Integral sliding mode control for ship main cooling systems

1.1 Research Background and Motivation

As energy efficiency and emissions reduction have emerged as critical challenges in the maritime and shipbuilding industries, the integration of the control and operation of auxiliary ship systems has become increasingly influential in shaping the overall energy profile and reliability of ships [1][2][3][4]. In particular, the proliferation of electric propulsion and shipboard microgrids has made the management of voltage quality and system stability necessary to mitigate rapid, short-term load fluctuations [5][6]. Consequently, it is essential to ensure sufficient safety margins through real-time control interventions.

As shown in Figure 1, the main cooling system plays a vital role in ensuring the thermal safety of engines and power electronic devices. This system is a multi-input single-output (MISO) system because of the combined effects of several interacting factors, including the operation of variable-speed seawater pumps and three-way valves, nonlinear heat transfer characteristics of the heat exchanger, temperature disturbances at seawater and freshwater inlets, and equivalent time delays from pipelines and sensors. Therefore, a control strategy that considers both temperature-tracking performance and energy consumption (seawater pump power) is required. However, traditional PI or internal model control (IMC) methods struggle to stably handle disturbances and uncertainties while meeting saturation constraints [7].

Figure 1:

Main cooling system

Sliding mode control (SMC), a representative method in robust control, provides structural robustness against matched uncertainties and has been systematically extended to nonlinear systems through the fuzzy Takagi–Sugeno model [8][9][10][11]. Furthermore, theoretical and design frameworks that incorporate practical constraints such as output feedback, actuator saturation, state delays, and singular system dynamics have been developed, broadening the applicability of SMC to real-world processes, plants, and transportation systems [12].

Recent process control studies have introduced data-driven robust control schemes that also combine fuzzy neural network identification, sliding observers, and disturbance observers to dynamically estimate and compensate for disturbances while adaptively tuning the switching gains [13][14]. Such developments provide a solid foundation for addressing the variations in thermal–hydraulic nonlinear coupling and disturbances caused by changes in navigation conditions.

The main cooling system of a ship inherently involves nonlinearity and model mismatch, disturbances (e.g., seawater/freshwater temperature and engine load variations), time delays, and actuator saturation with output constraints. Therefore, the development of a control strategy that simultaneously achieves stable and energy-efficient operations is required.

1.2 Previous Studies and Limitations of Existing Controllers

Traditional PI and IMC-based cooling controllers frequently perform poorly when faced with model discrepancies or fluctuations in disturbances. These types of controllers often cause considerable overshoot during transient responses, steady-state errors, and energy waste due to unwarranted pump overflows. The presence of actuator saturation and time delays necessitates cautious tuning, which ultimately reduces performance [15][16].

SMC provides strong tracking performance and robustness against matched uncertainties; however, its discontinuous switching action causes chattering, which may accelerate the wear of valves or pump components and introduce undesirable noise and vibration. To mitigate these issues, smooth control methods, higher-order sliding surfaces, and adaptive gain-adjustment techniques have been proposed. However, overestimation of the disturbance magnitude can still lead to unnecessary switching activities and energy waste [17][18].

The integral SMC (ISMC) approach introduces an integral term into the sliding surface to eliminate steady-state error and ensure bounded state behavior under model uncertainties [19][20]. Nevertheless, several challenges remain when these methods are applied to real systems. Specifically, the controller must simultaneously address actuator saturation, variable time delays, and complex nonlinear thermal-hydraulic coupling.

To handle actuator saturation and output constraints, output-feedback SMC and singular value decomposition-based dimensionality reduction techniques have been proposed [21][22]. However, their experimental validation remains insufficient in complex environments such as marine cooling systems, where multiple interacting variables and nonlinear constraints coexist.

For systems with time delays and singularities, stabilization methods based on vector integral sliding surfaces have also been proposed [23]. Although theoretically effective, few studies have experimentally validated these methods in scenarios with multiple delay sources, such as thermal inertia, sensor delays, and pipeline transport delays while maintaining the system state on the sliding surface.

Additionally, methods that smoothly approximate actuator saturation and guarantee fixed-time convergence have been presented. However, research applying these approaches to thermo-fluid coupled systems, where thermal and flow dynamics are interdependent, or analyzing them from an energy consumption perspective, remains limited [24]. Furthermore, although adaptive SMC techniques that ensure the stability and continuity of the sliding manifold under time-delay conditions have been explored, further investigation is required to generalize them to distributed delay and multi-disturbance environments, such as marine cooling networks.

Consequently, an experimentally validated integrated control framework that simultaneously addresses the practical operational conditions and navigation disturbances of the main cooling system of a ship while achieving both temperature regulation and power reduction is lacking.

1.3 Research Objective

This research focused on creating a control framework that enhances the performance and energy efficiency through simulations utilizing real temperature disturbance data from an actual shipping route, shown in Figure 2.

Figure 2:

Ship navigation route

The proposed approach employs ISMC to eliminate steady-state errors and maintain stable operation within defined limits amid model uncertainties. It also ensures resilience against input discrepancies and matched uncertainties. By integrating an integral sliding surface with a departure-prevention control law that accounts for response delays and dynamic constraints, the proposed framework ensures both the reachability and stability of the closed-loop system.

2.1 Seawater Pump and Three-Way Valve

In the initial model, it is assumed that the flow rate of the seawater supply pump is directly related to its rotational speed, while any time delays induced by piping and other components are ignored. The pump operates via an induction motor, and its rotational speed fluctuates based on the supply frequency from the electric power source. Assuming that the density of seawater remains constant, the correlation between the pump performance curve and motor frequency is derived in the following. First, the flow rate of the seawater supply is described by Equation (1).

Furthermore, as shown in Figure 1, the amount of freshwater flowing from the three-way valve to the heat exchanger is directly influenced by the valve opening, which varies with the stem movement of the three-way valve. When the total freshwater flow to the three-way valve remains constant, the flow rate to the heat exchanger can be represented by Equation (2).

Simultaneously, the rate of freshwater flowing past the heat exchanger is calculated by subtracting the amount supplied to the heat exchanger from the total freshwater flow entering the three-way valve. This can be represented as Equation (3).

Next, the outlet temperature of the central cooling system is established by combining the high-temperature freshwater flowing through the bypass side of the three-way valve with the cooled freshwater that has passed through the heat exchanger. The temperature variation within the three-way valve can be represented as the difference between the total thermal energy of the bypassed freshwater and heat-exchanged freshwater entering the valve and the thermal energy of the freshwater exiting the valve. This relationship can be described mathematically as Equation (4).

The mathematical representation of the heat exchanger is subsequently formulated as two state equations. Based on the principle of energy conservation, the change in energy on the freshwater side corresponds to the difference between the energy that enters and exits the freshwater flow in addition to the energy conveyed to the seawater side via the heat exchanger.

Similarly, the change in energy on the seawater side is related to the disparity between the energy entering and leaving the seawater flow along with the energy moved from the freshwater side via the heat exchanger. These connections can be represented mathematically as shown in Equations (5) and (6).

Assuming that the temperature distribution within the seawater is constant, with no changes in the internal seawater temperature, substituting Equation (1) into Equation (6) and rearranging the terms leads to Equation (7).

By substituting Equation (7) into Equation (5) and organizing the terms concerning the variation in freshwater temperature, the resulting formula can be derived as Equation (8).

The rate of freshwater flow entering the heat exchanger, as defined in Equation (2), can be integrated into Equation (8), enabling it to be rearranged to create the first state equation, that is, Equation (9).

By substituting Equations (2) and (3) into Equation (4) and organizing the terms according to the changes in the three-way valve outlet temperature, the resulting formula is the second state equation, which can be expressed as Equation (10).

These two state equations describe the temperature dynamics of the freshwater and the mixed outlet, forming a nonlinear MISO thermal system.

To design the controller, the state-space form of Equations (9) and (10) can be expressed as

where $$ x={\left[T_f, T_{out}\right]}^T$$, u₁ denotes the seawater pump frequency input, u₂ is the three-way valve opening, and f(x) represents the uncontrolled system dynamics. In addition, g₁(x) and g₂(x) are functions that express the response of the system to control inputs u₁ and u₂, respectively. Finally, d(t) represents temperature disturbances. This formulation enables the application of SMC theory described in the next section.

2.2 Definition of the Sliding Surface

An analysis of the state equations (Equations (9) and (10)) reveals that the system dynamics fluctuate based on the inputs and external disturbances. Furthermore, this system is a MISO system, where the two control inputs consist of the power supply frequency for the seawater pump motor and the position of the three-way valve opening. Moreover, the single output is the freshwater outlet temperature from the three-way valve.

To simplify the control design, the control objectives are divided into two components, $$ T_f^{\mathrm{*}}$$ and $$ T_{out}^{\mathrm{*}}$$, and a control strategy is designed such that $$ T_f^{\mathrm{*}}$$ is equal to or lower than $$ T_{out}^{\mathrm{*}}$$. Since the target temperature is constant and the inlet freshwater and seawater temperatures, which act as disturbances, do not change abruptly, the sliding surface s(t) is defined in terms of the temperature tracking error of the freshwater outlet temperature, as expressed in Equation (12).

Here, λ is a positive design parameter (λ > 0) that determines the bandwidth of the error dynamics on the sliding surface. When the system state reaches the sliding manifold (i.e., s(t) = 0), the error dynamics are governed by the first-order differential equation $$ \dot{e}\left(t\right)+\lambda e\left(t\right)=0$$. Consequently, λ dictates the exponential convergence rate of the temperature tracking error to zero. The reaching law adopts the exponential form in Equation (13) [25],

which is standard in SMC.

2.3 Controller

3.1 Model

The simulation model was implemented in MATLAB/Simulink, as shown in Figure 3, which replicates the main cooling system of an actual ship. The model configuration follows the practical layout described in [26].

Figure 3:

Simulink simulation model. S.W.: seawater; F.W.: freshwater

The setup includes two heat exchangers, a three-way valve, and a pair of seawater supply pumps. Each pump and heat exchanger can manage 50% of the designated load, allowing both units to function simultaneously. The speeds of both seawater-supply pumps is regulated using a single controller. Figure 4 shows the variable-speed motor drive incorporated in the model. While the actual ship utilizes induction motors for the seawater supply pumps, a DC power supply library block was implemented to facilitate electrical power calculations.

Figure 4:

Seawater supply pump motor driver part

The connections between the motor speed, current usage, and torque were defined using the data sheet of the 45 kW, 4-pole, 60 Hz seawater-pump motor from the modeled vessel.

Consequently, the parameters in the Motor & Drive (System Level) blocks shown in Figure 4 were configured as reported in Table 2, and the minimum supply frequency was set to 30 Hz, in alignment with the reference vessel.

Table 2:

Parameter settings of the motor and drive model

Because the supply voltage in the model is DC, the instantaneous electrical power and total consumed energy are calculated using Equations (28) and (29), respectively.

The heat exchanger was modeled based on the manufacturer’s data sheets using the Heat Exchanger (Thermal Liquid–Thermal Liquid) block in Simulink. Its parameter settings are listed in Table 3.

Table 3:

Parameter settings of the heat exchanger model

3.2 Simulation Conditions and Controller Parameters

The simulation utilized the system model illustrated in Figure 3, which was executed in MATLAB/Simulink 2025a. The overall duration of the simulation was 48 h, representing the equivalent ship operation time. The primary goal of the cooling system control was to sustain a steady freshwater outlet temperature of 36 °C.

The freshwater and seawater inlet temperatures, acting as disturbances, were varied according to the real ship operational data, as shown in Figure 5. In this study, rather than changing the reference temperature, these two inlet temperatures were used as time-varying disturbance inputs to evaluate the robustness of the controller under realistic operating conditions.

Figure 5:

Operational data of freshwater (F.W.) and seawater (S.W.) inlet temperatures

Two controller types were applied and compared under identical environmental conditions. First, a PI–PI controller was designed using the IMC tuning method to ensure simple implementation and stable response characteristics. Second, an ISMC was developed based on the mathematical model derived in Section 2 with the aim of enhancing robustness against parameter uncertainties and external disturbances.

The performance comparison of these two controllers focused on three key aspects. First, the freshwater outlet temperature response was examined to evaluate control accuracy and stability. Second, the rotational speed and corresponding power consumption of the seawater-supply pump motor were analyzed to assess energy efficiency. Finally, the stem displacement of the three-way valve was monitored to evaluate the actuator effort and mechanical stability of the system.

The PI–PI controller was tuned according to the IMC method proposed by Jeon and Jung [26]. The gains listed in Table 4 were selected to achieve a stable closed-loop response without excessive overshoot while ensuring sufficiently fast recovery from load disturbances.

Table 4:

Parameters of PI-PI controller

The ISMC parameters are listed in Table 5. The values of the heat transfer coefficients, heat exchanger area, fluid properties, and mass flow rates were determined using the manufacturer’s data sheets and the design specifications of the modeled vessel.

Table 5:

Parameter values for SMC

Author Contributions

Conceptualization, T. Y. Jeon; Methodology, S.T. Kim and T. Y. Jeon; Software, T. Y. Jeon; Formal Analysis, T. Y. Jeon; Investigation, S.T. Kim; Resources, T. Y. Jeon; Data Curation T. Y. Jeon; Writing-Original Draft Preparation, S.T. Kim; Writing-Review & Editing, Y.C. Lee and S.T. Kim; Visualization, S.T. Kim; Supervision, author’s name; Project Administration, Y.C. Lee; Funding Acquisition, Y.C. Lee.