adopts a comprehensive “source-grid-load-storage-export” paradigm, supported by an integrated process-equipment-system modeling approach that enables seamless coordination across day-ahead scheduling, intraday optimization, real-time control, and frequency stabilization. A distinctive feature of our methodology is the implementation of DC bus voltage deviation/ripple as a real-time hard constraint, ensuring operational stability while maximizing renewable energy utilization.

When compared to existing optimization approaches for industrial energy systems, our framework demonstrates significant advantages in several dimensions. Conventional methods often rely on simplified physical models that cannot adequately capture the complex dynamics between energy supply and industrial processes. Similarly, while recent deep reinforcement learning applications show promise in specific areas like distribution network optimization, they typically lack the comprehensive multi-time-scale coordination essential for energy-intensive industrial applications. In contrast to PV-direct-current integration systems that focus primarily on energy conversion efficiency, our solution achieves superior performance through holistic optimization across temporal scales and system components.

The effectiveness of the proposed strategy is validated through extensive simulation and hardware-in-the-loop testing on a real-world park in Southwest China. Implementation results confirm remarkable performance metrics: achieving a 53.33% PV penetration rate with 33.33% average utilization, while reducing thermal unit output by 24.61% and maintaining grid exchange power at 7.14 MW. These outcomes demonstrate the framework’s capability to balance multiple objectives including cost reduction (total system cost minimized to 9.5949 million CNY), frequency stability maintenance, and production deviation suppression without violating critical process constraints.

The proposed framework exhibits substantial potential for broader applications in industrial energy systems. Its scalable architecture supports extension to integrated energy management encompassing electricity, thermal, cooling, steam, and gas systems. Future enhancements through online parameter calibration, digital twin implementation, and distributed robust control will further strengthen its capability to support long-term, large-scale stable operation. The methodology establishes a replicable foundation for transforming conventional energy-intensive industries into efficient, renewable-integrated smart parks, contributing significantly to industrial decarbonization efforts.

This research bridges critical gaps between renewable energy integration and industrial process optimization, offering both theoretical contributions in multi-time-scale coordination and practical value through validated engineering applications. The demonstrated success in achieving high PV penetration while maintaining operational stability provides a compelling template for similar energy-intensive industries seeking to accelerate their renewable energy transition.

7. Prospects

While this study presents a comprehensive multi-time-scale framework for PV-integrated electrolytic aluminum parks, several limitations warrant attention in future research. The current model’s reliance on forecast data introduces inherent uncertainties, particularly regarding long-term PV generation patterns and load fluctuations. Additionally, the framework would benefit from enhanced robustness against extreme weather events and grid disturbances.

Future work should focus on developing adaptive prediction models incorporating machine learning techniques to improve forecasting accuracy. The integration of artificial intelligence for real-time parameter calibration could significantly enhance system responsiveness. Furthermore, exploring blockchain technology for secure energy trading within industrial parks represents a promising direction for decentralized energy management.

Another critical area involves expanding the multi-energy coupling approach to include hydrogen storage and waste-heat recovery systems, creating more diversified energy pathways. Research should also investigate scalability issues to adapt the framework for different industrial contexts and geographic regions. The implementation of advanced cybersecurity measures will be essential as these systems become increasingly digitalized and interconnected.

Long-term studies are needed to validate the framework’s performance under varying market conditions and policy environments. Collaboration with industry partners will be crucial for testing these enhancements in real-world settings, ultimately contributing to more sustainable and resilient industrial energy systems worldwide.