Frequency and Temperature-Dependent Power Cable Modelling for Stability Analysis of Grid-Connected Inverter

This paper presents a RLC circuit model of long transmission cable (LTC) with consideration of frequency and temperature-dependent characteristics. Per-unit-length (p.u.1.) impedance of the LTC considering frequency-dependent characteristics are first calculated and fitted by Vector Fitting (VF) algorithm. Then, the fitted model of the p.u.1. impedance is represented by a Π section, which consists of a series of RL branches and two parallel capacitors. Impedance model of LTC with arbitrary length can be established by cascading several Π sections. In addition, the effect of ambient temperature variation on electrical performance of LTC are investigated. Simulation results show that the proposed RLC circuit model is able to reveal practical frequency and temperature characteristics, which may be used to investigate effect of power cable on stability of grid-connected inverter.


I. INTRODUCTION
Offshore wind power generation has been paid increasing attentions in recent years [1], [2].Compared with onshore wind power plant, offshore wind farms are commonly integrated into power grid by offshore substation and long transmission cable (LTC) due to remote transmission distances [1].Voltage source converters (VSCs) have been widely applied in offshore wind power plant [2]- [4].
However, oscillation phenomena have been frequently reported in offshore wind power plants with LTC, where harmonic-frequency resonance may be triggered at multiple frequencies due to distributed parasitic capacitance of LTC [5].It's urgent to find out the effect of LTC on resonance phenomena of grid-connected inverter (GCI) system.
Previous work about modelling of power cable has been presented in [6]- [9].LTC is commonly modelled as a simple inductor, which fails to reveal practical frequency characteristics and perform inaccurate stability assessment [6].Furthermore, frequency-dependent circuit models of LTC considering distributed parasitic capacitance are established in [7]- [9], in which multiple cascaded-Π section circuit model is used.However, resistance and inductance of per-unit-length (p.u.l.) LTC are modelled as a fixed resistor and inductor, which cannot reveal the practical frequency-dependent characteristics and corresponding damping characteristics.To simulate practical frequency characteristics of LTC, extra RL branches are introduced, which are paralleled with each Π section [10], [11].The number of parallel RL branches and cascaded Π sections can be optimized based on the length of LTC.
In addition, electrical characteristic of LTC may be affected due to ambient temperature variation.Severe temperature fluctuation can be up to 20 • C under severe weather conditions [12].The temperature variation may cause electrical parameters perturbation along LTC [13], which thus influences terminal impedance characteristics.In [14], the consequences of 20% increased uncorrected line resistance caused by ambient temperature are shown.However, temperaturedependent characteristics have been paid limited attentions in existing frequency-dependent models.The effect of cable resistance perturbation caused by temperature fluctuation on high frequency resonance phenomena of DFIG system has been studied in [9].However, temperature-dependent LTC model is not established.A temperature-dependent LTC model considering longitude temperature variation is established in [13]- [16].However, the established temperature-dependent LTC model merely focuses on state estimation, e.g.line power transmission capability [13], [16], which was not applied to stability analysis.Therefore, this paper develops a RLC circuit model of LTC with consideration of frequency and temperature-dependent characteristics, which may be used to investigate effect of cable electrical characteristics on stability of GCI.
The main contributions of this paper can be briefly explained as follows.(1) Frequency and temperature-dependent characteristics of LTC is established in RLC circuit model, where 978-1-5386-8257-9/18/$31.00 © 2018 European Union the effect of temperature variation on electrical parameters perturbation is simulated by correcting electrical parameters of each Π section.(2) The impact of LTC length variation on stability of GCI is investigated.

II. IMPEDANCE MODELLING OF GCI
Fig. 1 shows diagram of GCI with LTC, which consists of LCL filter, current controller in dq frame, synchronous reference frame phase-locked loop (SRF-PLL) and modulator.The inverter side current feedback is used, since the current is the superposition of the grid side and filter capacitor currents (Capacitor current feedback can provide damping effect) [17], [18].The grid is emulated as an ideal voltage source V g , and LTC is used to connect the GCI and grid.
Small signal stability of GCI can be identified by means of impedance-based stability criterion, which plots the Nyquist diagram of the minor loop gain (MLG) T (s) and counts the number of encirclements [6], shown as follows, where Z inv cl (s) is GCI impedance, and Z g cl (s) is equivalent grid impedance which includes C f , L f 2 and LTC, as shown in Fig. 1.The GCI part in Fig. 1 can be transferred into Fig. 2, and the Norton equivalent circuit in red dotted box of Fig. 3 can be established on the basis of Fig. 2. The parameters of the Norton equivalent circuit are as follows, where G idq is the transfer function of the PI controller, G d is digital delay which includes one sampling computational delay and 0.5 sampling modulation delay [19], By substituting (3) into (2), Z inv cl can be represented as follows, For PI controller in rotating dq frame, the impact of parameter K i on Z inv cl in high-frequency range can be ignored.The frequency characteristics of terminal impedance of GCI can be obtained by substituting s = jw into (4), shown as follows, (5) It's obvious that the real part of Z inv cl (jω) is negative in the following frequency range, and thus the terminal impedance appears non-passive characteristics.
where f s is the sampling frequency.In practice, the frequency range below Nyquist frequency f s /2 is considered (k = 1).Thus, the non-passivity area is f N P A = (f s /6, f s /2).The non-passivity area is actually the potential unstable frequency range [7].Thus, a circuit model which can reveal the practical impedance characteristics of LTC in the area should be established.However, previous works seldom take the practical frequency and temperature-dependent characteristics into account, due to the complicated modelling process.Section III will explain the practical frequency and temperature-dependent characteristics of LTC, followed by the proposed RLC circuit modelling for LTC.

III. PROPOSED MODELLING METHOD FOR LTC
In this section, RLC circuit model of LTC with consideration of frequency and temperature-dependent characteristics is established, and detailed implementation procedure is given.The practical frequency and temperature characteristics of LTC are established in the form of RLC circuit, and then are incorporated into circuit model of power cable.A. Practical terminal impedance characteristics of LTC 1) Frequency-dependent characteristics of LTC: The terminal impedance characteristics of LTC can be reproduced by a two-port network shown in Fig. 4. The two important parameters are given as [20], where Z S and Y P are the series impedance and shunt admittance; z(ω) and y(ω) are the p.u.l.impedance and admittance.γ(ω) = (z(ω)y(ω) is propagation constant, and L is the length of LTC.z(ω) and y(ω) can be represented as, where r(ω), l(ω), g(ω) and c(ω) are p.u.l.resistance, inductance, conductance and capacitance.For LTC, the values of r(ω) and l(ω) change with frequency varying.g(ω) can be ignored and c(ω) is regarded as constant [20].
2) Temperature-dependent characteristics of LTC: The p.u.l.resistance of power cable depends on the resistivity of cable conductor and the area of cross-section, which can be affected by ambient temperature.And the inductance has the similar temperature-dependent behavior [13], [15].The temperature-dependent characteristics of p.u.l.resistance and inductance are shown as, where r(T c ) and l(T c ) are the p.u.l resistance and inductance at temperature T c .r(T 0 ) and l(T 0 ) are the p.u.l resistance and inductance at temperature T 0 .Besides, α r and α l are the temperature coefficients of resistance and reactance of power cable, which are independent on temperature and are determined by material property of power cable conductor.Here α r and α l are regarded as 0.00393/ • C. Fig. 5 shows the variation characteristics of p.u.l.resistance and inductance as functions of frequency and temperature.The p.u.l.resistance increases if frequency or temperature  increases, and the p.u.l.inductance increases if frequency decreases or temperature increases.The p.u.l.impedance at 20 • C calculated using ( 8) is shown as the blue line in Fig. 6.

B. Implementation of the proposed RLC circuit modelling method
In the proposed modelling method, VF algorithm is employed to fit frequency characteristics of p.u.l.series impedance z(ω).The mathematical representation can be obtained as [21], where f (s) is the fitted transfer function, N is the order of f (s), B n and A n are the nth residue and pole pair.D is nonzero if the order of the numerator polynomial is not lower than the order of denominator polynomial.And E indicates the transfer function is improper [21].In this paper, the p.u.l.impedance shown as the blue line in Fig. 6 will be fitted into form (10) using VF.Fig. 7 shows the detailed implementation procedure of the proposed RLC circuit modelling method, which consists of three steps.Step 1 is to establish a RLC circuit model incorporating frequency-dependent characteristics.Step 2 is to derive the temperature profile along the LTC, and the Calculate p.u.l.values (r,l,z) (Fig. 6) Fit RL circuit according to p.u.l.impedance (Fig. 8) Generate multi-section RLC circuit model (Fig. 9) Plot longitude temperature profile (Fig. 10) Divide LTC based on temperature profile (Fig. 10) Calculate the average temperature in each section (Fig. 10) Correct electrical parameters using actual temperature Step 1 Step 2 Step 3 Establish the frequency and temperature-dependent model Fig. 7. Implementation procedure of the proposed RLC circuit modelling method. .....
LTC is divided into several sections based on the temperature distribution.Finally, step 3 is to adjust the obtained RLC circuit parameters to incorporate the temperature variation information. 1) Step 1: The p.u.l.resistance, inductance and impedance at T = 20 • C are first calculated, shown as in Fig. 5 and Fig. 6.Then, the p.u.l.impedance is fitted by a transfer function in the form (10) using VF algorithm.In detail, when VF is implemented, the order of the transfer function increases gradually until the trade-off between accuracy and complexity is obtained.In this case, a 5-order (N = 5) transfer function is fitted, and its Bode diagram is plotted as the red dotted line in Fig. 6.As discussed in Section II, the potential unstable frequency range is from 1.67kHz to 5kHz if sampling frequency f s is 10kHz.It can be seen that the fitted results accurately match practical impedance characteristics of LTC in the frequency range.(10) is then represented by a RL network, shown as in Fig. 8.In principal, the electrical parameters of the RL circuit can be calculated from (10), Finally, multi-segment lumped-parameter RLC circuit model is adopted to represent terminal impedance characteristics of LTC, as shown in Fig. 9.The number of Π sections will be increased as extension of cable length.And the number of RL branches in each Π section will be increased if the frequency range of interest widens.The parameters of each Π section can be calculated on the basis of LTC length L, the number of Π sections m and the fitted parameters in (11), shown as follows, where c is the constant p.u.l.capacitance.

2)
Step 2: In step 1, the longitude temperature is regarded as constant (T = 20 • C), and the p.u.l.resistance and inductance in ( 8) are calculated at the fixed temperature.In step 2, the p.u.l.resistance and inductance are corrected based on temperature characteristics of LTC, shown as in Fig. 5.
The temperature-based LTC segmentation method is illustrated in Fig. 10.Temperature profile along the LTC is first measured, shown as the red line (For simplicity, linear temperature variation is considered).Then, the LTC is divided into several sections based on the measured temperature distribution, shown as [0, For each section, the following requirement should be satisfied, where i and j are the arbitrary two points in arbitrary section, and ∆ T is the pre-defined temperature threshold value.The section number varies with the temperature threshold value, and more accurate model can be obtained if the value decreases.
Finally, the average temperatures in all sections are calculated, shown as T av1 , T av2 , T av3 , T av4 , T av5 in Fig. 10, which will be used in step 3.
3) Step 3: The values of resistors and inductors of all Π sections in Fig. 9 are corrected using (9) by the updated average temperatures obtained in step 2.
The circuit model of LTC considering practical frequency and temperature-dependent characteristics has been established based on the aforementioned three steps.Simulation will be performed in Section IV.

IV. SIMULATION VERIFICATION
To validate correctness of the frequency and temperaturedependent RLC circuit model, the simulation in Matlab/Simulink is performed.The advanced frequency-dependent WideBand Line model is provided by OPAL-RT ARTEMiS-SSN library, and it can be easily integrated into Matlab/Simulink model for time-domain simulation [22], where comparative analysis of original WideBand Line model, frequency-dependent RLC circuit model and non-frequencydependent RLC circuit model is performed.Based on this, the impact of longitude temperature variation on system stability is analysed.The electrical and control parameters of the GCI ... [ ] x km simulation model is shown in Table I, and the p.u.l.electrical parameters of the LTC simulation model is shown in Fig. 5.    will be used to verify the impact of different models on smallsignal stability analysis conclusion.
The equivalent grid impedance Z g cl is the combination of Z C f , Z L f 2 and Z LT C , shown as in Fig. 3.The Bode plots of the equivalent grid impedance using the dependent circuit model with 20 Π sections Z g cl1 , using the non-frequency-dependent circuit model with 20 Π sections Z g cl2 and using the WideBand Line model Z g cl3 can be calculated from Fig. 12, as shown in Fig. 13.And the terminal impedance of GCI with f s = 10kHz is also plotted.It can be seen that Z g cl1 can approximate Z g cl3 well up to the upper limitation 5kHz of the non-passivity area.However, the non-frequency-dependent circuit cannot reproduce the inherent damping characteristics of LTC.It's clear that there are 7 impedance interaction points between non-frequency-dependent cable model and GCI, of which the phase differences are all larger than 180 • .It means that the GCI-LTC system tends to oscillate at multiple frequencies.However, the conclusion that the system tends to be stable will be obtained if WideBand Line model and frequency-dependent model are adopted.
Fig. 14 shows time-domain grid current waveforms and corresponding frequency spectrum when non-frequencydependent circuit model of 40km LTC is used.It can be seen that there are multiple oscillation frequencies, among which the magnitude of 2562Hz frequency component is largest, since the frequency is located at the non-passivity region [10kHz/6,10kHz/2] and its phase margin is smallest, shown as in Fig. 13.By reducing sampling frequency f s from 10kHz to 4kHz, the time-domain grid phase currents and frequency spectrum are shown in Fig. 15.Compared with Fig. 14, the dominant oscillation frequency reduces from 2562Hz to  1870 Hz (located in [4kHz/6, 4kHz/2]).It means that the sampling frequency f s can influence the phase margin of the interaction points, and thus influence which oscillation frequency is dominant.
Similarly, the time-domain grid phase current waveforms and frequency spectrum using WideBand Line model and frequency-dependent circuit model are shown in Fig. 16.It can be seen that grid phase currents are stable and no divergence of grid phase current is observed.
It can be concluded that using the non-frequency-dependent LTC circuit model for stability analysis tends to lead to inaccurate conclusions, due to the lack of representation of inherent damping characteristics.

B. Impact of LTC length on stability analysis
To analyze the impact of LTC length on stability analysis, LTC length is reduced from 40km to 0.5km with sampling frequency f s = 10kHz.The Nyquist plot of the MLG is shown in Fig. 17

C. Impact of longitude temperature variation on stability analysis
To further investigate effect of longitude temperature variation on system stability, temperature-dependent circuit model is developed, where electrical parameters of circuit model are revised according to temperature characteristics shown in Fig. 5.The terminal impedances of these frequency-dependent circuit models at different uniformly-distributed temperatures are plotted in Fig. 18.It's clear that the impedance curve shifts leftwards in higher temperature and shifts rightwards in lower temperature.
As an example, time-domain current waveforms and frequency spectrum when the longitude temperature is -50 • C are shown in Fig. 19.Compared with Fig. 17 (b), (c), the dominant oscillation frequency moves rightwards from 3078Hz to 3240 Hz, which results from the rightward shift of the impedance curves interaction points between GCI and LTC at lower temperature.It can be understood that ambient temperature variation may affect system oscillation phenomenon.Thus, temperature-dependent characteristics is important for stability analysis of GCI with LTC.

V. CONCLUSIONS
This paper presents a frequency and temperature-dependent circuit model of LTC for small signal stability analysis, where    the detailed implementation procedure of modelling is given.P.u.l.impedance of power cable is first obtained by Vector Fitting (VF) algorithm.Then, cascaded-Π circuit model is established to represent terminal impedance of LTC.Finally, the electrical parameters of each Π section are adjusted to according to temperature variation and temperature-dependent model of power cable is formulated.Comparative analysis among WideBand Line model, frequency-dependent model and non-frequency-dependent model is performed.Simulation results show that, the proposed frequency-dependent RLC circuit model can reveal practical damping characteristics of LTC.In addition, electrical parameters of LTC may be perturbed due to variation of longitude temperature.The temperaturedependent characteristics of LTC has to be considered for more accurate small-signal stability analysis of GCI with LTC.

Fig. 5 .
Fig. 5.The variation of p.u.l.resistance and inductance of LTC with frequency and temperature.

Fig. 11
Fig.11shows the terminal impedances of WideBand Line model, the frequency-dependent RLC circuit with different number of Π sections (1, 5, 10 and 20) for a 40km LTC.It can be seen that fitting accuracy become higher as increase of the number of cascaded Π sections in frequencydependent circuit model.On the other hand, Fig.12shows the terminal impedances of WideBand Line model, the frequencydependent RLC circuit with 20 Π sections and the nonfrequency-dependent RLC circuit model with 20 Π sections for the same 40km LTC.It can be seen that the non-frequencydependent circuit model has a poorer approximation.The frequency-dependent circuit model with 20 Π sections and the non-frequency-dependent circuit model with 20 Π sections

Fig. 13 .
Fig. 13.Bode diagrams of terminal impedances of GCI Z inv cl with fs = 10kHz and equivalent grid impedances using different circuit models.
(a).It can bee seen that, different from Fig.17 (a), the Nyquist plot encircles (-1, j0) point two times, indicating the system is unstable.Fig.17 (e),(f) shows corresponding timedomain grid phase current waveforms and frequency spectrum.It can be seen that the dominant frequency component appears at 3078Hz, which is located in the non-passivity region of GCI output impedance [10kHz/6, 10kHz/2].It can be seen that the stability analysis results are different for the same type of LTCs in different lengths. 0

Fig. 14 .
Fig. 14.Time-domain waveforms and frequency spectrum of grid currents using non-frequency-dependent circuit model for 40km LTC with fs = 10kHz.(a) Time-domain waveforms; (b) Frequency spectrum.

Fig. 15 .
Fig. 15.Time-domain waveforms and frequency spectrum of grid currents using non-frequency-dependent circuit model for 40km LTC with fs = 4kHz.(a) Time-domain waveforms; (b) Frequency spectrum.

TABLE I .
SYSTEM PARAMETERS OF THE EXEMPLIFIED GCI.