Let x[n] and y[n] represent the input and output signals, respectively, of a discrete-time and causal nonlinear system. The Volterra series expansion for y[n] using x[n] is given by:

$$ y(n)={\displaystyle {h}_0}+{\displaystyle {\sum}_{{\displaystyle {m}_1}=0}^{\infty }{\displaystyle {h}_1}\Big[}{\displaystyle {m}_1}\left] x\left[ n-{\displaystyle {m}_1}\right]+{\displaystyle {\sum}_{{\displaystyle {m}_1}=0}^{\infty }{\displaystyle {\sum}_{{\displaystyle {m}_2}=0}^{\infty }{\displaystyle {h}_2}\Big[{\displaystyle {m}_1},{\displaystyle {m}_2}}}\right] x\left[ n-{\displaystyle {m}_1}\right] x\left[ n-{\displaystyle {m}_2}\right]+.. $$

(1)

In (1), h_{p} [m_{1}, m_{2,}…, m_{p}] is known as the p-the order Volterra kernel of the system, Without any loss of generality, the kernels can be assumed to be symmetric. In general any kernel \( {\mathrm{h}}_{\mathrm{p}} \) [m_{1}, m_{2}. …, m_{p}] can be replaced by symmetric one by simple setting.

$$ \left[{\displaystyle {h}_p^{sym}}\left[{\displaystyle {m}_{1,}}{\displaystyle {m}_2},....{\displaystyle {m}_p}\right]=\frac{1}{n!}{\displaystyle {\sum}_{\left({\displaystyle {m}_{{\displaystyle {i}_1}}},{\displaystyle {m}_{{\displaystyle {i}_2}}}.....{\displaystyle {m}_{{\displaystyle {i}_p}}}\right){\displaystyle {\varepsilon}_{\mathrm{s}}}}{\displaystyle {h}_p}\left[{\displaystyle {m}_{{\displaystyle {i}_1}}},{\displaystyle {m}_{{\displaystyle {i}_2}}}.....{\displaystyle {m}_{{\displaystyle {i}_p}}}\right]}\right] $$

(2)

Where s is set of all permutations of m_{1}, m_{2,}.... m_{p}. The Volterra series is a power series with memory. This can be checked by changing the input by a gain factor d so that the new input is dx(t). By using (2), the new output is

$$ y(n)={\displaystyle {h}_0}+{\displaystyle {d}^m}\left[{\displaystyle {\sum}_{{\displaystyle {m}_1}=0}^{\infty }{\displaystyle {h}_1}\left[{\displaystyle {m}_1}\right]} x\left[ n-{\displaystyle {m}_1}\right]+{\displaystyle {\sum}_{{\displaystyle {m}_1}=0}^{\infty }{\displaystyle {\sum}_{{\displaystyle {m}_2}=0}^{\infty }{\displaystyle {h}_2}\left[{\displaystyle {m}_1},{\displaystyle {m}_2}\right]}} x\left[ n-{\displaystyle {m}_1}\right] x\left[ n-{\displaystyle {m}_2}\right]+....\right] $$

(3)

This is a power series with amplitude factor d. The integrals are convolutions it shows that series having memory. As an effect of its power series features, it has some limitations associated with the application of the Volterra series to nonlinear problems. The convergence of Volterra series is one of the major limitations. One can think of the Volterra series expansion as a Taylor series expansion with memory. The limitations of the Volterra series expansion are similar to those of the Taylor series expansion both expansions do not do well when there are discontinuities in the system description [5].

### 2.1 Volterra kernels estimation by exponential method

Numerous methods have been explored in the literature for determining the kernels or the associated transfer functions. Among them, the method of exponential inputs specifically chosen for calculating Volterra kernel in this paper. Let us consider the Volterra series expansion of a nonlinear system of the form

$$ y(t)={\displaystyle {\sum}_{p=1}^{\infty }{\displaystyle \underset{0}{\overset{{\displaystyle {t}_1}}{\int }}....}{\displaystyle \underset{0}{\overset{{\displaystyle {t}_p}}{\int }}{\displaystyle {h}_p}\left({\displaystyle {m}_{1,}}{\displaystyle {m}_2},....{\displaystyle {m}_p}\right)}} x\left( t-{\displaystyle {m}_1}\right).... x\left( t-{\displaystyle {m}_p}\right) d{\displaystyle {m}_1}...... d{\displaystyle {m}_p} $$

(4)

Let the input x(t) be a sum of exponentials

$$ x(t)={\displaystyle {e}^{{\displaystyle {s}_1} t}}+{\displaystyle {e}^{{\displaystyle {s}_2} t}}+......{\displaystyle {e}^{{\displaystyle {s}_q} t}} $$

Where \( {\mathrm{s}}_1,{\mathrm{s}}_2,\dots {\mathrm{s}}_{\mathrm{q}} \) are rationally independent. This means that there are no rational numbers \( {\upbeta}_1,{\upbeta}_2,\dots .{\upbeta}_{\mathrm{q}} \) such that the sum \( {\upbeta}_1{\mathrm{s}}_1+{\upbeta}_2{\mathrm{s}}_2+\dots {\upbeta}_{\mathrm{q}}{\mathrm{s}}_{\mathrm{q}} \) is rational. Then (4) becomes

$$ y(t)={\displaystyle {\sum}_{p=1}^{\infty}\left[{\displaystyle {\sum}_{{\displaystyle {q}_1}}^q....}{\displaystyle {\sum}_{{\displaystyle {q}_p}=1}^q{\displaystyle {H}_p}\left({\displaystyle {s}_{{\displaystyle {q}_1}}}....{\displaystyle {s}_{{\displaystyle {q}_m}}}\right)}{\displaystyle {e}^{\left({\displaystyle {s}_{{\displaystyle {q}_1}}}+....{\displaystyle {s}_{{\displaystyle {q}_m}}}\right) t}}\right]} $$

(5)

if each \( {\mathrm{s}}_{\mathrm{i}} \) occurs (\( {s}_{q_1\dots \dots .}{s}_{q_m} \)), \( {\mathrm{r}}_{\mathrm{i}} \) times, then there are

$$ \frac{\mathrm{p}!}{{\mathrm{r}}_1!{\mathrm{r}}_2!\dots ..{\mathrm{r}}_{\mathrm{q}}!} $$

Then (4) can be written in the form

$$ y(t)={\displaystyle {\sum}_{p=1}^{\infty }{\displaystyle {\sum}_r\frac{p!}{{\displaystyle {r}_1}!{\displaystyle {\mathrm{r}}_2}!....{\displaystyle {\mathrm{r}}_{\mathrm{q}}}!}}{\displaystyle {H}_p}\left({S}_{q_1}....{S}_{q_m}\right)}{\displaystyle {e}^{\left({\displaystyle {s}_{{\displaystyle {q}_1}}}+....{\displaystyle {s}_{{\displaystyle {q}_m}}}\right) t}}\Big] $$

(6)

where r under the summation sign indicates that the sum includes all the distinct vectors \( \Big({r}_1,{r}_2,\dots .{r}_q \)) such that \( \sum_{\mathrm{i}=1}^{\mathrm{q}}{\mathrm{r}}_{\mathrm{i}}=\mathrm{p} \) . If \( {r}_1={r}_2=\dots ={r}_q=1 \) then the amplitude associated with the exponential term \( {\mathrm{e}}^{\left({\mathrm{s}}_{{\mathrm{q}}_1+\dots }{\mathrm{s}}_{{\mathrm{q}}_{\mathrm{m}}}\right)\mathrm{t}} \) is q \( !{\mathrm{H}}_{\mathrm{q}}\left({\mathrm{s}}_1,\dots .{\mathrm{s}}_{\mathrm{q}}\right) \). Now just calculating the transfer function of the system, we can calculate Volterra series kernel.

Volterra series having application in various fields of engineering and physics and can be practically classified into two distinct categories. In the first classification, a model of an observed dynamical phenomenon is build using Volterra series and the estimation of Volterra frequency response function requires experimental and numerically generated data. The second classification comprises the analysis of dynamical systems that are already represented by an analytic model, such as differential equation in mathematics. The harmonically excited nonlinear systems’ behavior can be investigated better using Volterra series representation [6].

Volterra series expansion exists for nonlinearity and non-stationarity. Even though PQ events are non-stationary in nature so this technique is applicable to power quality disturbances, Volterra system models have been successfully employed in PQ events detection and localization applications in this paper and such models can also be implemented in real time PQ events monitoring system which can change the present scenario of PQ events monitoring system.