Discharge-Based DC-Bus Voltage Link Capacitor Monitoring with Repetitive Recursive Least Squares Method for Hybrid-Electric Aircraft

Abstract

Hybrid-electric aircraft require a reliable power distribution architecture. The electrical drive system is connected to the power source via a DC-link composed mostly of capacitors—one of the faultiest power electronic components. In order to ensure the safe operation of the aircraft, DC-link capacitor condition monitoring is needed. The main requirements for such an algorithm are low data consumption and the possibility to use it in generator- or battery-powered systems. The proposed discharge-based repetitive recursive least squares (RRLS) method provides satisfactory estimates utilizing small data packages. Its execution during capacitor discharge makes it independent from the power source type. Based on the capacitor’s physical parameters, the computational complexity of the estimation process is reduced. Simulation validation and experimental tests were conducted. An analysis was carried out in a capacitance range between 705 μF and 1175 μF. The effective range of the algorithm is 881 μF–1044 μF, with an estimation error of less than 5%. Additionally, a range of changes in the time constant of the multiplier of 0.1–10 was tested in the simulation study.

1. Introduction

Around 3% of the total greenhouse gas (GHG) emissions within the European Union (EU) are produced by aircraft traffic [1]. Regional flights that cover distances no longer than 500 km have the highest carbon density—carbon dioxide per revenue-passenger-km [2,3]. One of the efforts in reducing regional aircraft GHG emissions is implementing hybrid-electric propulsion systems. Such solutions require electrical power distribution systems [4] with DC voltage buses that transfers energy to the propulsion inverters via a DC-link. Capacitors, which are a main component of the DC-link, have some of the highest failure rates among power electronics [5]. Capacitor failure/degradation can cause higher DC voltage ripples, leading to propulsion motor torque pulsations [6] and faster ageing of the insulation [7]. Hence, condition monitoring of DC-link capacitors is important for hybrid-electric aircraft safety.

In this paper, a hybrid-electric drive system is considered—an electric motor is powered by a converter (Figure 1) which can utilize both the battery and the generator as its energy source [8]. The DC-link in this architecture is between the inverter and the rectifier/battery.

1.1. A Review of the Methods for Condition Monitoring

One symptom of capacitor degradation is a drop in capacitance over time [9]. It is possible to utilize voltage ripple harmonics in order to estimate the capacitance value of the DC-link [10,11,12,13]. This approach requires low-frequency ripples to be present. These low-frequency signals correspond to the capacitance-dominant impedance of the capacitor. Extracting the voltage and current signals of the capacitor in the dominant harmonic of the DC voltage ripple allows the capacitance value to be calculated. It is worth mentioning that in [13], the voltage imbalance failure mode was taken into consideration. As these solutions are applied only in power distribution systems where energy is provided by a rectified AC source, they cannot be implemented in systems with both a generator and a battery. Another method where only the AC source is considered is based on neural networks with transfer learning [14]. This method utilizes simulation data for initial training of the network (recognizing symptoms) and experimental measurements for system and noise adaptation. This approach reduces the need for experimental training data. Overall, artificial intelligence methods require training data [15,16], a sufficient amount of which may not be possible to obtain experimentally, or a detailed power electronics model [14] which poses the risk of revealing important know-how on aviation industry solutions. A different approach is presented in [17,18,19], where signal transients are obtained in order to perform capacitor health monitoring—such condition monitoring could be implemented in a hybrid-electric power distribution. These approaches introduce signal transients via stopping inverter transistor switching, resulting in an increase in the DC-link voltage transient. During this stage, the capacitance can be estimated based on the capacitor’s current integral and the value of the voltage change. However, forcing transients into an airplane could introduce a potential safety risk during flight. The electrical signal transient of the capacitor that occurs during the normal operation of an electrical drive is shown in [20]. In that article, the regeneration mode of a motor drive was considered—that signal will be mitigated using a breaking resistor. It is worth mentioning that the authors proposed an RLS algorithm for increased robustness to noise.

After a hybrid-electric airplane lands, the DC-link capacitor discharge procedure will be implemented. This type of transient does not affect the power distribution system in flight and can be used regardless of the type of power source. Discharge-based capacitor diagnostics are described in [21,22]. Both referenced methods consist of discharging the DC-link using resistors and assessing the capacitor’s condition based on the discharge time constant. According to [22], in order to improve the capacitance estimation accuracy, the following factors have to be taken into consideration: the optimal sampling time, the discharge history, and temperature.

The Repetitive Recursive Least Squares (RRLS) method [23] is less computationally complex than the AI-based solutions and can be implemented in diagnostic systems where the data sizes and computational resources are constrained due to the abundance of condition monitoring algorithms. However, the simplification of DC-link capacitor health monitoring results in a lower estimation accuracy compared to that of other various solutions. Tests on varying the capacitance have proven the RRLS method’s robustness to a change in the DC-link parameters within the expected range.

The RRLS solution presented in this article provides a discharge-based algorithm that can be implemented in both generator- and battery-supplied propulsion systems, unlike [23], where only the implementation of generator-powered systems was considered. Additionally, discharge RRLS does not require signal preprocessing, making it less complex not only in comparison to in-flight RRLS [23] but also to other discharge-based methods [22].

1.2. The Purpose of the Article

The aim of this article is to present a DC-link capacitor health monitoring method that utilizes the capacitor discharge in order to obtain satisfactory capacitance estimates for the implementation of aluminum electrolytic capacitors. The solution described below requires simple tuning and has been tested with regard to the data input size, as airplane diagnostic systems are very complex and a low data bus load is expected of every health monitoring algorithm. These two requirements are crucial for hybrid-electric aircraft diagnostics, as multiple condition monitoring procedures are carried out simultaneously. Using small data packages and simple numerical operations is substantial for conducting airplane monitoring as fast as possible. Additionally, as a safety measure, discharge RRLS utilizes a post-flight procedure—not interfering with the aircraft’s normal operation.

The system in consideration (Figure 2) for both the simulations and the experiment consists of a DC power source, a two-level DC-link, a discharging circuit, an inverter, and an induction motor. The energy source can be disconnected via a switch, and the discharging procedure can be initiated by closing the respective transistors in the discharging circuit for each level.

2. The Proposed Method

The DC-link health monitoring solution provided in this paper utilizes an RRLS approach with the capacitor voltage U_C and the current I_C during discharge as the input signals.

2.1. The Condition Monitoring Process

In order to obtain a DC-link capacitance estimate, the procedure presented in Figure 3 is implemented. When the inverter is not in operation, both levels of DC-link capacitors (C1 and C2) are disconnected from the power source discharge via designated resistors (R1 and R2). Discharge is possible when transistors T1 and T2 are opened. During the process, the required amount of U_C and I_C samples is acquired. As the DC-link is discharged only via resistors and no additional power is provided, the health monitoring procedure can be carried out in the same way for both the generator and the battery source. Afterwards, the obtained dataset is transferred to the RRLS algorithm—the main idea of this approach is to repetitively feed the same input to the RLS with the purpose of correcting the estimated capacitance value after each iteration. Such a solution allows for smaller data inputs, resulting in a reduced data bus load.

2.2. The Capacitor Model During Discharge

For low-frequency signals, the capacitor’s impedance consists mostly of the capacitance component [9]. Due to the discharge process being aperiodic, the voltage–current relation in DC-link capacitors can be described by the following equation:

$$U_C\left(n\right)=U_C\left(n-1\right)+{\displaystyle \frac{T_p}{2C}}·\left(I_C\left(n\right)+I_C\left(n-1\right)\right),$$

(1)

where Tp is the sampling time, and n is the sample number.

During DC-link discharge, relation (1) does not include the Equivalent Series Resistance (ESR) and Equivalent Series Inductance (ESL), making the calculations simpler. Additionally, the discharging resistance value is not taken into account because of the usage of both electrical signals, not voltage exclusively.

2.3. Recursive Least Squares

In the RLS method, C is estimated at each measurement step in order to minimize the squared error of the DC-link model (1)’s output U_{C_mod} and the measured U_C. The capacitor model input consists of the measured I_C value (Figure 4).

The RLS for DC-link capacitance estimation can be described by

$$S(n)={\displaystyle \frac{T_p}{2}}\left(I_C\left(n-1\right)+I_C\left(n\right)\right),$$

(2)

$$G(n)={\displaystyle \frac{\left(\left(P\left(n-1\right)S\left(n\right)\right)\right)}{\left(R+{\left(S\left(n\right)\right)}^{2}P\left(n-1\right)\right)}},$$

(3)

$$P\left(n\right)=\left(1-G\left(n\right)S\left(n\right)\right)P(n-1),$$

(4)

$$C{\left(n\right)}^{-1}=C{\left(n-1\right)}^{-1}+G\left(n\right)\left(U_C\left(n\right)-\left(S\left(n\right)C{\left(n-1\right)}^{-1}+U_C\left(n-1\right)\right)\right).$$

(5)

A singular-parameter capacitor model allows the values of the state-to-measure S, gain G, state covariance P, and measurement covariance R to be singular instead of matrices in multi-parameter models. By using this approach, the RLS procedure is made less complex.

S is derived directly from the capacitor model (1). Its role is to perform the change from the state (C) to the measurement (U_C) based on the I_C and T_P values. G is the parameter responsible for gauging how much the present input should affect the estimated C (5). This gain is calculated with help of P and R (3). P describes the uncertainty in the C estimate (process noise), while R corresponds to the measurement uncertainty (measurement noise). P is updated during the RLS operation. R is a constant that needs to be tuned in order to secure a satisfactory estimation performance.

During the algorithm’s initialization, two measurements are required to calculate S (2) and proceed to the next steps of algorithms Equations (3)–(5). The initial values of R and P must be stated—R needs to be chosen via tuning, and P is equal to 1. G is initially equal to 1 as well. The initial C value should be similar to a healthy DC-link capacitance (the highest value expected).

3. Simulation Validation

For the purpose of validating the method described in the section above, a digital model was made using the MATLAB/Simulink environment with the Simscape Electrical toolbox. The solver for the simulations is based on the backwards Euler method with a step size of 10⁻⁷ seconds. In this section, the influences of two key discharging circuit parameters, the DC-link capacitance and discharge resistance, were taken into consideration. The relationship between the discharge-based RRLS performance and the parameter change was investigated.

3.1. The Digital Model

A DC/AC converter was modeled (Figure 5). The system’s structure is similar to that presented in Figure 2. However, in the digital model, a three-phase load with resistance and inductance elements represents the induction motor. During simulations, the inverter remains in the zero-vector state, which means that one level has all three transistors opened and the other has all of them in the conducting state. This state allows the electrical current to flow through the discharge circuit resistors (R1 and R2), neglecting the influence of the load parameters on the DC-link condition monitoring algorithm.

In

Table 1. Digital model parameters.

Parameter	Value	Unit
Power source voltage	560	V
Discharge resistor resistance	1000	Ω
Load resistance	100	Ω
Load inductance	5	mH

, the most important model parameters that do not change throughout the validation process are presented.

3.2. Validation Results

Simulations that provided a proof of concept for the discharge RRLS approach were conducted using the sequence presented in Figure 3. The discharging procedure was carried out using five different DC-link capacitance values. The U_C and I_C values (Figure 6) were measured with a sampling frequency of 10 kHz and a sample count of 750 in order to represent the data bus parameters of the diagnostic system. The whole simulation represents 75 ms of capacitor discharge. For better clarity of the plots, Figure 6 consists only of a zoom (10 ms) into the whole discharging procedure. As can be seen, the lower the DC-link capacitance, the faster discharge occurs. The discharge trend may seem linear, while in reality, it is exponential—this is due to the fact that a small time range is showcased in Figure 6 in relation to the complete discharge time. Differences between the initial voltage values occur because of the different speeds of the capacitor self-discharging in the time between disconnecting from the power source to opening the discharging transistors. These differences also affect the initial I_C values.

The main advantage of the presented RRLS approach is its ability to provide more accurate estimates than those of RLS (1st iteration) with the same amount of input data. As can be seen in Figure 7, the DC-link capacitance estimation stabilizes as the iteration count increases. The estimate output number throughout one iteration is 749, with the U_C and I_C samples being 750. This is due to model (1), which requires initial values of the capacitor’s voltage and current. In both the simulations and experiments, the algorithm performs 50 iterations.

Validation of RRLS consisted of calculating the relative error values for the estimations with five different DC-link capacitance values, as shown in

Table 2. DC-link capacitor estimation value errors in relation to capacitance values.

Capacitance [μF]	Estimation Error [%]
705	0.18
881	0.80
940	0.03
1044	0.24
1175	0.11
Mean	0.27

. These capacitances were chosen as a capacitance drop of 20% is the end-of-life criterion for aluminum electrolytic capacitors [9]. The difference between the lowest and highest capacitance in Table 2 is 40%, allowing for a proper investigation of the method’s performance. For the algorithm, the R parameter (3) was manually tuned so that the mean estimation error was minimized. The R parameter directly influences the algorithm’s performance and needs to be tuned properly for the respective capacitance range.

As mentioned above, RLS corresponds to the 1st iteration of RRLS—in order to compare both conventional RLS and discharge RRLS, the estimation errors in the capacitance for the both the 1st and 50th (final) iterations are presented in

Table 3. Estimation error values for 1st and 50th iterations of simulated discharge-based RRLS in relation to DC-link capacitance.

Capacitance [μF]	Estimation Error of 1st Iteration [%]	Estimation Error of 50th Iteration [%]
705	11.11	0.18
881	7.36	0.80
940	5.43	0.03
1044	3.35	0.24
1175	0.41	0.11

. The RRLS approach provides lower errors with the same data input size compared to RLS.

3.3. The Influence of the Time Constant

The time constant during DC-link capacitor discharge is the product of the capacitance and discharge circuit resistance. During simulation tests, different resistance values for the discharging circuit were investigated in the form of multipliers, where the R parameter (3) was tuned in order to minimize the estimation error in the original resistance value (with the multiplier equal to 1). In

Table 4. DC-link capacitor estimation value errors in relation to discharge circuit resistance.

Resistance Multiplier	Estimation Error [%]
0.1	4.55
0.2	0.21
1	0.04
5	0.18
10	0.65

, it can be noticed that a 10 times smaller resistance gives a significantly higher error than a 10 times higher resistance.

This occurrence is probably connected to the decrease in the time constant, resulting in a faster discharge process. From a discharge-based RRLS perspective, the data input is continuous throughout the iterations (Figure 8). If the discharge is more rapid, the difference between the last value recorded in the ongoing iteration and the first value in the next one is bigger. This kind of U_C and I_C behavior is not compatible with model (1), where the signals have smooth transitions. Thus, the dynamics of the DC-link discharge process should be taken into consideration while applying discharge-based RRLS, keeping in mind that fast discharge may inflict higher error rates.

4. Experimental Tests

The experiment was conducted using a stand designed for testing DC-link condition monitoring methods—the DC-link capacitance is adjustable. The capacitance values under test were the same as those in the simulation (Table 2). The sampling frequency was 10 kHz. Different numbers of samples were fed into the discharge RRLS algorithm in order to observe the influence of the measurement count. A total of 300 individual measuring series were taken (60 for each capacitance value) at two different starting U_C values (30 for each capacitance value) of 79 V and 101 V. An example of the discharge process for the five different DC-link capacitance values is presented in Figure 9. As the initial voltage U_C value differs due to the usage of an AC rectified voltage source, the initial value was removed for an easier comparison. Figure 9 proves that real systems behave in the same way as the digital model—the lower the DC-link capacitance, the faster the discharge (Figure 6). Differences in the initial capacitor voltage compared to that in the simulations (Section 3.2) occur due to the constraints applied while conducting the experiments. In the simulation, a rectified three-phase voltage of around 560 V is assumed. During the experiments, two different, initially lower U_C values are taken into consideration for the purpose of minimizing the risk of capacitor parameter drift caused by degradation during the tests. The time intervals in Figure 9 are larger than those in Figure 6 because of the fact that additional system resistance and inductance, which slow down the discharging process, are not taken into account during the simulation study. As a result of the differences mentioned above, the I_C values and dynamics also differ between the simulations and experiment.

4.1. The Experimental Stand

The stand that was used in the experiment consisted of (Figure 10) two electric motors coupled by a rigid shaft, with each motor having an independent power supply. The main side (the Sh 80X-4C motor) includes a custom-made AC/DC/AC converter with current and voltage sensors. The control and measurement signals are operated using a fast prototyping ds1103 device. On the load side, the electrical motor (Sh 80X-4D) is connected to a Danfoss FC302 converter.

The AC/DC/AC converter (Figure 11) on the main side of the stand gives the ability to connect and disconnect resistors, which are in series, to DC-link capacitors, which are grouped into two levels—C1 and C2. Connecting such resistors to the capacitors cancels out the influence of the capacitance of the component, making it possible to change the overall capacitance of the DC-link. As can be noticed, the DC-link discharge current is measured using LEM LA 25-NP/SP13 transducers. U_C values were measured using an external Pintek DP-35 differential voltage probe. The rectifier consists of diodes, and the inverter is made of Insulated-Gate Bipolar Transistors (IGBTs). The measurements were taken at room temperature.

Table 5. Key parameters of experimental bench.

Parameter	Value
Singular electrical motor power [W]	1500
DC-link capacitance per capacitor [μF]	470
Discharge circuit resistance per resistor [Ω]	470
Current transducer measuring range [mA]	±500
Voltage probe DC measuring range [V]	±800
Sampling frequency [Hz]	10,000
Transistor switching frequency [Hz]	10,000

showcases the key parameters of the whole experimental set-up, including the operational and sensor parameters.

4.2. The Test Results

During the test, different counts of samples of U_C and I_C served as the data input to the diagnostic algorithm. For each amount of samples, the R parameter (3) was tuned in order to obtain the lowest mean estimation error. Regarding this work, the R value was chosen manually. However, automated tuning is also possible using, for example, metaheuristic optimization algorithms [24,25,26]. The capacitance values estimated using discharge RRLS (a description of this algorithm is given in Section 2) were compared with the real DC-link values, and their relative estimation error was calculated.

In Figure 12, the mean error values (of the same capacitance sets as those in Table 2) depending on the amount of samples are presented. As can be seen, the overall lowest estimation error is present at 1000 samples. This is due to an insufficient data size below 1000 samples, while a higher measurement count leads to a higher signal difference between iterations, as described in Section 3.3.

The estimation accuracy for a dataset of 1000 samples of U_C and I_C is shown in the

Table 6. DC-link capacitor estimation value errors in relation to capacitance values for experimental test with input of 1000 samples.

Capacitance [μF]	Estimation Error [%]
705	7.64
881	2.19
940	1.79
1044	3.93
1175	6.85
Mean	4.48

: the error for a capacitance between 881 μF and 1044 μF is below 5%. This result allows for proper aluminum electrolytic capacitor health monitoring [9]. For extreme values of 705 μF and 1175 μF, the estimation error is over 5% but still below 8% and may be acceptable in some cases. This part of the study shows the range limitations of the discharge RRLS algorithm.

In comparison to the results of the simulation validation (Table 2), those presented in Table 6 are less accurate because of the presence of noise in the measured signals and the capacitance uncertainties of the DC-link capacitor.

In order to investigate the proposed solution further, a box plot of the estimation errors in relation to the capacitance values was made for the 1000 samples (Figure 13). As can be seen below, the median error value for capacitances between 881 μF and 1044 μF is less than 5%. At 881 μF and 940 μF, the upper adjacent does not exceed a 5% estimation error, proving the effectiveness of discharge-based RRLS in this range. For a capacitance of 1044 μF, the performance is still satisfactory, as the upper limit of the interquartile range (IQR) is slightly above 5% (5.24%). Both 705 μF and 1175 μF have a median estimation error significantly above the desired threshold. No outliers are observed—this proves the robustness of the discharge-based RRLS algorithm to noise and its stability.

5. Discussion

Discharge-based RRLS allows for DC-link capacitor health monitoring using small amounts of input data—these results are obtained via repetitively feeding the same signal packages into RLS, which updates the estimations with each iteration. Measurements are conducted during the discharge procedure while the inverter is not in operation. This approach can be applied to a converter powered by both an AC-rectified source and a battery. The usage of both U_C and I_C removes the necessity of assessing the discharging resistance. The transient nature of discharging allows the capacitor model to be simplified and the computational complexity and tuning of the whole algorithm to be reduced further.

Thanks to the reduction in the data pack size required in comparison to that with conventional RLS, discharge RRLS does not constantly occupy the hybrid-electric aircraft’s data bus, allowing other diagnostic algorithms in the system to acquire data and speeding up the whole post-flight condition monitoring procedure. The non-complex mathematical structure without any matrices can be implemented in simple computing units within aircraft or reduce the computational load from more advanced ones.

The time constant of the discharge process needs to be taken into consideration, as fast transient changes cause higher estimation inadequacies (Table 4). The amount of change in U_C and I_C in a singular data pack has an impact on the algorithm error—the number of acquired samples needs to be chosen properly (Figure 12). The R parameter value (3) needs to be tuned in order to obtain satisfactory capacitance estimates. It is recommended to tune it using discharge measurements from the maximal and minimal acceptable DC-link capacitance values. It is worth noting that even after the R parameter is tuned, an accurate capacitance estimation range is limited (Table 6).

Future research regarding this method may include modified RLS in a repetitive configuration or rapid DC-link fault detection. Moreover, implementing signal processing may improve the RRLS performance. The replacement of an RLS core with a Kalman Filter could increase the effective range of the algorithm. However, such an improvement would introduce another parameter that requires tuning besides R.