Energy Optimization of Motor-Driven Systems Using Variable Frequency Control, Soft Starters, and Machine Learning Forecasting

Abstract

This paper presents a unified modeling framework for quantifying power and energy consumption in motor-driven systems operating under variable frequency control and soft starter conditions. By formulating normalized expressions for voltage, current, and power factor as functions of motor speed, the model enables accurate estimation of instantaneous and cumulative energy use using only measurable electrical quantities. The effect of soft starter operation during startup is incorporated through ramp-based profiles, while variable frequency control is modeled through dynamic speed modulation. Analytical results show that variable speed control can achieve energy savings of up to 36.1% for sinusoidal speed profiles and up to 42.9% when combined with soft starter operation, with the soft starter alone contributing a consistent 8.6% reduction independent of the power factor. To support energy optimization under uncertain demand scenarios, a two-stage stochastic optimization framework is developed for motor sizing and control assignment, and four physics-guided machine learning models—MLP, LSTM, GRU, and XGBoost—are benchmarked to forecast normalized energy ratios from key electrical parameters, enabling rapid and interpretable predictions. The proposed framework provides a scalable, interpretable, and practical tool for monitoring, diagnostics, and smart energy management of industrial motor-driven systems.

1. Introduction

Motor-driven systems—including pumps, compressors, blowers, and fans—are critical components in industrial and infrastructure applications. Collectively, they account for over 40% of electricity consumption in the manufacturing sector globally [1]. These systems are prevalent across sectors such as HVAC, water treatment, chemical processing, and mining. However, their performance is often compromised by fixed-speed operation, which ignores real-time variations in load demand, leading to energy inefficiency, excessive mechanical stress, and limited operational flexibility. With increasing regulatory pressure for energy conservation, rising electricity costs, and the adoption of energy management frameworks such as ISO 50001 [2] there is a growing demand for scalable, interpretable, and measurement-compatible models that can evaluate and optimize the performance of motor-driven systems.

To address these challenges, significant attention has been given to technologies like variable-frequency drives (VFDs) and soft starters, which offer improved control over motor speed and reduce mechanical and electrical stress during startup and operation [3,4]. The energy-saving potential of these technologies is well-established, particularly in systems with variable loads or frequent on–off cycling. Case studies consistently show that employing variable speed drives instead of constant-speed operation yields significant energy savings across diverse settings, including water purification plants [5], industrial pumping stations, and distribution networks [6,7,8], while soft starters have likewise been shown to reduce inrush current and improve efficiency during frequent start–stop operation [9,10,11]. However, despite widespread adoption, quantifying the actual energy benefit of such control strategies remains difficult in the absence of system-specific performance models. This limitation is especially evident in industrial settings where only electrical parameters—voltage, current, and power factor—are readily available, making it challenging to assess energy performance or benchmark efficiency across systems of different sizes and configurations.

Several studies have proposed modeling approaches for motor-driven systems, leveraging either empirical relations or simplified analytical frameworks. Wang [12] introduced a frequency- and voltage-based estimation approach for VFD-motor-pump systems that leveraged available electrical measurements to approximate input power. Wang et al. [13] further integrated sectional voltage and frequency regulation with fuzzy logic for dynamic load matching in oilfield beam pumping motors, achieving measurable improvements in load balancing and energy use. Broader programmatic efforts, such as the motor challenge program [14] and its evolution under the U.S. Department of Energy’s MEASUR platform, have emphasized industrial motor efficiency but primarily focused on promoting best practices rather than developing predictive models.

More recent research has advanced model-based optimization and control strategies for motor systems. Yang et al. [15] introduced a rigid–flexible electromechanical coupling model for linear motors, incorporating adaptive genetic algorithms to optimize PID performance under mechanical deformation. Diao et al. [16] developed a multiobjective optimization approach for switched reluctance motor drives, employing subspace partitioning and finite-element analysis to balance efficiency and torque ripple. Complementary studies reinforce the practical benefits of deploying VFDs and soft starters in industrial systems [8,11], yet these evaluations are often context-specific and do not provide scalable models for predicting savings across diverse applications [14,17].

Building on physics-based models, data-driven methods have begun to be explored for their potential in energy forecasting and optimization. Cheng et al. [17] reviewed surrogate modeling techniques, highlighting their potential to accelerate electric motor optimization through statistical regressors and deep learning. Extending this direction, physics-informed neural networks (PINNs) have been applied to energy-related systems such as diesel engines, where they enable simultaneous parameter estimation and dynamic prediction from operational data [18], and to nonlinear energy supply–demand systems, where they approximate solutions of coupled differential equations and capture dynamics beyond the reach of conventional numerical solvers [19]. While PINNs represent one subset of physics-guided learning, our focus is on the broader class of physics-guided machine learning (PGML) architectures for motor-driven systems, encompassing both neural (MLP, LSTM, GRU) and non-neural (XGBoost) models.

Beyond these approaches, recent deep learning studies have demonstrated strong predictive capabilities across diverse energy applications. Lv et al. [20] integrated deep neural networks with robust model predictive control to improve adaptability in combined heat and power and hydrogen systems, while Bai et al. [21] proposed DeepGreen-Opt, which combines LSTM forecasting with adaptive particle swarm optimization for real-time industrial energy management under carbon neutrality constraints. In the transportation domain, Niranjani et al. [22] employed an EMD–CNN–SOA framework for accurate electric vehicle demand prediction, and Lai et al. [23] developed a hybrid SSA–LSTM model to enhance forecasting accuracy in industrial parks. Complementary efforts include short-term load forecasting with customized LSTM–CNN ensembles [24] and comparative evaluations of multiple ML and DL algorithms for EV energy consumption [25]. Collectively, these studies highlight the predictive power of machine learning and deep learning in energy systems, but they remain largely domain-specific, data-intensive, and often adopt black-box formulations that limit interpretability and scalability in practice.

However, despite these methodological advances, several practical challenges continue to hinder industrial deployment. Specifically, (i) many methods require detailed motor parameters or high-resolution mechanical data that is rarely available for plants [26,27]; (ii) purely data-driven models are computationally demanding and not suited to real-time or retrofit contexts; and (iii) black-box architectures reduce interpretability, a known barrier in industrial AI applications [28]. Furthermore, most models are designed for domain-specific applications (e.g., EVs, buildings, renewable forecasting) and do not generalize readily to motor systems under VFD or soft-start control.

In this context, the objective of this paper is to address these limitations by developing a unified, physically grounded, and normalized modeling framework for quantifying power consumption in motor-driven systems. The framework leverages real-time electrical signals—voltage, current, and power factor—to model energy consumption under both variable-frequency and soft-start control modes. Building on this foundation, the study introduces a two-stage stochastic optimization strategy for motor sizing and load allocation under uncertainty, and benchmarks four physics-guided machine learning (PGML) models—MLP, LSTM, GRU, and XGBoost—for forecasting normalized energy ratios from control parameters. In doing so, the work contributes (i) a normalized physics-based formulation compatible with readily available electrical measurements, (ii) a robust stochastic optimization framework for motor selection and control allocation, (iii) a comparative evaluation of diverse PGML architectures for interpretable forecasting, and (iv) validation against analytical references through sensitivity studies, demonstrating both accuracy and scalability. Together, these contributions advance the state of the art by bridging physics-based modeling with machine learning, yielding a practical and deployable framework for monitoring, diagnostics, and energy optimization in industrial motor-driven systems. Typical application scenarios include variable torque systems such as pumps and fans, as well as constant-torque systems such as conveyors and hoists.

The remainder of this paper is organized as follows. Section 2 develops a normalized, physics-based modeling framework for motor-driven systems, incorporating variable frequency control and soft starter effects using measurable electrical parameters. Section 3 presents a two-stage stochastic optimization strategy for motor sizing and control selection, along with the development and comparative evaluation of the four PGML models for rapid energy forecasting. Finally, Section 4 summarizes the main findings, discusses the limitations, and outlines future research directions.

2. Power Consumption Modeling

2.1. Variable-Frequency Power Controls

The instantaneous electrical power drawn by a motor-driven system, such as a pump or fan, is governed by the relationship between mechanical output and conversion efficiency. This relationship can be expressed as

$$P\left(t\right)=N·{\displaystyle \frac{L\left(t\right)}{\eta \left(t\right)}},$$

(1)

where $P\left(t\right)$ is the instantaneous electrical input power (in Watts), and N is the number of identical motors in operation. The term $L\left(t\right)$ denotes the instantaneous mechanical load on the motor shaft at time t, expressed as a fraction of the rated mechanical output, and $\eta \left(t\right)$ represents the instantaneous electromechanical efficiency. In practical industrial applications, the mechanical load factor $L\left(t\right)$ can be approximately estimated from the ratio of the actual drawn current to the rated current provided on the motor nameplate, subject to appropriate correction factors accounting for variations in power factor and efficiency [29].

To express the mechanical load in a dimensionless form, it is normalized by the rated mechanical power output of the motor. The instantaneous mechanical load is written as $L\left(t\right)=\widehat{L}\left(t\right)·P_r$, where $\widehat{L}\left(t\right)$ is the normalized load factor, and $P_r$ is the rated mechanical power of each motor. In terms of torque and rotational speed, $\widehat{L}\left(t\right)$ is defined as

$$\widehat{L}\left(t\right)={\displaystyle \frac{\mathcal{T}\left(t\right)·\mathcal{W}\left(t\right)}{P_r}},$$

(2)

where $\mathcal{T}\left(t\right)$ is the motor shaft torque in newton-meters, and $\mathcal{W}\left(t\right)$ is the angular velocity in radians per second. The motor efficiency $\eta \left(t\right)$ is defined as the ratio of mechanical output power to the electrical input power, and can be determined from measured electrical parameters under the assumption of a balanced three-phase system:

$$\eta \left(t\right)={\displaystyle \frac{\mathcal{T}\left(t\right)·\mathcal{W}\left(t\right)}{\sqrt{3}·V\left(t\right)·I\left(t\right)·cos\left(\varphi \left(t\right)\right)}},$$

(3)

where $V\left(t\right)$, $I\left(t\right)$, and $cos\left(\varphi \right(t\left)\right)$ represent the line voltage, line current, and power factor, respectively. Substituting Equations (2) and (3) into Equation (1) and simplifying yields the widely known and used power relation for three-phase motor systems:

$$P\left(t\right)=N·\sqrt{3}·V\left(t\right)·I\left(t\right)·cos\left(\varphi \left(t\right)\right).$$

(4)

This form is advantageous in industrial monitoring contexts, as it relies solely on electrical measurements that can be acquired using power meters, ammeters, or motor control systems. For cross-system comparison and performance tracking, it is useful to define the normalized power consumption. The normalized power $\widehat{P}\left(t\right)$ is defined as the ratio of instantaneous power to the rated total electrical input power, given by $\widehat{P}\left(t\right)=P\left(t\right)/\overline{P}$, where the total rated power $\overline{P}$ is computed as $N·P_r$. Using Equation (4), we get the normalized instantaneous power equation:

$$\widehat{P}\left(t\right)={\displaystyle \frac{\sqrt{3}·V\left(t\right)·I\left(t\right)·cos\left(\varphi \left(t\right)\right)}{P_r}},$$

(5)

which provides a dimensionless representation of energy demand, independent of motor size or quantity. To evaluate long-term performance or average energy use, we integrate the power expression to obtain the normalized energy consumption over a time interval $[0,T]$ as follows:

$$\widehat{E}={\displaystyle \frac{\sqrt{3}}{P_r}}{\int }_0^{T}V\left(t\right)·I\left(t\right)·cos\left(\varphi \left(t\right)\right)dt.$$

(6)

This formulation enables direct analysis of energy consumption trends and system behavior over time using only plant-level electrical measurements and is fully compatible with variable-speed motor operation. For systems controlled by a variable-frequency drive, the motor speed is adjusted dynamically to match the load, and the normalized speed can be expressed as $\widehat{r}\left(t\right)=RPM\left(t\right)/RP{M}_r$. Using this, and modeling voltage, current, and power factor as functions of $\widehat{r}\left(t\right)$, we get

$$V\left(t\right)=V_r·\widehat{r}\left(t\right),I\left(t\right)=I_r·{\widehat{r}}^2\left(t\right),cos\left(\varphi \left(t\right)\right)=cos\left({\varphi }_r\right)-\delta \left(1-\widehat{r}\left(t\right)\right),$$

where $V_r$, $I_r$, and $cos\left({\varphi }_r\right)$ are rated values, and $\delta$ is a small empirical correction factor. The angle ${\varphi }_r$ is the rated power factor angle corresponding to full-load operation, typically yielding $cos\left({\varphi }_r\right)\in [0.85,0.95]$, while $\delta \in [0.05,0.15]$ reflects the observed degradation in power factor at reduced speeds due to increased reactive current draw.

Substituting the expressions for voltage, current, and power factor in terms of the normalized motor speed $\widehat{r}\left(\tau \right)$, and applying the time normalization $\tau =t/T$, the normalized energy consumption becomes

$$\widehat{E}={\displaystyle \frac{\sqrt{3}·V_r·I_r·T}{P_r}}{\int }_0^1{\widehat{r}}^3\left(\tau \right)\left[cos\left({\varphi }_r\right)-\delta \left(1-\widehat{r}\left(\tau \right)\right)\right]d\tau ,$$

(7)

where $\widehat{r}\left(\tau \right)$ is the normalized speed profile. This unified form captures the dynamic influence of voltage, current, and power factor variations due to variable-frequency drive-controlled speed modulation and can be evaluated for any assumed motor speed profile. For constant-speed operation without variable-frequency drive control, where the motor runs continuously at full-rated speed (i.e., $\widehat{r}\left(t\right)=1$), the normalized energy consumption simplifies to

$${\widehat{E}}_0={\displaystyle \frac{\sqrt{3}·V_r·I_r·cos\left({\varphi }_r\right)·T}{P_r}}.$$

(8)

To quantify the impact of variable frequency-based operation relative to baseline constant-speed control, the ratio of normalized energy consumption is defined as

$${\displaystyle \frac{\widehat{E}}{{\widehat{E}}_0}}={\displaystyle \frac{1}{cos\left({\varphi }_r\right)}}{\int }_0^{1}f\left(\widehat{r}\left(\tau \right)\right)d\tau .$$

(9)

where $f\left(\widehat{r}\left(\tau \right)\right)={\widehat{r}}^3\left(\tau \right)\left[cos\left({\varphi }_r\right)-\delta \left(1-\widehat{r}\left(\tau \right)\right)\right]$. The dimensionless ratio $\widehat{E}/{\widehat{E}}_0$ reflects the relative energy savings (if $\widehat{E}/{\widehat{E}}_0<1$) or excess consumption (if $\widehat{E}/{\widehat{E}}_0>1$) due to speed modulation under variable frequency control. It depends solely on the normalized speed profile $\widehat{r}\left(\tau \right)$ and the motor’s rated power factor $cos\left({\varphi }_r\right)$.

In the absence of measured real-time motor speed data, a normalized velocity profile is assumed to characterize the effect of speed modulation under variable frequency control. Two representative profiles for the normalized motor speed $\widehat{r}\left(\tau \right)$ over the normalized time interval $\tau \in [0,1]$ are considered:

$${\widehat{r}}_1\left(\tau \right)=1-ϵ{sin}^2\left(n\pi \tau \right),{\widehat{r}}_2\left(\tau \right)=1-ϵ\left(1-{tanh}^2\left(k(\tau -0.5)\right)\right)$$

(10)

Here, $ϵ$ denotes the normalized speed dip, n is the number of modulation cycles in the sinusoidal profile, and k controls the steepness of the transition in the tanh-based profile. Both formulations ensure that the profile starts and ends at full speed ($\widehat{r}=1$) and exhibit smooth, symmetric speed reduction and recovery, consistent with typical variable-frequency drive applications. The sinusoidal profile yields greater theoretical energy savings due to broader modulation, while the tanh-based profile results in lower but more localized energy savings with a sharper dip shape.

Figure 1 presents the assumed normalized speed profiles and the corresponding normalized energy consumption $\widehat{E}/{\widehat{E}}_0$ across a range of rated power factor values $cos\left({\varphi }_r\right)$ for varying normalized dips $ϵ$. The sinusoidal profile produces broader speed modulation, while the tanh-based formulation results in sharper, more localized speed dips. In both cases, energy consumption decreases consistently with increasing $ϵ$, indicating that deeper modulation leads to greater savings under variable frequency control. For the sinusoidal profile, energy savings increase from 7.4% at $ϵ=0.05$ to 36.1% at $ϵ=0.30$, while the tanh-based profile yields lower but smoother savings ranging from 3.0% to 14.9%. Grouped bars represent different power factor values, and the annotated values indicate the average energy savings, calculated as $(1-\widehat{E}/{\widehat{E}}_0)\times 100$. Although the sinusoidal profile includes a modulation frequency parameter n, the overall energy consumption remains invariant with respect to n, since the mean of ${sin}^2\left(n\pi \tau \right)$ over $\tau \in [0,1]$ is constant. Thus, the total energy savings are primarily governed by the normalized speed dip $ϵ$, while the shape of the modulation profile also contributes noticeably, particularly at higher normalized speed dips.

2.2. Impact of Soft Starters on Power Consumption

To further refine the energy modeling framework, we now account for the impact of soft starters, which are commonly used in motor-driven systems to mitigate the inrush current and reduce mechanical stress during startup. Unlike direct-on-line starting, which applies full voltage immediately, soft starters increase the voltage gradually over a short time interval, resulting in a ramped torque and smoother acceleration.

The duration of this ramp-up period, denoted $t_s$, is typically short compared to the total operational time T. In this analysis, we define the soft start time as a fixed fraction of the operating duration, using $t_s=0.05·T$, or equivalently, ${\tau }_s=0.05$ in normalized time.

During the startup phase $0\le \tau \le {\tau }_s$, the voltage, current, and power factor follow simplified ramp profiles, consistent with typical soft starter behavior:

$$V\left(t\right)=V_r·{\displaystyle \frac{t}{t_s}},I\left(t\right)=I_r·{\left({\displaystyle \frac{t}{t_s}}\right)}^2$$

$$cos\left(\varphi \left(t\right)\right)=cos\left({\varphi }_s\right)+\left(cos\left({\varphi }_r\right)-cos\left({\varphi }_s\right)\right)·{\displaystyle \frac{t}{t_s}}$$

where $cos\left({\varphi }_s\right)$ represents the initial power factor during the early stages of the ramp, typically in the range 0.3 to 0.5, and $cos\left({\varphi }_r\right)$ is the rated full-load value.

Using (5) and the time scale $\tau =t/T$, the instantaneous power during this startup interval follows the three-phase formulation and yields a normalized startup energy term:

$${\widehat{E}}_{\mathrm{start}}={\displaystyle \frac{\sqrt{3}·V_r·I_r·t_s}{P_r}}\left({\displaystyle \frac{cos\left({\varphi }_s\right)}{4}}+{\displaystyle \frac{cos\left({\varphi }_r\right)-cos\left({\varphi }_s\right)}{5}}\right).$$

(11)

This startup energy is then combined with the energy consumed during normal operation (for $\tau >{\tau }_s$) as defined in (7) to compute the total normalized energy with a soft starter as follows:

$${\widehat{E}}_s={\widehat{E}}_{\mathrm{start}}+{\displaystyle \frac{\sqrt{3}·V_r·I_r·(T-t_s)}{P_r}}{\int }_{{\tau }_s}^{1}f\left(\widehat{r}\left(\tau \right)\right)d\tau .$$

(12)

To quantify the influence of the soft starter, we define its relative impact with respect to the constant-speed baseline as defined in (8):

$${\displaystyle \frac{{\widehat{E}}_s}{{\widehat{E}}_0}}={\displaystyle \frac{1}{cos\left({\varphi }_r\right)}}\left[{\tau }_s\left({\displaystyle \frac{cos\left({\varphi }_s\right)}{4}}+{\displaystyle \frac{cos\left({\varphi }_r\right)-cos\left({\varphi }_s\right)}{5}}\right)+(1-{\tau }_s){\int }_{{\tau }_s}^{1}f\left(\widehat{r}\left(\tau \right)\right)d\tau \right],$$

(13)

and also relative to variable-speed operation without soft start:

$${\displaystyle \frac{{\widehat{E}}_s}{\widehat{E}}}={\tau }_s·{\displaystyle \frac{{\widehat{E}}_{\mathrm{start}}}{\widehat{E}}}+(1-{\tau }_s)·{\displaystyle \frac{{\int }_{{\tau }_s}^{1}f\left(\widehat{r}\left(\tau \right)\right)d\tau }{{\int }_0^{1}f\left(\widehat{r}\left(\tau \right)\right)d\tau }},$$

(14)

where the integrand is given by $f\left(\widehat{r}\left(\tau \right)\right)={\widehat{r}}^3\left(\tau \right)\left[cos\left({\varphi }_r\right)-\delta \left(1-\widehat{r}\left(\tau \right)\right)\right]$ and remains unchanged from earlier definitions. In the case of constant-speed operation without a variable frequency control, where the motor runs continuously at full-rated speed (i.e., $\widehat{r}\left(\tau \right)=1$), we obtain the energy ratio for using a soft starter (applicable primarily for large units):

$${\displaystyle \frac{{\widehat{E}}_{s0}}{{\widehat{E}}_0}}={\displaystyle \frac{{\tau }_s}{cos\left({\varphi }_r\right)}}\left({\displaystyle \frac{cos\left({\varphi }_s\right)}{4}}+{\displaystyle \frac{cos\left({\varphi }_r\right)-cos\left({\varphi }_s\right)}{5}}\right)+{(1-{\tau }_s)}^2,$$

(15)

This formulation ensures that the soft start contribution is fully decoupled from the main operational profile and allows a direct comparison with both idealized constant-speed and realistic variable-speed operation. Although ${\tau }_s$ is relatively small (e.g., 0.05), its impact is non-negligible in systems with frequent startups or short duty cycles. Therefore, including the soft starter behavior not only improves energy estimation accuracy but also provides a more complete representation of motor performance under practical control conditions.

The results in Figure 2 show that combining variable frequency control with a soft starter significantly improves energy efficiency relative to constant-speed operation. As the normalized speed dip $ϵ$ increases, the normalized energy consumption ${\widehat{E}}_s/{\widehat{E}}_0$ decreases substantially, reaching up to 42.9% savings for sinusoidal profiles and 22.8% for tanh-based profiles at $ϵ=0.30$. This trend confirms that deeper speed modulation enhances efficiency by extending the low-speed operating duration, where power input is inherently reduced due to its cubic relation to speed. Moreover, higher-rated power factors $cos\left({\varphi }_r\right)$ yield slightly lower energy ratios, indicating that motors operating closer to ideal conditions experience lower reactive losses, further amplifying energy savings. Therefore, both the normalized speed dip and power factor contribute to maximizing the benefit of hybrid control strategies.

The bottom plots in Figure 2 highlight the incremental benefit of adding a soft starter to variable frequency control, as measured by the normalized ratio ${\widehat{E}}_s/\widehat{E}$. The additional savings range from 9.0% to 10.6% for sinusoidal profiles and from 8.7% to 9.3% for tanh-based profiles, showing consistent gains across all values of $ϵ$ and $cos\left({\varphi }_r\right)$. These improvements result from suppressing inrush current and torque overshoot during startup, which are particularly relevant in systems with frequent cycling or high mechanical inertia.

Table 1. Normalized energy consumption (15) and energy savings using a soft starter for different values of $cos\left({\varphi }_r\right)$, with ${\tau }_s=0.05$ and $cos\left({\varphi }_s\right)=0.4$.

$cos\left({\mathit{\varphi }}_{\mathit{r}}\right)$	0.75	0.80	0.85	0.90	0.95	1.00
${\widehat{E}}_{s0}/{\widehat{E}}_0$	0.9138	0.9138	0.9137	0.9136	0.9136	0.9135
Savings (%)	8.62	8.63	8.63	8.64	8.64	8.65

reinforces this observation, reporting approximately 8.6% energy savings for soft starter-only operation with ${\tau }_s=0.05$ and $cos\left({\varphi }_s\right)=0.4$, with only negligible variation across the studied range of power factors. Thus, while variable frequency control is the primary driver of energy efficiency, the soft starter adds measurable value by reducing transient losses and improving overall system performance.

To further confirm the reliability of these findings, our predictions were compared against empirical results reported in the literature. Prior studies have demonstrated energy savings between 20% and 60% with variable-frequency drive applications in motor-driven systems [6,7], while soft starters have been shown to provide approximately 5% to 11% savings by reducing inrush currents and iron losses, particularly under low-load conditions [9]. In addition, recent experimental investigations involving frequency-based control strategies have reported substantial reductions in energy losses and RMS current [30], further supporting the effectiveness of speed modulation approaches. These observations are consistent with the trends identified in the present study, reinforcing the validity and broader applicability of the proposed normalized framework for hybrid control strategies.

3. Power Consumption Forecasting and Optimization

3.1. Optimization for Variable Frequency Controls

In the two-stage stochastic framework, the second stage determines the optimal speed profile for each realized workload scenario, minimizing normalized energy consumption while ensuring the required mechanical load is satisfied. To evaluate the minimum normalized energy consumption under variable frequency control, we solve the following discretized optimization problem derived using Equation (9):

$$\underset{{\widehat{r}}_1,\dots ,{\widehat{r}}_K}{min}{\displaystyle \frac{\widehat{E}}{{\widehat{E}}_0}}={\displaystyle \frac{1}{cos\left({\varphi }_r\right)}}\sum _{k=1}^K\Delta \tau ·f\left({\widehat{r}}_k\right),$$

(16)

where $f\left({\widehat{r}}_k\right)={\widehat{r}}_k^3\left[cos\left({\varphi }_r\right)-\delta (1-{\widehat{r}}_k)\right]$, and $\Delta \tau =1/K$ with K denoting the number of uniform time discretization points. The speed profile satisfies

$$\sum _{k=1}^K\Delta \tau ·{\widehat{r}}_k=\overline{L},0\le {\widehat{r}}_k\le 1.$$

(17)

Here, $\overline{L}=W/\left(P_r·T\right)\in (0,1]$ is the normalized average mechanical load over the time horizon, where W is the remaining mechanical energy required, $P_r$ is the motor’s rated power, and T is the duration of operation.

The corresponding first-stage (here-and-now) decision addresses motor sizing, i.e., determining the number and rating of motors in a new system design, with or without variable frequency control [31,32]. Uncertainty in workload demand and operating conditions is represented by a finite set of scenarios $\mathsf{\Omega }$, each with a probability of $p_{\omega }$. The first-stage sizing decisions are fixed across all scenarios, while the second-stage speed profiles adapt optimally within each scenario. The decision variables include the number of motors N, and the rated power per motor $P_r$. The objective is to minimize the expected total electrical energy consumption:

$$\underset{N,P_r}{min}\sum _{\omega \in \mathsf{\Omega }}{p}_{\omega }·E^{\omega }$$

(18)

where the energy consumption $E^{\omega }$, for variable frequency control in scenario $\omega$ is computed as

$$E^{\omega }=E_{\mathrm{fixed}}^{\omega }+N·{\displaystyle \frac{\sqrt{3}{V}_r{I}_{r}T}{P_r}}·\left({\displaystyle \frac{1}{cos\left({\varphi }_r\right)}}\sum _{k=1}^K\Delta \tau ·f\left({\widehat{r}}_k^{\omega }\right)\right)$$

(19)

where $E_{\mathrm{fixed}}^{\omega }$ denotes the total energy consumption contributed by the pre-assigned motors, and the integral term is obtained by solving the second-stage speed optimization problem defined in (16) and (17), and yields a value in the range $(0,1]$. For constant-speed control, the integral term is fixed at unity, since the system runs continuously at full rated speed. The design variables are chosen from discrete motor options:

$$P_r\in \{P_1,P_2,\dots ,P_M\},M\in {\mathbb{Z}}_{+}$$

(20)

and the mechanical workload constraint is enforced in discretized form:

$$\sum _i{N}_{\mathrm{fixed},i}{P}_{\mathrm{fixed},i}+N·P_r·\sum _{k=1}^K\Delta \tau ·{\widehat{r}}_k^{\omega }\ge W^{\omega },\forall \omega \in \mathsf{\Omega }$$

(21)

where $W^{\omega }$ represents the total mechanical energy demand in scenario $\omega$, and $N_{\mathrm{fixed},i}$ and $P_{\mathrm{fixed},i}$ denote the number and rating of fixed motors, respectively. Electrical sizing and energy consumption calculations subsequently account for motor and drive system efficiency. Under uncertain operating conditions, this extended framework ensures that mandatory motor assignments and optimized selections are consistently incorporated into the system design.

As an illustrative case, consider a system with a total mechanical energy requirement of $W=350\mathrm{kWh}$ over a one-hour period. The mechanical energy requirement W is converted to the expected electrical energy input by dividing by the assumed motor-drive system efficiency $\eta$. For this example, $350\mathrm{kWh}/0.8\approx 437.5\mathrm{kWh}$. Two fixed motors are pre-assigned: one rated at 40 kW and another at 100 kW. The remaining load is satisfied by optional motors selected from a candidate pool with ratings of 30, 40, and 50 kW. The rated voltage and current are assumed to be 400 V and 50 A, respectively, with a rated power factor of 0.85. Using the proposed two-stage stochastic optimization framework, the minimized total expected electrical energy consumption is found to be approximately 437.50 kWh, with an optimal configuration of nine motors each rated at 30 kW. The corresponding optimal speed profile maintains a near-constant normalized value of approximately 0.778 throughout the operation, reflecting effective adjustment to maximize energy savings under the given constraints.

3.2. Physics-Guided Forecasting Using Machine Learning

A set of physics-guided machine learning models (PGMLs) was developed to forecast normalized energy consumption in motor-driven systems operating under variable frequency and soft starter controls [25,33]. The forecasting model learns a functional mapping from three key electrical control parameters— normalized speed dip $ϵ$, rated power factor $cos\left({\varphi }_r\right)$, and torque deviation factor $\delta$—to a set of four energy ratios that quantify energy performance under different operating conditions. These energy ratios are defined as

$$f_{\mathrm{PGML}}:\underset{\mathrm{input}}{\underbrace{[ϵ,cos\left({\varphi }_r\right),\delta ]}}⟶\underset{\mathrm{energy}\mathrm{ratios}}{\underbrace{\left[\widehat{E}/{\widehat{E}}_0,{\widehat{E}}_s/{\widehat{E}}_0,{\widehat{E}}_s/\widehat{E},{\widehat{E}}_{s0}/{\widehat{E}}_0\right]}}$$

where ${\widehat{E}}_s$, $\widehat{E}$, ${\widehat{E}}_0$, and ${\widehat{E}}_{s0}$ represent normalized energy consumptions under different operation modes, as defined in Equations (9) and (13)–(15).

The normalized speed dip $ϵ$, introduced as a surrogate for the motor speed profile $\widehat{r}\left(t\right)$, serves as a physically meaningful and measurable proxy for load modulation. Rather than requiring access to the full time-dependent motor speed profile, $ϵ$ can be estimated directly from observable electrical quantities. Under typical operating assumptions, it satisfies the following approximations:

$$ϵ\approx 1-\widehat{r}\left(t\right)\approx 1-\sqrt{{\displaystyle \frac{I\left(t\right)}{I_r}}}\approx 1-{\displaystyle \frac{P\left(t\right)}{P_r}}$$

(22)

where $I\left(t\right)$ and $P\left(t\right)$ denote the instantaneous current and real power, respectively, and $I_r$, $P_r$ are their rated values. This formulation enables the forecasting model to bypass the need for explicit speed profiles, streamlining both dataset generation and real-time deployment. In the forecasting model, the torque deviation factor $\delta$ is treated as a load-dependent parameter rather than a directly measurable input. It characterizes the variation of load torque with motor speed and plays a critical role in accurately predicting energy consumption under variable frequency control. Since $\delta$ cannot be directly obtained from standard electrical measurements, it is assigned based on the type of mechanical load connected to the motor. Typical values of $\delta$ are determined by the load class: variable torque loads such as pumps and fans are associated with higher torque deviations ($\delta \approx 0.10–0.15$), whereas constant torque loads such as conveyors and hoists exhibit lower deviations ($\delta \approx 0.05–0.08$). In practical implementation, users are prompted to select the load type, and the corresponding $\delta$ value is automatically assigned within the prediction framework. Advanced users may optionally specify a custom $\delta$ value if more detailed load information is available.

Normalized energy ratios were computed using physics-based formulations for two representative motor speed profiles: a sinusoidal profile and a hyperbolic tangent profile, as defined in Equation (10). These profiles were utilized solely for offline data generation and are not required during model deployment. The input feature space was discretized as follows: the normalized speed dip $ϵ$ was sampled uniformly from $[0.05, 0.50]$ at 100 discrete levels, the rated power factor $cos\left({\varphi }_r\right)$ from $[0.75, 1.00]$ at 10 levels, and the torque deviation factor $\delta$ from $[0.05, 0.15]$ at 10 levels. This discretization resulted in (100 × 10 × 10 = 10,000) unique input combinations for each speed profile. Two separate datasets were generated—one for the sinusoidal profile and one for the hyperbolic tangent profile—each containing 10,000 samples, with three input features and four target energy ratios per sample. Because the datasets were generated as Cartesian products of $ϵ$, $cos\left({\varphi }_r\right)$, and $\delta$, the input features are mutually independent with near-zero pairwise correlations. Both datasets were randomly split into 80% training and 20% validation sets and used consistently across all machine learning models considered in this work (MLP, LSTM, GRU, and XGBoost), ensuring a fair comparison across architectures.

To benchmark forecasting performance, four physics-guided machine learning models were trained on the generated datasets: a multilayer perceptron (MLP), long short-term memory network (LSTM), gated recurrent unit network (GRU), and gradient-boosted decision tree model (XGBoost) [25,34,35]. All models are physics-guided in the same sense: (i) their target outputs are derived directly from analytical energy equations, and (ii) their input features are physically interpretable electrical parameters ($ϵ,cos\left({\varphi }_r\right),\delta$). This design ensures that the learned mappings are constrained by established motor energy laws and remain aligned with system-level physics, regardless of the underlying learning architecture. The neural network implementations (MLP, LSTM, GRU) were trained on inputs normalized using Min–Max scaling and optimized with the Adam algorithm at a learning rate of ${10}^{-3}$. A weighted mean squared error (MSE) loss function was used, assigning higher weights to the ${\widehat{E}}_s/\widehat{E}$ and ${\widehat{E}}_{s0}/{\widehat{E}}_0$ ratios to prioritize forecasting fidelity for these quantities. Training was performed for up to 5000 epochs with early stopping based on validation loss, and the best-performing model checkpoints were retained. The XGBoost model was trained on the same datasets using a gradient-boosted regression framework with hyperparameters tuned for validation accuracy.

An ablation study was performed to assess the architectural sensitivity by comparing single-layer (L-1), two-layer (L-2), and three-layer (L-3) configurations for the neural network models, with results reported in

Table 2. Performance comparison of different learning architectures under sinusoidal (${\widehat{r}}_1\left(\tau \right)$) and hyperbolic tangent (${\widehat{r}}_2\left(\tau \right)$) speed profiles. Metrics are reported on the validation set.

Model	Layers	Sinusoidal Profile, ${\widehat{\mathit{r}}}_1\left(\mathit{\tau }\right)$			Tanh-Based Profile, ${\widehat{\mathit{r}}}_2\left(\mathit{\tau }\right)$
		MSE	MAE	RMSE	MSE	MAE	RMSE
MLP	L1	0.000249	0.008488	0.011453	0.000162	0.005402	0.006852
	L2	0.000069	0.004686	0.007968	0.000023	0.002810	0.004218
	L3	0.000103	0.006054	0.009778	0.000025	0.002924	0.004800
LSTM	L1	0.000068	0.003931	0.005989	0.000130	0.006437	0.009632
	L2	0.000040	0.002877	0.003813	0.000032	0.003008	0.004009
	L3	0.000015	0.002258	0.003586	0.000010	0.001749	0.002928
GRU	L1	0.000073	0.004095	0.005975	0.000046	0.003282	0.004705
	L2	0.000024	0.002577	0.003686	0.000028	0.003118	0.004755
	L3	0.000072	0.003996	0.005007	0.000059	0.003782	0.004835
XGBoost	–	—	0.000144	0.000218	—	0.000093	0.000145

. The optimal variants were MLP-2, LSTM-3, and GRU-2, each achieving the lowest error metrics across both sinusoidal (${\widehat{r}}_1\left(\tau \right)$) and hyperbolic tangent (${\widehat{r}}_2\left(\tau \right)$) speed profiles. The final architectures were chosen from the ablation study: MLP-2 (two hidden layers, 48 neurons each, Tanh activation, 5% dropout), LSTM-3 (three layers, 32 hidden units), and GRU-2 (two layers, 32 hidden units). Hyperparameters such as learning rate, dropout, and layer size were tuned empirically through grid search on the validation set to ensure stable convergence and avoid overfitting. For XGBoost, the number of trees, learning rate, and tree depth were tuned on the validation set, making it a strong non-neural benchmark. While recurrent architectures provided strong accuracy at greater depth, the multilayer perceptron remains conceptually more appropriate for this task, as it directly maps low-dimensional steady-state electrical parameters to scalar energy ratios without sequential dependencies. LSTM and GRU are therefore included primarily as benchmarking baselines to validate the robustness of the proposed framework. Validation loss is reported as weighted MSE, consistent with the training objective emphasizing ${\widehat{E}}_s/\widehat{E}$ and ${\widehat{E}}_{s0}/{\widehat{E}}_0$. For neural networks, MSE corresponds to the weighted training loss, whereas for XGBoost, only MAE and RMSE are reported.

The trained machine learning models were validated by comparing their predictions against direct physics-based calculations of normalized energy ratios. Figure 3 presents the results for the sinusoidal and hyperbolic tangent speed profiles, ${\widehat{r}}_1\left(\tau \right)$ and ${\widehat{r}}_2\left(\tau \right)$, as defined in Equation (10). The solid lines denote the analytical reference solutions, while the other curves correspond to predictions from the four physics-guided models (MLP-2, LSTM-3, GRU-2, and XGBoost). Across both profiles, all models closely follow the analytical trends for the four energy ratios—$\widehat{E}/{\widehat{E}}_0$, ${\widehat{E}}_s/{\widehat{E}}_0$, ${\widehat{E}}_s/\widehat{E}$, and ${\widehat{E}}_{s0}/{\widehat{E}}_0$. Among the models, XGBoost predictions were most closely aligned with the analytical solutions, including at larger values of $ϵ$. LSTM-3 also performed strongly, particularly for the ratio ${\widehat{E}}_s/\widehat{E}$, whereas GRU-2 exhibited slightly higher deviations in the same region. The MLP-2 remained competitive despite its simpler architecture, confirming that a feedforward model is sufficient to capture the smooth input–output mapping in this problem. Minor deviations across all models occur primarily at larger normalized speed dips, where load modulation effects are strongest. The ratio ${\widehat{E}}_{s0}/{\widehat{E}}_0$, which is theoretically independent of speed variation, is predicted consistently across models, further demonstrating the robustness of the forecasting framework. Energy savings percentages can be readily derived from these ratios, e.g., $(1-\widehat{E}/{\widehat{E}}_0)\times 100\%$ or $(1-{\widehat{E}}_s/{\widehat{E}}_0)\times 100\%$, depending on the operating scenario.

Figure 3 also illustrates sensitivities with respect to the other two input parameters, $cos\left({\varphi }_r\right)$ and $\delta$. Compared with $ϵ$, the influence of $cos\left({\varphi }_r\right)$ is relatively minor, producing only modest changes in the energy ratios over the studied range. In contrast, $\delta$ exerts a stronger effect, particularly on the ratio ${\widehat{E}}_s/\widehat{E}$, consistent with the physical role of load type (constant versus variable torque) in determining energy performance. Across all three parameters, the machine learning models reproduce the correct analytical trends with close agreement, with XGBoost and LSTM-3 showing the best alignment, followed by MLP-2 and GRU-2. This confirms that the proposed framework captures not only the overall mapping from electrical parameters to energy ratios but also the correct parameter sensitivities implied by motor physics.

4. Conclusions

This study developed a unified, physics-guided framework for modeling, forecasting, and optimizing the energy consumption of motor-driven systems operating under variable-frequency drive and soft starter controls. By deriving normalized energy formulations directly from measurable electrical parameters, the proposed model offers a robust and scalable basis for characterizing energy trends across a wide range of operating conditions without requiring detailed mechanical measurements.

Analytical evaluation of variable-speed control strategies revealed that significant energy savings can be achieved by modulating motor speed relative to load demands. Deeper speed modulation consistently resulted in greater savings, with reductions reaching up to 36.1% for sinusoidal profiles and 14.9% for hyperbolic tangent profiles under typical operating conditions. When combined with soft starter operation, additional efficiency gains of approximately 8% to 10% were realized, with total reductions reaching up to 42.9% for sinusoidal profiles and 22.8% for hyperbolic tangent profiles. The soft starter alone contributed a consistent 8.6% savings relative to constant-speed operation, largely independent of the motor’s rated power factor. These findings were validated against a broad range of power factor scenarios and were found to be consistent with empirical trends reported in the literature [6,7,8,9,30].

Building upon these insights, a two-stage stochastic optimization framework was introduced for system design, enabling optimal motor sizing and load allocation under uncertain demand scenarios. An illustrative case study demonstrated that coupling dynamic speed control with motor selection decisions substantially minimizes electrical energy consumption, offering a practical and flexible methodology for improving plant-wide efficiency under operational variability. To enable rapid deployment in industrial settings, a set of physics-guided forecasting model was developed to predict normalized energy ratios based solely on three electrical control parameters: normalized speed dip, rated power factor, and torque deviation factor. The trained model accurately replicated physics-based predictions across different operating scenarios, allowing energy forecasting without requiring time-dependent motor speed profiles during deployment. This approach offers significant practical advantages for real-time energy management and decision support.

Despite these advances, several limitations remain. The speed profiles considered in this study were idealized and may not fully capture the complexities of real-world motor behavior. The treatment of efficiency and power factor variations was empirical and did not account for temperature effects, inverter-induced harmonics, motor aging, or mechanical transients. Furthermore, the model was trained on synthetic datasets generated from assumed profiles, indicating that future validation against real-world operational data would further enhance the reliability and robustness of the framework.

Future work will focus on extending the framework to incorporate dynamic load variations, thermal effects, inverter losses, and facility-wide motor coordination strategies. Integrating field data into the training process, expanding the model to accommodate a broader range of motor types and control schemes, and coupling energy optimization with lifecycle cost analysis represent promising directions for enhancing the applicability and impact of the proposed framework. In addition, continual learning (CL) approaches [36,37] represent a promising avenue for future research, as they would enable the forecasting model to adapt incrementally to new operating conditions and aging effects without retraining from scratch. Future work will also explore probabilistic uncertainty quantification (e.g., Bayesian methods) to provide confidence bounds on forecasts, extending the scenario-based and sensitivity analyses included in this study. Overall, the results presented in this study provide a strong foundation for the systematic optimization and smart control of motor-driven systems, contributing to more sustainable and energy-efficient industrial operations.