The floating nuclear power plant grid is a system with multi branches and short cable lines. Since the system is grounded by high-resistance in the normal state, a large number of transient fault components will be generated when ground faults occur. Except for extreme HIFs, the transient fault current can meet the threshold value requirement. In order to make good use of the local fault characteristics of the transient signals, an adaptive protection method based on WTMM is proposed.

### Principle of wavelet transform singularity detection

If a function *f*(*x*) is discontinued somewhere on the domain, or its derivative is not continuous, it is regarded as singular. Whether the function *f*(*x*) is singular can be characterized by the Lipschitz function and the function can also be used for characterization of the signal singularity detection [18]. This refers to the detection and extraction of signal mutation points. Where *n* is a non-negative integer and *a* satisfies *n* ≤ *a* ≤ *n* + 1, if there are two constants A, *h*_{0} > 0, and an n-degree polynomial *P*_{n}(*x*), for any *h*∈[−*h*_{0}, *h*_{0}], the following can be established:

$$ \left|f\left({x}_0+h\right)-{P}_n(h)\right|\le A{\left|h\right|}^{\alpha } $$

(1)

Then the function *f*(*x*) is taken to be Lipschitz at the point *x*_{0}. The Lipschitz property of *f*(*x*) at point *x*_{0} indicates the singularity degree of the function at that point. The larger *a* is, the smoother the function *f*(*x*) is at that point, whereas the smaller *a* is, the more singular the function *f*(*x*) is. The singularity of the function *f*(*x*) can be characterized by its WTMM. The theory shows that the magnitude of the WTMM represents the strength of the signal mutation, and the polarity represents the direction of the signal mutation.

When ground faults occur in the nuclear power plant system, the transient fault components at the initial stage are all abrupt or singular. According to the principle of wavelet transform singularity detection, the singularity of the WTMM of the transient zero-sequence current corresponds to the original fault signals. The transient fault current of each line can be decomposed by wavelets, and then the WTMM can be solved after wavelet reconstruction. By comparing the polarities of the WTMM, it can determine whether the fault occurs on the bus or on the lines. For line faults, the line with the largest square value of the WTMM is regarded as the fault line.

### Optimal wavelet basis and decomposition scale selection method based on MUEER

The oprational environment of the floating nuclear power plant is complex, and various ground fault forms can occur. Wavelet transform can effectively extract the local characteristics of the transient fault current and has high accuracy in fault line selection. However, if the wavelet basis and decomposition scale are selected improperly, the WTMM cannot effectively reflect the fault characteristics, leading to misjudgment and affecting the safety of the system. Therefore, the corresponding evaluation criteria is formulated to select the best wavelet basis and decomposition scale of the fault signals. Based on the energy of the wavelet transform results, and the correlation between wavelet basis functions and fault transient signals, a fusion index which can guide the optimal wavelet basis and decomposition scale for fault line selection is proposed.

Performing wavelet decomposition on the fault signal *Y*(*i*) through the wavelet basis *db*(*k*) and decomposition scale *j* with the sampling length of *N* to obtain the wavelet coefficient *W*(*j,k*). Its energy *E* can be described as

$$ {E}_{j,k}=\sum \limits_{i=1}^N{\left|W\left(j,k\right)\right|}^2 $$

(2)

According to the Parserval theorem [19], the energy of the fault signal after orthogonal wavelet transform is equal to its origin. As WTMM is used to select the fault line, larger wavelet coefficient energy makes it easier to select the fault line accurately. Therefore, the energy value of the wavelet coefficient is used as an indicator of the energy of the wavelet transform.

The essence of the wavelet transform method is to extract local fault information from transient fault signals. When the center frequencies of the wavelet basis function and the transient fault signal are close, the wavelet transform results can effectively reflect the fault information. The concept of cross entropy in information theory is introduced to measure the correlation of the center frequency between transient fault signals and wavelet basis functions [20]. It can be used to describe the difference between two probability distributions. The expression of the cross entropy between *X* and *Y* can be described as

$$ H\left(X,Y\right)=-\sum \limits_{x\in X}\sum \limits_{y\in Y}p\left(x,y\right)\log p\left(x,y\right) $$

(3)

In this paper, the energy distribution perspective is used to define the two-dimensional cross entropy between the fault signal and wavelet basis function. The energy distribution probability of the wavelet basis function *db*(*k*) can be described as

$$ p(db)=\frac{{\left| db(k)\right|}^2}{\sum \limits_{j=1}^k{\left| db(k)\right|}^2} $$

(4)

The energy distribution probability of the fault signal *Y*(*i*) can be described as

$$ p(Y)=\frac{{\left|Y(i)\right|}^2}{\sum \limits_{i=1}^N{\left|Y(i)\right|}^2} $$

(5)

The joint energy distribution probability of the fault signal *Y*(*i*) and the wavelet basis *db*(*k*) can be defined as

$$ p\left({Y}_i,{db}_k\right)=\frac{{\left|Y(i)\right|}^2{\left| db(k)\right|}^2}{\sum \limits_{i=1}^N\sum \limits_{j=1}^k{\left|Y(i)\right|}^2{\left| db(k)\right|}^2} $$

(6)

Then the cross entropy can be defined as

$$ H\left({Y}_i,{db}_k\right)=-\sum \limits_Y\sum \limits_{db}p\left({Y}_i,{db}_k\right)\log p\left({Y}_i,{db}_k\right) $$

(7)

In fault line selection, the smaller the cross entropy, the higher the degree of fit of the wavelet basis function to the original fault signal and the more effective information can be extracted. Therefore, the cross entropy can be used as an indicator of the center frequency correlation between the wavelet basis functions and the fault signals.

In order to select the optimal wavelet basis function and decomposition scale, based on the cross entropy and the wavelet transform energy, a fault line selection method based on the united energy entropy ratio is proposed, it is

$$ R=\frac{E_{j,k}}{H\left({Y}_i,{db}_k\right)} $$

(8)

A larger *R* indicates larger correspondence of the wavelet coefficient amplitude to the fault signal, higher degree of similarity with the original fault signal, and better reflection of the transient local characteristics of the fault signal. Therefore, the optimal wavelet basis function and decomposition scale can be selected by comparing the values of *R*.

### Fault line selection method based on WTMM

The wavelet transform coefficient represents the frequency component of the fault signal at a certain decomposition scale. The magnitude of the WTMM indicates the mutation strength and the polarity represents the mutation direction.

The WTMM value of the zero-sequence current in the fault line is the largest and its polarity is opposite to other lines. This characteristic can be used to construct a protection criterion to distinguish bus and feeder faults. The fault line selection protection process based on WTMM is shown in Fig. 3.

After detecting the ground fault in the power system, the optimal wavelet basis function and decomposition scale of the transient zero-sequence fault current are selected by MUEER. The optimal wavelet basis function is used to perform wavelet decomposition and reconstruction of the transient zero-sequence current of each line, and the magnitude and polarity of the WTMM of the zero-sequence current of each line after the reconstruction are solved. By comparing the polarities of the WTMM, it can determine whether the fault occurs on the bus or the feeder. If the polarities are the same, the fault is on the bus, otherwise it is on the feeder.

The fault line selection method based on WTMM uses the transient fault signals to realize the protection function. For intermittent arc light grounding faults, the transient process is intense, and the amplitudes of the WTMM are large for all the lines. However, for low-impedance grounding faults, the WTMM amplitude of the fault line is usually larger than 1, while the WTMM amplitude is usually less than 1. Taking the square value of the WTMM amplitude can further enlarge the difference between the fault and the sound lines, so as to improve the protection sensitivity. In this way, the line with the largest square value of the WTMM is finally regarded as the fault line.