## 1 Introduction  
In distribution networks, issues such as harmonic pollution, low power factors, and three-phase imbalance have become increasingly severe. To enhance their power quality, management methods using active power filters (APFs) and other active switch compensation equipment based on power electronics technology have become the focus of much research, and the application of these management methods in power systems has become increasingly widespread. These methods are particularly effective for comprehensive power quality management in distribution networks [1, 2]. The fundamental approach involves detecting the reactive current of the target and then utilizing fully controlled power electronic devices to generate currents with the required amplitude, phase, and frequency. However, the amplitude of the compensation current is constrained by the peak current capacity of the power electronic device, and since the three-phase load is random, the reactive power compensation device cannot completely compensate for the reactive current. In full compensation mode, if the reactive load current surpasses the device's compensatory capacity, the device may quickly cease operation, or the current may even damage the device [3, 4, 5]. Calculating the reference current of a compensation device is an important factor affecting the compensation effect. In recent years, according to different power theories of power systems, many scholars have studied a variety of reference current calculation methods. Based on the type of analysis method, these power theories can be roughly divided into three categories: time-domain power theory, frequency-domain power theory, and time-frequency-domain power theory. Among the time domain power theories, the instantaneous reactive power theory proposed by Japanese scholar [PERSON] in 1983 has attracted much attention. This theory considers the three-phase system uniformly and is suitable for the three-phase system with and without a middle line [6; 7; 8]. Since then, the \(d\)-\(q\) method derived from this theory has been widely used in the design of power electronic devices [9; 10]. Other time domain power theories include Fryze power theory [11], Fryze-Buchholz-Dpenbrock (FBD) power theory [12; 13], conservative power theory (CPT) [14], etc. The main frequency domain power theory is the Budeanu power theory, while the main time-frequency domain power theory is the currents' physical component (CPC) power theory [15; 16; 17; 18; 19]. Various power theories are widely used in compensation devices. [PERSON] uses the modified instantaneous reactive power to improve the output current of the compensation device and suppress the instantaneous power oscillation of the active-reactive power comprehensive optimization [20]. In order to explore the suppression mechanism of APF on low-frequency oscillation, [PERSON] established the dq coordinate system impedance model of the vehicle-grid system with APF [21]. [PERSON] proposes an efficient and reliable three-phase shunt active power filter (SAPF) based on Fryze current minimization technology, which improves the power quality of wind-solar grid-connected power generation systems [22]. In order to improve the rapidity and accuracy of the compensation reference current detection in the compensation device, [PERSON] proposed an improved FBD reactive and harmonic current detection method based on voltage sequence decomposition [23]. [PERSON] adjusts the parameters of the passive compensator through the extended Budean theory, thereby reducing reactive power and distortion power [24]. [PERSON] calculated the parameters of the compensator through the CPC power theory and proved that in the three-phase, four-wire system under a non-sinusoidal periodic waveform, the reactive and unbalanced components in the current can be completely compensated by the reactive power compensator [25]. However, when the compensation device cannot completely compensate for the reactive power, the corresponding compensation algorithm is less studied.