In high-power and high-reliability applications such as aerospace and industrial megawatt-level power output, MSFC hybrid systems have become an important energy solution for long-term, high-power, and highly stable operation due to their superior overall performance (Eddine Halledj et al., 2025; Chen et al., 2026). Through a multi-energy architecture and intelligent energy management, this system significantly improves power output capability, system lifespan, and operational reliability.

As shown in Figure 2, the system utilizes a multi-module parallel architecture for power coordination and redundant control. Each fuel cell stack is connected to the DC bus via an independent unidirectional DC/DC converter, primarily responsible for providing continuous baseload power, thereby ensuring the basic operating time and endurance of the system. The battery pack, connected to the same bus via a bidirectional DC/DC converter, provides power regulation and dynamic support. It can rapidly absorb or release energy during load changes, startup, fault conditions, and EIS detection, maintaining bus voltage stability.

This hybrid system is highly modular and scalable, meeting the stringent weight, efficiency, and reliability requirements of aerospace applications as well as industrial-grade high-power applications exceeding megawatts, demonstrating excellent engineering applicability and control flexibility.

According to Figure 2, the bidirectional DC/DC converter of the lithium battery module can be modelled according to Formula 1: L b d dt i L , b = v b − i L , b r b − 1 − d b , 2 v cb C b d dt v cb = 1 − d b , 2 i L , b − i bus , b (1) where L b is the inductor value. i L , b is the inductor current, which is also the output current of the battery. v b is the output voltage of the battery. r b is the lumped parasitic resistance value, including the switch resistance and the inductor resistance. d b , 2 is the duty cycle value of the switch S b , 2 , v cb is the voltage value at the capacitor C b terminal, C b is the capacitance value, and i bus , b is the current value of the battery input bus.

The unidirectional boost converter of the MSFC module is modelled according to Formula 2: L fc , m d dt i fc , m = v fc , m − i fc , m r fc , m − 1 − d fc , m v bus , m = 1 , 2 , 3 , . . . , N ∑ m = 1 N C fc , m d dt v C fc , m = ∑ m = 1 N 1 − d fc , m i fc , m − i bus , fc , m = 1 , 2 , 3 , . . . , N (2) where L fc , m is the inductance of the converter corresponding to the mth FC, i fc , m is the mth inductor current of the converter corresponding to the mth FC (also the output current of the mth FC), v fc , m is the output voltage of the mth FC, r fc , m is the equivalent lumped parasitic resistance of the converter corresponding to the mth FC (including switch resistances and inductor resistances), d fc , m is the duty cycle of switch S fc , m , v C fc , m is the voltage across capacitor C fc , m , C fc , m is the capacitance of the converter corresponding to the mth FC, and i bus , fc is the total current injected into the bus by the FCs.

For the MSFC hybrid system, according to the law of conservation of energy, Formula 3 can be obtained: P bus , b + ∑ m = 1 N P bus , fc , m = P bus , m = 1 , 2 , 3 , . . . , N ∑ m = 1 N i bus , fc , m = i bus , fc , m = 1 , 2 , 3 , . . . , N i bus , fc + i bus , b = i bus , m = 1 , 2 , 3 , . . . , N (3) where P bus , b is the output power of the converter corresponding to the lithium battery, P bus , fc , m is the output power of the converter corresponding to the mth FC, P bus is the load power on the bus, i bus , fc , m is the current value of the mth FC, that is input to the bus, i bus , fc is the total current value of m FCs input to the bus, and i bus is the current value of the bus.

Since the MSFC and lithium battery are connected in parallel to form a hybrid power system, the voltage relations of each fuel cell, lithium battery, and the bus can be described with the following formula: v C fc , m = v Cb = v bus (4)

From the perspective of the bidirectional converter for the battery, the MSFC hybrid system satisfies the following energy conservation requirements: d dt E Cb = P b + ∑ m = 1 N P fb , m − P r − P bus , b − P loss (5) where E cb is the energy stored in the capacitor C b of the battery converter, satisfying E Cb = 1 2 C b v Cb 2 . According to Formula 4, to stabilize v bus , it is only necessary to ensure the stability of v Cb . P b is the power provided by the lithium battery, satisfying P b = i L , b v b . P fb , m is the power from the m t h FC in the MSFC that enters the battery terminal via the bus. P r is the energy loss due to the lumped parasitic resistance in the bidirectional converter, satisfying P r = r b i b 2 . P loss represents other power losses.

In the MSFC hybrid system, to realize online EIS detection of MSFC systems and maintain DC bus voltage stability, this paper designs a control scheme as shown in Figure 3. Considering that the fuel cell itself has a slow response characteristic due to the electrochemical process, it is not suitable for performing frequent and rapid power compensation tasks. However, as the main energy source of the system, it is suitable as the power source to receive the EIS excitation signal. To ensure that the injected excitation current signal can track the specific frequency sinusoidal reference signal given by the EIS algorithm with high precision and low static error, a proportional resonant (PR) controller is used in the current loop of the DC/DC converter on the fuel cell side.

The bus voltage fluctuation caused by EIS excitation is regarded as an observable and compensable deterministic disturbance. Lithium batteries are suitable for this type of disturbance suppression task due to their high power density and fast dynamic response characteristics. To this end, a dual-loop control structure consisting of an inner loop of inductor current and an outer loop of bus voltage is adopted in the converter of the lithium battery side. The voltage outer loop is designed as an active disturbance rejection controller based on the MRESO, and its output serves as the reference signal for the current inner loop. The current inner loop adopts a PI controller to achieve fast tracking of the inductor current. The controller generates a duty cycle signal based on the inductor current feedback, which is then modulated through the PWM module to ultimately generate the corresponding switching signal to drive the converter.

Based on Formulas 12, 13, the transfer function from the total disturbance to its estimated error can be obtained as: G e s = z 2 M R s − x 2 s x 2 s = − s s + β 1 s s + β 1 + β 2 1 + s ∑ m = 1 3 G PR , m s (20)

Figure 6 shows the Bode plots of the system described by Formula 10 and Formula 20, when the EIS detection frequencies of the three stacks are 30 Hz, 50 Hz, and 100 Hz, respectively. The corresponding parameters in Figure 6a are set as: ω o = 400 and k p = 100 , while the corresponding parameters in Figure 6b are set as: k r , m = 1 and k p = 100 . As can be seen from Figure 6a, in the low-frequency and high-frequency regions, the MRESO demonstrates observation performance similar to that of the traditional ESO ( k r , m = 0 ). At the set resonant frequency ω r , m and its adjacent frequency bands, the MRESO can still accurately maintain a low gain amplitude. This shows that it has good estimation accuracy for multi-frequency periodic disturbances. Figure 6b further shows that increasing the observer bandwidth ω o can comprehensively improve the disturbance estimation capability within the entire frequency band. However, increasing the resonant gain k r , m can only promote the estimation accuracy at the EIS excitation frequency and its nearby frequency bands, and will cause the estimation accuracy to decrease at the excitation frequency cut-in and cut-out stages.

Based on Formulas 8, 9, 12, 14, the transfer function from the total disturbance to the energy of capacitor, which is located at the output end of the battery converter, can be expressed as: G h s = E cb s h s = s s + β 1 s + k p s 2 + β 1 + β 2 ∑ m = 1 3 G PR , m s s + β 2 (21)

Figure 7 presents the Bode plots of the system described by Formulas 10, 21 at three EIS detection frequencies (30 Hz, 50 Hz, and 100 Hz). The corresponding parameters in Figure 7a are set as: ω o = 400 and k p = 100 , while the corresponding parameters in Figure 7b are set as: k r , m = 1 and k p = 100 . As can be seen from Figure 7a, in the low-frequency and high-frequency regions, the anti-interference ability of the MRESO is similar to that of the traditional ESO. Near the set resonant frequency ω r , m (30 Hz, 50 Hz, and 100 Hz), due to the introduction of the multi-resonant controller, the system exhibits a higher interference suppression ability. Further analysis shows that increasing the resonant gain k r , m can enhance the anti-interference performance near the EIS characteristic frequency. The results in Figure 7b reveal that increasing the observer bandwidth ω o can improve the interference suppression ability of the system over the entire frequency range. However, too high an observer bandwidth can easily lead to problems such as excessively high noise sensitivity, difficulty in discretization, and decreased control quality.

Based on Formulas 8, 9, 12, 14, the transfer function from measurement noise to the energy of capacitor, which is located at the output end of the battery converter, can be expressed as: G d s = E cb s d s = s G h s − 1 (22)

Figure 8 shows the Bode plots of the system described by Formulas 10, 22 under the conditions of three EIS detection frequencies (30 Hz, 50 Hz, and 100 Hz). The corresponding parameter in Figure 8a is set as: ω o = 400 and k p = 100 , and the corresponding parameter in Figure 8b is set as: k r , m = 1 and k p = 100 . The analysis results show that in the low-frequency range, the noise resistance performance of MRESO is relatively close to that of traditional ESO. However, as the frequency increases to the medium and high frequency bands, boosting the resonant gain k r , m or enhancing the observer bandwidth ω o will lead to a decrease in the noise resistance performance of the system. It is worth noting that after the multi-resonance unit is embedded, the noise resistance performance of the system in the area near the EIS characteristic frequency exhibits a certain degree of improvement, although the improvement effect is not significant.

The voltage control system of the lithium battery includes inner loop current control and outer loop voltage control. As shown in Formula 23, the inner-loop current control is modeled as a first-order inertial link: i L , b s I L , b _ ref s = 1 T c i s + 1 (23) where T c i is the time constant of the inner loop control, which is generally used to evaluate the performance of the current controller. The voltage control of the outer loop is designed based on MRESO-ADRC. The stability of the closed-loop control system constructed can be analyzed by its characteristic polynomial, that is, Formula 24. T c i s 3 s + β 1 + s + k p s 2 + β 1 s + β 2 1 + s ∑ m = 1 3 G PR , m s (24)

To simplify the evaluation process of k r , m , a step-by-step design method is adopted. k r , i i = 1 , 2 , 3 is designed as one of the resonators separately and k r , m m ≠ i = 0 is fixed for the other two. The critical resonance gain k r , m c i r t is determined corresponding to each scanning frequency, and finally take k r , m = k r = min α 1 k r , 1 c i r t , α 2 k r , 2 c i r t , α 3 k r , 3 c i r t , where α m = 0.3 ∼ 0.7 is taken as the safety factor. Based on this, the equivalent open-loop transfer function with unit negative feedback is derived as shown in Formula 25: G k s = k r , i 2 s + k p ω c , i β 2 s 2 T c i s 3 s + β 1 s 2 + 2 ω c , i s + ω r , i 2 + s + k p s 2 + β 1 s + β 2 s 2 + 2 ω c , i s + ω r , i 2 (25)

Based on the generalized root locus method, the critical stability value of the resonant gain k r , i for different EIS frequencies and observer bandwidths can be obtained, as shown in Figure 9. Figure 9a corresponds to parameter k p = 100 and T c i = 0.01 m s , while Figure 9b corresponds to parameter k p = 100 and T c i = 0.05 m s . The results show that in the low-frequency region, the critical stability value of k r , i is significantly affected by the observer bandwidth ω o . As the EIS frequency ω r , i increases, the influence of the observer bandwidth gradually decreases. Meanwhile, the EIS frequency itself also has an important influence on the stability boundary of the resonant gain. The comparisons between Figures 9a,b shows that reducing the current inner loop time constant T c i or lowering the observer bandwidth ω o can help improve the critical stability value of k r , i . Therefore, in actual control system design, it is necessary to reasonably select the resonant gain based on the specific EIS detection frequency to ensure stable system operation.

The effectiveness of the proposed MRESO was verified in the MATLAB/Simulink simulation environment. During actual operation, fuel cells usually undergo three polarization processes: activation polarization, ohmic polarization, and concentration polarization, and have the characteristics of small ohmic resistance and large interfacial double-layer capacitance. The equivalent impedance of the fuel cell is modeled using the Randles model (Harel et al., 2025), and its structure is shown in Figure 10, where R m represents the internal membrane resistance, R p represents the polarization impedance, and C dl represents the double-layer capacitance at the electrolyte interface. The specific circuit parameters of the MSFC hybrid system are shown in Table 1. In the MSFC system, the rated voltage of each PEMFC is 24 V. However, due to the existence of equivalent impedance, the actual terminal voltage is usually lower than the rated value. The output voltage of the MSFC system is increased to 48 V by the Boost circuit. The rated voltage of the lithium battery is 24 V, and it is connected to the load side through a bidirectional Buck-Boost converter. The DC bus voltage is set to 48 V, the load impedance is 8Ω, and the switching frequency is set to 50 kHz.

In the control strategy comparison and verification, the PI controller, the traditional ESO-based active disturbance rejection controller, and the MRESO-based active disturbance rejection controller were used for performance evaluation. The parameters of the voltage loop and current loop of the PI controller are designed based on experience, where the voltage loop has a proportional coefficient of 0.01 and an integral coefficient of 50, and the current loop has a proportional coefficient of 5 and an integral coefficient of 10. Meanwhile, the voltage loop parameters of the MRESO-ADRC controller are set as follows: k p = 100 , ω o = 400 , and ω c , m = 2 % ω r , m , and the current loop uses PI regulation, with a proportional coefficient of 5 and an integral coefficient of 10. When the resonant gain k r , m = 0 , the MRESO structure degenerates into a traditional ESO structure. To facilitate fair comparison, other parameters of traditional ESO are kept consistent with those of MRESO.

Figure 11 shows a schematic diagram of the system components with the MRESO-ADRC control scheme, displaying the Simulink model of the overall system components.

The test for step disturbance of bus current is shown in Figure 12. At 1 s, a step disturbance of 0.5 A was applied to the bus current, and the response of the bus voltage is shown in Figure 12b. Facing step tests, MRESO-ADRC ( k r , m = 0.24 ) has the smallest fluctuation range, but the fluctuation time is relatively long. PI control has the fastest response speed, but the largest fluctuation range.

Figures 13, 14 show the simulation results of a MSFC hybrid system under load current disturbance conditions, where k r , m = 0.24 . Figure 13 shows the dynamic response after simultaneously injecting sinusoidal current disturbances with an amplitude of 0.2 A and frequencies of 30 Hz, 50 Hz, and 100 Hz into the system load at t = 1 s. In the stage of t > 1 s, the bus current can be expressed as: i bus = 6 + 0.2 ⁡ sin 60 π t − 1 + 0.2 ⁡ sin 100 π t − 1 + 0.2 ⁡ sin 200 π t − 1 A.

The simulation results show that under the PI control and traditional ESO-based active disturbance rejection control strategies, the bus voltage fluctuations cannot be effectively suppressed, and the fluctuation ranges are 46.25∼49.58 V and 46.83∼49.18 V, respectively. However, when the proposed MRESO combined with active disturbance rejection control, the bus voltage fluctuations only briefly occur during the initial phase of disturbance injection, ranging from 47.64 to 48.42 V. Compared with traditional PI control and traditional ESO-ADRC, the voltage fluctuation amplitude is reduced by 76.58% and 66.81%, respectively. The system stabilizes after approximately 0.08 s, and the steady-state voltage fluctuation range further narrows to 47.88–48.15 V, with no significant oscillations. This performance improvement is primarily attributed to the ability of MRESO to accurately observe multi-frequency disturbance signals and the resulting rapid compensation of the lithium battery inductor current. Simulation results further verify the effectiveness of the MRESO in improving system robustness.

Figure 14 shows the simulated response of a MSFC hybrid system under sequential injection of load current disturbances, where k r , m = 0.24 . Starting at t = 1 s, sinusoidal current disturbances with an amplitude of 0.2 A and frequencies of 30 Hz, 50 Hz, and 100 Hz were sequentially injected into the system load, with each disturbance injection interval lasting 0.25 s. When 1 s < t < 1.25 s, the bus current is: i bus = 6 + 0.2 ⁡ sin 60 π t − 1 A; when 1.25 s ≤ t < 1.5 s, the bus current is: i bus = 6 + 0.2 ⁡ sin 60 π t − 1 + 0.2 ⁡ sin 100 π t − 1.25 A; and when 1.25 s ≤ t < 1.5 s, the bus current is: i bus = 6 + 0.2 ⁡ sin 60 π t − 1 + 0.2 ⁡ sin 100 π t − 1.25 + 0.2 ⁡ sin 200 π t − 1.5 A.

Simulation results show that bus voltage fluctuations are not effectively suppressed under both PI control and the traditional ESO-ADRC strategy. Specifically, during the 1∼1.25 s period, the voltage fluctuation range under PI control is 47.12∼48.85 V, and under ESO-ADRC strategy is 47.48∼48.49 V. Meanwhile, the MRESO-based ADRC control limits the fluctuation range to 47.69∼48.27 V. In the 1.25∼1.5 s stage, the voltage fluctuation expanded to 46.53∼49.46 V under PI control, 47.12∼48.88 V under ESO-ADRC, and 47.73∼48.19 V under MRESO-ADRC. After 1.5 s, the voltage fluctuation ranges under PI control, ESO-ADRC, and MRESO-ADRC strategies are 46.25∼49.58 V, 46.85∼49.18 V, and 47.85∼48.21 V, respectively. Compared with the previous two control strategies, MRESO-ADRC reduces the voltage fluctuation amplitude by 89.19% and 84.55% respectively. This performance improvement is mainly due to the ability of MRESO to accurately observe multi-frequency disturbances and the rapid compensation mechanism of the lithium battery inductor current. By comparing the simulation results in Figures 13, 14, it can be further found that under the MRESO control strategy, the sequential disturbance injection method results in a better fluctuation suppression effect than the simultaneous disturbance injection method.

Figures 15, 16 show the simulation results of a MSFC hybrid system under EIS measurement disturbance conditions. Figures 15a–c show the inductive current of the fuel cell and other resulting signal responses during simultaneous EIS detection under MRESO-ADRC, ESO-ADRC and PI control conditions. At t = 1 s, EIS measurements were simultaneously initiated for all three fuel cell stacks. The DC operating current was 14.5 A, and the AC excitation signal amplitude was 1 A, with frequencies of 30 Hz, 50 Hz, and 100 Hz, respectively. Figure 15 shows that before the disturbance was introduced, all three controllers were able to maintain the bus voltage stable near the 48 V reference value. After the EIS excitation was introduced, the bus voltage under PI control and traditional ESO control exhibited significant oscillations, fluctuating between 46.86∼48.96 V and 47.03∼48.85 V, respectively. In contrast, the MRESO-based controller only caused slight voltage fluctuations during the initial EIS phase, ranging from 47.78 to 48.29 V, and the fluctuation difference only accounts for 1.06% of the rated voltage. The obtained result was 75.71% lower than that of PI control and 71.98% lower than that of traditional ESO. The system stabilized after approximately 0.08 s, and voltage fluctuations were negligible thereafter. These results demonstrate the significant advantages of MRESO in suppressing bus voltage disturbances caused by synchronized multi-stack EIS measurements, demonstrating its robust performance and control effectiveness.

Figures 16a–c show the inductor current of the fuel cell and the resulting responses of other signals during sequential EIS testing under MRESO-ADRC, ESO-ADRC and PI control conditions. At t = 1 s, the three fuel cell stacks were sequentially activated for EIS measurements with a 0.2 s interval. Their DC operating current was 14.5 A, and their AC excitation amplitude was 1 A, with frequencies of 30 Hz, 50 Hz, and 100 Hz, respectively. Figure 16 shows that before the disturbance was introduced, all three controllers were able to stabilize the bus voltage near the target value of 48 V. After the EIS measurement excitation was introduced, the bus voltage under both PI and traditional ESO control exhibited significant oscillations. However, under the MRESO-based control strategy, the voltage fluctuation range was only 47.84∼48.22 V, and the fluctuation difference is only equivalent to 0.79% of the rated voltage, and no significant oscillations were observed. By comparing the observed variable Z_2MR and the lithium-ion battery inductor current under each controller, it can be seen that MRESO, by embedding multiple resonant units, enhances its ability to observe specific frequency disturbances. This allows the lithium-ion battery inductor current to achieve a faster and more flexible dynamic response, effectively compensating for bus voltage fluctuations, suppressing the interference introduced by sequential EIS measurements of multiple stacks, and improving the power quality of the system. It should be noted that the excitation frequency selected in the simulation is only used to verify the effectiveness of the controller, and the proposed method also has the ability to suppress disturbances of other frequencies.

Table 2 presents the suppression of periodic disturbances achieved by MRESO-ADRC, ESO-ADRC, and PI control. It should be emphasized that all three control schemes employ identical circuit component parameters. Nevertheless, only the MRESO-ADRC scheme is capable of maintaining the bus voltage steadily at 48 V during EIS detection, while the other schemes fail to deliver comparable performance. This result confirms that the enhancement in voltage stability genuinely originates from the improvement of the control strategy itself, rather than from the inherent characteristics of the circuit components.

Table 3 shows the comparison of complexity between the proposed implementation and existing solutions.