Operating under high temperature (> 700 o C), solid oxide fuel cell (SOFC) devices can transform chemical energy from H 2 , ammonia, natural gas and other fuels into electricity and heat with high efficiency. Despite these advantages, the commercial deployment of SOFC systems faces significant challenges in maintaining safe, stable and optimal operation under dynamic power demands. These operational challenges stem from two primary factors: (1) external fuel composition variations, particularly in natural gas grid supplies with temporal hydrocarbon concentration fluctuations, and (2) internal system performance degradation over extended operation periods. To address these challenges, real-time optimization (RTO) emerges as an option for SOFC system control and optimization. RTO encompasses various optimization strategies that integrate process measurements with models to achieve the system performance optimum (efficiency, power output or heat output). Two-step approach, the conventional RTO approach in industry, first uses plant measurements to update the model parameters via nonlinear regression and then solves the optimization problem with the updated model to generate new plant inputs. This approach presents critical limitations: (1) potential constraint violation risks due to model uncertainty, and (2) inherent structural plant-model mismatch that prevents from obtaining true system optimum. Another member in the RTO family is called modifier adaptation, which can overcome the disadvantages of the two-step approach. The most important feature of this method is that it can enforce the plant to reach the optimum even in the presence of structural plant-model mismatch. Thanks to the correction terms to the cost and constraint, the modified optimization problem can match the necessary conditions of optimality with no change in the model. The difference between two methods is illustrated in Figure 1a. Notably, since the optimality of the SOFC systems is expected to be fully determined by active plant constraints (on, e.g., maximum fuel utilization, maximum stack temperature, etc.), constraint adaptation (CA) can be used instead, that is, a simpler version of modifier adaptation that does not require the estimation of plant gradients. In this study, we applied CA-based RTO algorithm to maximize the electrical efficiency of commercial SOFC system manufactured by new enerday GmbH under different power targets. SOFC system mainly consists of (1) a catalytic partial oxidation reactor (CPOX), (2) an afterburner to burn the unreacted fuel, (3) an air heat exchanger to preheat the sweep air of the stack, (4) an air mixing valve to control the ratio of non-superheated air, (5) a water heat exchanger for waste heat recovery, and (6) a stack with power capacity from 550 to 850 W power at nominal condition. As is depicted in Figure 1b, natural gas (mainly CH 4 ) was partially oxidized in the CPOX with air to maintain the stack negatrode inlet temperature around 790 ℃. The positrode air first passes through the air mixing valve, with a portion of cold air bypassing the air heat exchanger and instead passing through the startup burner at around 100 ℃, while the remaining air is heated to around 750 ℃. Two air flows are then mixed before entering the stack positrode, and adjusting the air mixing valve ratio allows for rapid control of the stack inlet air temperature, ensuring a safe stack operating temperature. Unreacted fuel from the stack is completely burned with the surplus O 2 from the positrode in the afterburner, and the exhaust is used as the heating medium to heat up both air and water for waste heat recovery. Three system inputs are manipulated, including (1) natural gas blower workload, (2) CPOX air blower workload, and (3) stack current. Constrained process variables include (1) fuel utilization (FU), (2) carbon to oxygen ratio of CPOX, (3) stack voltage, (4) CPOX outlet temperature, (5) stack outlet temperature, (6) stack inlet air temperature, and (7) afterburner outlet temperature. The experimental protocol comprised initial baseline operation (780 W) followed by dynamic power tracking (840→800→760→720→800 W) with RTO algorithm activated. For each power target, the objective is to reach the power setpoint while maximizing the system electrical efficiency with RTO iterations performed ever 3 minutes. Notably, the electrical efficiency is defined as the ratio between net power generation after deduction of the power consumption of the blowers and the chemical energy input from natural gas. The results are shown in Figure 2. The system inputs and outputs are plotted in Figure 2. Overall, the system electrical efficiency increased from 33% at the initial condition to 35.7% at 840 W and 34.6% at 800 W. Three constraints were always active, confirming that CA is enough to meet the optimum, since the activation of these three constraints sets the value of the three degrees of freedom, that is the three aforementioned systems inputs. Note that the constraint on CPOX air blower workload is active for all power setpoints – and that the same obviously holds for the constraint on electrical power. This is different for the upper bound on FU (79%) and the lower bound on stack voltage (42V): at high powers, the constraint on FU is active, but at lower powers the constraint on stack voltage becomes more restrictive and must therefore be activated .This is caused by the heavy thermal inertia of the system. The stack outlet temperature slowly and continuously decreased with power target fixed at 800 W, which hence decreased the stack voltage. When the stack voltage hit the lower bound, the RTO algorithm increased the natural blower workload to maintain the stack voltage at its lower bound of 42 V, which simultaneously decreased the FU. In Figure 2a, the system inputs changed immediately after the activation of the RTO algorithm. Switching from power at 780 W to power target at 840 W, current increased gradually to 20 A with unchanged natural gas blower workload. CPOX air blower workload decreased step by step to its lower bound of 60% and remain unchanged at different power targets. With unchanged chemical input energy, the power generated was increased from 780 W to 840 W, while the power consumption of CPOX air blower workload was decreased, contributing to an increased system electrochemical efficiency. Simultaneously, the natural gas blower workload fluctuated with time to cope with the natural gas composition variation. In Figure 2b, it was observed that the system power output reached the target within 30 minutes. This could be further reduced by increasing the RTO frequency. Combining with the record of BoP and stack temperatures in Figure 2c, all the constraints were respected during the experiment, indicating the capability of RTO algorithm to keep the system operating in the safe region. In conclusion, the proposed CA-based RTO algorithm successfully optimized the system performance with 8% relative improvement. When switching between different power setpoints, the target can be reached quickly in 30 minutes, which can be further accelerated by reducing the iteration time. The system has been kept in the safe operating region with all the constraints respected when RTO algorithm is activated. Figure 1 Figure 2