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Fault Detection of Power Electronic Circuit Using Wavelet Analysis in Modelica Fault Detection of Power Electronic Circuit using Wavelet Analysis in Modelica Jianbo Gao*, Yang Ji**, Johann Bals**, Ralph Kennel* *Technische Universität München Arcisstr. 21, 80333 Munich, Germany [email protected], [email protected] ** German Aerospace Center Muenchner Str. 20, 82234 Wessling, Germany [email protected], [email protected] obtain maximum flight reliability and minimum Abstract maintenance efforts, advanced failure analysis tech- nologies shall be applied to ensure correct and quick In more electric aircrafts (MEA) the electric pow- er network is important for the reliability. To prevent fault detection and isolation. It is well known that an severe faults it is the key issue to identify the faults output voltage regulated DC/DC power converter in the early stage before a complete failure happens. supplying constant power loads could de-stabilize In this paper an early stage fault detection method the network stability due to the degraded perfor- using wavelet multi-resolution analysis (MRA) for a mance of its input filter. The sensitivity study of in- regulated buck DC-DC converter is studied. Specifi- put filter parameters concerning the network stability cally, the electrolyte input capacitor is diagnosed. addressed in [2] reveals that the observation of de- The study was carried out using simulation with graded degree of the capacitor in the input filter can Modelica / Dymola. The fault features that were ex- significantly increase the network reliability. tracted from different levels of wavelet decomposi- tion provided clear information for both fast and 1.2 State of the art slow occurring faults. This method showed signifi- cant advantages compared with filter techniques. It is Reviewing the considerable development of fault concluded that wavelet transform is a suitable tool diagnosis techniques and many successful applica- for early stage fault detection of the power electron- tions attached to them in the last time [3] [4] [5], ics in MEA. In addition, the simulation language power systems keep a challenge for fault detection. Modelica provides a convenient possibility for the For overcoming this challenge intelligent methods quick design of fault detection strategy. like artificial neural networks have shown their pos- sibilities in this field [6]. Besides, the analytical Keywords: power electronics; DC-DC converter; model based technology is also obtaining more atten- fault detection; wavelet; Modelica; Dymola tion [7] [8] [9]. Signal-based methods, e.g. Fourier transform and 1 Introduction wavelet transform, also provide other possibilities to perform the fault detection and isolation. With the rash development of the new mathematical tool, 1.1 Motivation wavelet transform [10], a great amount of studies The concept of More Electric Aircraft (MEA) is have been done in different fields for fault detection. attracting increasing interest in the aircraft industry Some attempts have also been made in power elec- not only because of its potential in energy optimiza- tronics for fault detection [11] [12]. The implementa- tion, but also due to its significant advantages con- tion of wavelet transform for the post processing of cerning weight, maintenance requirements, liability Modelica simulation data has been seen, for exam- and passenger comfort [1]. For this, the electrical ple, in a study of vehicle steering, where wavelet power distribution network is playing a more im- transform was carried out in the software Matlab for portant role and facing increasing challenges in the calculating power spectra [13]. prognosis and accurate localization of faulty units in an even more complex power network. In order to DOI Proceedings of the 9th International Modelica Conference 513 10.3384/ecp12076513 September 3-5, 2012, Munich, Germany Fault Detection of Power Electronic Circuit using Wavelet Analysis in Modelica 1.3 Main contributions are used for the transformation, wavelet transform uses different wavelet functions, which can be se- Modelica was developed as a free, object-oriented lected according to the specific problems from a and equation-based modeling language. It has signif- principally unlimited set. Nevertheless, the wavelet icant benefits such as easy reusability of models and function must fulfill some conditions; namely, it multi-domain modeling capability. In combination must be an orthonormal function. The precise math- with the simulation environment Dymola, a conven- ematical description of orthonormality is easily ient platform is provided to the complete model- found in almost every book about wavelet transform, based design and the integration of MEA systems e.g. [10], and is not repeated here. [14]. In contrast to the excellent performance in Parameter, a, defines the width and height of the modeling and simulation, Modelica only supports wavelet function, ψ. A smaller scale, a, makes ψ nar- limited signal analysis features [15], which are actu- rower; thus the wavelet represents fast changes and ally crucial for the fault analysis and virtual testing the transform focuses on the high frequency compo- activities in the verification and validation phase of nents of the signal. Parameter, b, shifts the wavelet the system development. function along the time axis, so that the transform This work focuses on the fast design of a fault de- represents different locations of the signal. Using tection strategy of on board power supply units in different values of scale, a, and position, b, it is able MEA with wavelet transform using Modelica simu- to observe the signal at different position and in dif- lation. To realize this, a wavelet library for Modelica ferent frequency range with only one transformation. is being developed. Multi-Resolution Analysis Thanks to these special properties, wavelet transform (MRA) of wavelet technology is applied to detect the is especially suitable for analyzing changing pro- failure of electrolyte capacitors in a very early stage. cesses. In addition, the design process using Modelica Two forms of wavelet transform are available. simulation shows high flexibility and efficiency. It is They are continuous wavelet transform (CWT) and possible to identify the most important failure fea- discrete wavelet transform (DWT). In CWT both tures and helps to design a effective fault detection scale and position parameters are continuous real strategy within only a short time. values. CWT expresses the signal changes in contin- uous details. It is more suitable for visual examina- tion. In this work only DWT is used, which will be 2 Wavelet transform described in more detail in the next section. 2.1 Definition 2.2 Discrete wavelet transform Wavelet transform could be considered as a fur- In DWT only discrete values of the scale and lo- ther development of Fourier transform, or more pre- cation parameters are used. The values are selected cisely, of short time Fourier transform (STFT) [16]. in a discrete form, namely Using STFT, people try to localize the signal chang- (2) ing with time by selecting suitable time window. , This transformation, however, is limited in time- where and . The transform re- frequency resolution capability due to the uncertainty sults, i.e. the wavelet coefficients, are therefore also principle. Wavelet transform overcomes this prob- discrete. lem. This transform is defined as [10]: In the numeric calculation of DWT, an extra scal- ing function, in addition to the wavelet function, is . (1) used to carry out a complete and efficient DWT. The It is described as the wavelet transform of the scaling function represents the low frequency com- square-integrable function, f, using wavelet function, ponents of the signal. It is orthogonal to its own dis- ψ, at dilation (or scale), a, and position (or transla- crete translations and to all wavelet functions. The tion), b. The bar above function, ψ, stands for conju- wavelet and the scaling functions with the discrete gation. For the given a and b, the transform result is scaling and translation parameters build a complete a single real number, a wavelet coefficient. orthogonal basis of the Hilbert space. The DWT is Obviously wavelet transform is the integral of the thus another representation of the time signal. multiplication of the signal, f, with a wavelet func- As an example, Figure 1 shows the form of the tion, ψ. It has the same form as the STFT. However, third order Daubechies scaling and wavelet functions not like STFT, where only sine and cosine functions and their Fourier transforms [17]. 514 Proceedings of the 9th International Modelica Conference DOI September 3-5, 2012, Munich Germany 10.3384/ecp12076513 Session 4D: Electromagnetic Systems II 2 1 different frequency ranges is obtained, as shown in Scaling function Scaling Figure 3: 1 function Wavelet function h ↓2 cA (k) 0 d0 3 . hd0 ↓2 cA2(k) -1 Wavelet function 0 h ↓2 cA (k) h ↓2 cD (k) 0 1 2 3 4 t 0 1 2 3 4 f d0 1 d1 3 (a) (b) f(k) hd1 ↓2 cD2(k) Figure 1: The third order Daubechies scaling and wavelet func- h ↓2 cD (k) tions (a) and their Fourier transforms (b) d1 1 Figure 3: Multi-resolution analysis using DWT From the Fourier transforms it can be seen that This is the wavelet multi-resolution analysis the scaling function covers lower frequency range (MRA). The output of this operation, cD , cD , …, while the wavelet function stretches in a higher fre- 1 2 cD and cA , are different levels of DWT coeffi- quency range. From this point of view, DWT is actu- n n cients, representing the signal components from ally the division of the time signal into different fre- higher to lower frequencies. Here the original signal quency bands. Thus, it is straightforward to under- is treated as the lowest level of approximation coef- stand that the calculation of DWT is realized using ficients. This analysis provides a convenient tool to filter banks. In inverse DWT the calculation is simi- observe different frequency components of the signal lar.
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