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1 Introduction

With the development of smart electricity, interoperability has become the core technology of smart grid. The interoperability includes the interaction between the electricity side and supply side, and the power distribution equipment and electrical equipment [1]. Many research groups are promoting the standardization work of the interoperability, such as SGIP (Smart Grid Interoperability Panel) PAP (Priority Action Plan) 17 facility smart grid information standard [2] and IEC (International Electrotechnical Commission) PC (Project Committee) 118 smart user interface. Electrical equipment status information sensing technology has become a new research focus in grid demand forecast [3]. In Contemporary sensor method includes two types, mainly, a shared sensor method and another dedicated sensor method. It is in the shared sensor method that electricity equipment status is determined by the electricity load. For example, the power meter is determining the electrical equipment status based on the load of the electrical equipment. The meter is the shared sensor. This method is simple and of low cost, but can’t determine the status of each device, which has the same load. Another method is what a device corresponds to a sensor. The electricity equipment is connected to grid by use the smart socket. Each smart socket is used to monitor the status of each device. The accuracy device information is obtained by this method. But the method needs many smart sockets in more of the device. Thus this method costs higher. Low-cost, accurate, and effective information perception technology becomes a bottleneck restricting the development of smart grid.

The electric equipments that are connected to the power network will transmit EMI (electromagnetic interference) to the power network, such as conducted disturbance, voltage dip, electrical fast transient burst, and so on. This ranges from influencing the power supply quality of power grid to threatening the work of other equipments unit sharing the same grid. Even it can damage or threaten the safety of power grid [4]. So the TC (Technical Committee) 77 work group and CISPR (International Special Committee on Radio Interference) have been established especially by IEC to evaluate the EMI’s emission and immunity. Moreover the corresponding standards and regulations have been worked out, such as CISPR Pub series standards and IEC 61,000 series standards. Also some other standards just like EN and ITU are included. In SGIP, the EMI and immunity problem of power grid are being studied specially by EMIWG (Electromagnetic interoperability working group).

Most of the researchers carry out researches mainly aimed at decreasing the emission intensity of EMI sources, blocking the interference approaches and increasing the immunity of sensitive equipment [5]. The outstanding feature of this kind of problems is that the frequency band of EMI signal is very wide and there is great difference in frequency spectrum of different electric equipments’ disturbance signal. Moreover because of the positions that the electric equipments connected to the power network are different, the transmission routes of EMI are different too [6].

In this chapter, the power supply grid with double disturbance sources and double observation points is the research object. The main research contents include three parts. First of all the high frequency lumped parameter model of power supply network is discussed, and the functional relation of observed signal and identification objects is given on this basis. Secondly the identification objects’ partial differential function is solved by using multiple-scales perturbation method. Finally verify the applicability and reliability of this method according to the experiment.

2 Lumped Parameter Modeling

The circuit diagram of power supply network is shown in Fig. 1. In the figure, the power supply combinations are expressed by L and N. The current monitoring equipments Sensor1 and Sensor2 are separately connected to power supply combinations and power supply network ends. The electric equipments in the power supply grid are shown by EUT1 and EUT2.

Fig. 1
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Pathway of power supply network

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As for the switchgears connected to the power grid, while they are absorbing power, they are transmitting EMI to the power grid. Moreover they will divorce from the wire’s constraint gradually and transmit EMI to space as frequency increasing.

Generally low frequency EMI spread along the wires, so according to the test standard of EMI emission, the EMI in the frequency band of 150 kHz to 30 MHz is defined as conduction EMI (CEMI). Its pathway is shown as Fig. 1. Sensor1 and Sensor2 are used to collect CEMI in the power supply grid. Sensor consists of 50 Ω impedance and current sensor. A 50 Ω impedance is used to match the impedance of power line, in order to decrease the signal reflection of CEMI. And current sensor is used to record CEMI.

In order to describe the electrical parameter relationship of the above power supply grid, the high frequency lumped parameter circuit model of CEMI is set up. Its circuit topology is shown as Fig. 2 [7]. The emission sources EUT1 and EUT2 are modeled by using Thevenin equivalent circuit. That is, series circuits consist of voltage source and internal impedance, respectively, for series connection of u 1 and Z 4 and series connection of u 2 and Z 7. The high frequency equivalent circuits of power line are modeled by using lumped parameter impedance, respectively, for Z 2, Z 3, Z 5, Z 6, Z 8, and Z 9. Sensor1 and Sensor2 are modeled by using matching impedance, respectively, for Z 1 and Z 10. The current is expressed by i 1 and i 3.

Fig. 2
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Equivalent circuit topology

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According to the circuit topology expressed in Fig. 2, the equations of voltage and current of each branch can be set up by using mesh method:

$$ \left\{ {\begin{array}{llll} {{Z_1}{i_1}+{Z_2}{i_1}+{Z_3}{i_1}+{Z_4}{i_1}-{Z_4}{i_1}+{u_1}=0} \cr {-{Z_4}{i_1}+{Z_4}{i_2}+{Z_5}{i_2}+{Z_6}{i_2}+{Z_7}{i_2}-{Z_7}{i_3}+{u_2}=0} \cr {-{Z_7}{i_2}+{Z_8}{i_3}+{Z_9}{i_3}+{Z_{10 }}{i_3}+{Z_7}{i_3}-{u_2}=0} \\\end{array}} \right. $$
(1)

In the equations, i 2 is the current between EUT1 and EUT2. The equations are arranged to

$$ \left\{ {\begin{array}{lll}{{Z_4}{i_2}-{u_1}=\left( {{Z_1}+{Z_2}+{Z_3}+{Z_4}} \right){i_1}} \cr {\left( {{Z_4}+{Z_5}+{Z_6}+{Z_7}} \right){i_2}-{u_2}={Z_4}{i_1}+{Z_7}{i_3}} \cr {{Z_7}{i_2}+{u_2}=\left( {{Z_7}+{Z_8}+{Z_9}+{Z_{10 }}} \right){i_3}} \\\end{array}} \right. $$
(2)

The state equations are as follow:

$$ \left[ {\begin{array}{lll} {{Z_4}} & {-1} & 0 \\{{Z_4}+{Z_5}+{Z_6}+{Z_7}} & 0 & {-1} \\{{Z_7}} & 0 & 1 \\\end{array}} \right]\left[ {\begin{array}{lll} {{i_2}} \\{{u_1}} \\{{u_2}} \\\end{array}} \right]=\left[ {\begin{array}{llll} {({Z_1}+{Z_2}+{Z_3}+{Z_4}){i_1}} \\{{Z_4}{i_1}+{Z_7}{i_3}} \\{({Z_7}+{Z_8}+{Z_9}+{Z_{10 }}){i_3}} \\\end{array}} \right] $$
(3)

Solution:

$$ \left[ {\begin{array}{lll} {{i_2}} \\{{u_1}} \\{{u_2}} \\\end{array}} \right]=\left[ \begin{array}{lll} {{{{{Z_4}}} \left/ {{\left( {{Z_4}+{Z_5}+{Z_6}} \right)}} \right.}{i_1}-{{{\left( {{Z_8}+{Z_9}+{Z_{10 }}} \right)}} \left/ {{\left( {{Z_4}+{Z_5}+{Z_6}} \right)}} \right.}{i_3}} \cr \begin{array}{lll} {-\left( {{{{{Z_4}\left( {{Z_5}+{Z_6}} \right)}} \left/ {{{Z_4}+{Z_5}+{Z_6}}} \right.}+{Z_1}+{Z_2}+{Z_3}} \right){i_1} }\end{array} \cr \quad { -{{{{Z_4}\left( {{Z_8}+{Z_9}+{Z_{10 }}} \right)}} \left/ {{\left( {{Z_4}+{Z_5}+{Z_6}} \right)}} \right.}{i_3}} \cr \begin{array}{lll} {-{{{{Z_4}{Z_7}}} / {{( {{Z_4}+{Z_5}+{Z_6}} )}} }{i_1}+( {{Z_8}+{Z_9}+{Z_{10 }}+{{{{Z_7}( {{Z_4}+{Z_5}+{Z_6}} }}}}}\cr \quad {{ {{ { +{Z_8}+{Z_9}+{Z_{10 }}} )}} / {{( {{Z_4}+{Z_5}+{Z_6}} )}} }} ){i_3} \end{array}\end{array} \right] $$
(4)

When Z 4 → ∞, i 2 = i 1, u 2 = −Z 7 i 1 + (Z 7 + Z 8 + Z 9 + Z 10)i 3; it means EUT1 is open circuit and is not connected to the power grid. And only EUT2 is transmitting CEMI to power grid. The signal is u 2.

When Z 7 → ∞, i 2 = i 3, u 1 = Z 4 i 3−(Z 1 + Z 2 + Z 3 + Z 4)i 1; it means EUT2 is open circuit and is not connected to the power grid. And only EUT1 is transmitting CEMI to power grid. The signal is u 1.

When Z 4 and Z 7 are all limited values, it means EUT1 and EUT2 are all connected to the power grid. Without loss of generality, suppose its internal impedance is 50 Ω [8]. The impedance of power line is expressed by using transmission line model [9], that is, pure inductive reactance model. The phase function equation of (4) is

$$ B\left[ {\begin{array}{lll} {{u_1}} \\{{u_2}} \\\end{array}} \right]=\left[ {\begin{array}{lll} {{A_{11 }}} & {{A_{12 }}} \\{{A_{21 }}} & {{A_{22 }}} \\\end{array}} \right]\left[ {\begin{array}{lll} {{i_1}} \\{{i_3}} \\\end{array}} \right] $$
(5)

In the equation

$$ \begin{array}{lll} B ={R_4}+\left( {{l_5}+{l_6}} \right)s \hfill \\{A_{11 }} =-{R_4}{R_1}-\left( {\left( {{R_4}+{R_1}} \right)\left( {{l_5}+{l_6}} \right)+{R_4}\left( {{l_2}+{l_3}} \right)} \right)s-\left( {{l_2}+{l_3}} \right)\left( {{l_5}+{l_6}} \right){s^2} \end{array} $$
$$ \begin{array}{lll}{A_{12 }} =-{R_4}{R_{10 }}-\left( {{l_8}+{l_9}} \right)s \\ {A_{21 }} =-{R_4}{R_7}\\ \begin{array}{lll} {A_{22 }} ={\ R_4}{R_{10 }}+\left( {{R_4}+{R_{10 }}} \right){R_7}+\left( {\left( {{R_4}+{R_7}} \right)\left( {{l_8}+{l_9}} \right)+\left( {{R_7}+{R_{10 }}} \right)\left( {{l_5}+{l_6}} \right)} \right)s \cr \quad\quad +\left( {{l_8}+{l_9}} \right)\left( {{l_5}+{l_6}} \right){s^2} \end{array}\end{array}$$

From this, the state equation is

$$ \left\{ {\begin{array}{ll} {{{\dot{u}}_1}+a{u_1}={c_1}} \\{{{\dot{u}}_2}+a{u_2}={c_2}} \\\end{array}} \right. $$
(6)

In the equation, \( \dot{u} \) is the first derivative of u. The other parameters are as follows.

$$ \begin{array}{lll} a ={{{{R_4}}} \left/ {{{l_5}+{l_6}}} \right.} \hfill \\{c_1} ={\begin{array}{lll} - {{{{R_4}{R_1}}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}{i_1}-\left[ {\left( {{R_4}+{R_1}} \right)+{{{{R_4}\left( {{l_2}+{l_3}} \right)}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}} \right]{{\dot{i}}_1}-\left( {{l_2}+{l_3}} \right){{\ddot{i}}_1} \end{array}} \cr \qquad { -{{{{R_4}{R_{10 }}}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}{i_3}-{R_4}{{{\left( {{l_8}+{l_9}} \right)}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}{{\dot{i}}_3} } \hfill \\{c_2} =\begin{array}{lll} -{{{{R_4}{R_7}}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}{i_1}+{{{\left( {{R_4}{R_{10 }}+\left( {{R_4}+{R_{10 }}} \right){R_7}} \right)}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}{i_3} \end{array} \cr \qquad +\left[ {{{{\left( {{R_4}+{R_7}} \right)\left( {{l_8}+{l_9}} \right)}} \left/ {{\left( {{l_5}+{l_6}} \right)}} \right.}+{R_7}+{R_{10 }}} \right]{{\dot{i}}_3}+\left( {{l_8}+{l_9}} \right){{\ddot{i}}_3}\end{array} $$
(7)

In the equations, \( \dot{i} \) and \( \ddot{i} \) separately are first derivative and second derivative of i. In the practical application, equivalent can be set up by using first-order central difference and second-order central difference separately.

According to (6), it can be known that the corresponding homogeneous equations of the two first order nonhomogeneous differential equations are the same. It means their general solutions are the same. The nonhomogeneous terms of the two differential equations are different. It means the difference between them is on the special solutions. Here the analytic solution of (6) is given as:

$$ \left\{ {\begin{array}{lll} {{u_1}={{e}^{-at }}+{c_1}/a} \\{{u_2}={{e}^{-at }}+{c_2}/a} \\\end{array}} \right. $$
(8)

According to (8), the analytic form of source CEMI can be known. And u 1 and u 2 are separate. Moreover their change rates are the same. Their characteristics are reflected by the initial value of the time when t = 0. So the working condition of equipments under test can be identified by c 1 and c 2.

3 Analysis and Verification of Test

The test case is arranged according to Fig. 1. Parallel double wires are used to supply power at LN. The voltage is 220 V. EUT1 and EUT2 are switching mode power supply. When they are under the rated condition, their carrier frequencies are, respectively, 30 and 40 kHz. And their rated power is 100 w. The same oscilloscope connecting to Sensor1 and Sensor2 is used to test. Its sampling frequency is 1 GHz. The wire between Sensor1 and EUT1 is as long as 5 m. The wire between EUT1 and EUT2 is as long as 1 m. And the wire between EUT2 and Sensor2 is as long as 2 m. Agilent 4395A is used to test the equivalent inductance of single conductor in the frequency of 10 Hz to 1 MHz. The results are shown in Table 1. Because the structure and material of each wire are the same in double wires, it is supposed that parameters of equivalent circuit are the same too.

Table 1 Test results of wires’ equivalent inductance
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The test results of voltage at Sensor1 and Sensor2 are shown as Fig. 3. According to the figure, it can be seen that the differences between the results of the two voltages are not obvious. Moreover because they contain fundamental component of 50 Hz, the change trends when they are around 0 V are the same. The results are shown as Fig. 4 when calculating (10) using the test results. According to Fig. 4a, b, it can be seen that the magnitude orders of amplitude of di 1/dt and di 3/dt are around 104. And according to Fig. 4c, d, it can be seen that the magnitude orders of amplitude of d2 i 1/d2 t and d2 i 3/d2 t are around 1014. And four results are all much larger than i 1 and i 3. Analyzing (8), it can be known that the calculated results of c 1 and c 2 are influenced by di 1/dt, di 3/dt, d2 i 1/d2 t, and d2 i 3/d2 t conspicuously while inconspicuously by i 1 and i 3.

Fig. 3
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Voltage waveform at observation points. (a) Voltage of Sensor1 and (b) voltage of Sensor2

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Fig. 4
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Time series of the first and second derivatives of the observed current. (a) First derivative of i 1, (b) first derivative of i 3, (c) second derivative of i 1 and (d) second derivative of i 3

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When EUT1 and EUT2 are under full load condition, the calculated results of c 1 and c 2 are shown as Fig. 5. Their change laws are conspicuously different. In order to quantize their features, the norm of the time series of c 1 and c 2 is calculated out, respectively, \( \parallel {c_1}\parallel =9.59\times {10^{13 }} \) and \( \parallel {c_2}\parallel =3.03\times {10^{13 }} \).

Fig. 5
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Time series of c 1 and c 2 when EUT1 and EUT2 are under full load condition (a) c 1 and (b) c 2

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When EUT1 still works under full load condition, adjust EUT2 to work under 50 % load condition. Then the calculated results of time series of c 1 and c 2 are shown as Fig. 6. \( \parallel {c_1}\parallel =9.59\times {10^{13 }} \) and \( \parallel {c_2}\parallel =3.03\times {10^{13 }} \). Comparing Figs. 5a and 6a, it can be seen that the calculated results under two conditions are different conspicuously. But the norm of c 1 and c 2 is invariable. It shows that c 1 and c 2 are insensitive to load variation. They just reflect the first and second derivatives of current. c 1 and c 2 can be used to represent CEMI and be one of the CEMI’s quantitative indexes because they will not conspicuously influence the high-frequency component of CEMI under different conditions. This is consistent with the analysis before. That is, the norms of the inhomogeneous term of conducted EMI differential equations are invariable when the switching frequency of electric equipments is invariable.

Fig. 6
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Time series of c 1 and c 2 when EUT1 works under full load condition and EUT2 works under 50 % load condition (a) c 1 and (b) c 2

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4 Conclusion

This chapter discusses an online-observation method for conducted EMI of electric equipments. This method provides a modeling of power supply grid with single power and double electric equipments, and gives the analytic expressions of inhomogeneous differential equations for conducted EMI of electric equipments. The analysis of differential equations shows that the general solutions of these analytic expressions for conducted EMI of electrical equipment are the same, while the special solutions of them are different. Finally, these analytic expressions are used to take an experiment on two different switching power supplies with the carrier frequency of 30 and 40 kHz each. When the switching frequency does not change, the load of electric equipment two will be reduced by 50 %. And the norms of inhomogeneous differential equations of conducted EMI will be invariable. The result of this experiment shows that the inhomogeneous term norms of conducted EMI of electric equipments can be used to identify electric equipments, which are consistent with the theoretical analysis of the results. This method uses the characteristics of conspicuous differences of conducted EMI from electric equipments to identify electric equipments. The recognition effect is well. And less calculation amount is suitable for using online. Moreover using fewer sensors can save sensors when the number of electric equipments is large. But to non-switching equipments, this method is useless because of the low rate of change of current and voltage. This method is just suitable for switching equipments such as variable frequency converter and power electronic equipments. Using this method at smart grid field is helpful in improving the flexibility and intelligent level of demand forecast, as well as decreasing the cost and complexity of installing.