Power System Series Resonance Studies by Modified Admittance Scan

Power System Series Resonance Studies by Modified Admittance Scan Felix O. Kalunta, MNSE, Frank N. Okafor, FNSE, Member, IEEE, Osita U. Omeje Abstract This paper presents a modified method of formulating the loop admittance matrix which is deployed to identify the series resonant frequencies in large electrical networks involving numerous shunt capacitances. Three matrices were assembled from the network R, L and C elements and later synthesized to obtain the network loop admittance matrix. Its application to a sample network has shown the practicability and effectiveness of this method. This paper also seeks to demonstrate the impact of certain factors like cable capacitance and skin effect on the value of resonant frequencies. Index Terms Harmonic Resonance, Loop Admittance Matrix, Power Quality and Series Resonance. I. INTRODUCTION The application of capacitor banks in the power industry has yielded some utility benefits such as power factor correction, voltage support and release of system capacity. However, their interaction with system inductive supply circuit [1, 2] causes power quality problems by way of amplifying high order harmonics. These could lead to overheating, failure of the capacitor banks themselves or blowing of power transformer units resulting in constant interruption in production schedules. Another concern is that the broadband spectrum emitted in the process could result in emission limits being exceeded for non-characteristic harmonics and for inter-harmonics. The challenge is how to theoretically predict the resonant frequencies based on which appropriate mitigation measures can be included at the planning stage or the magnitude of amplified currents for the power equipment to be de-rated in order to withstand this exigency. There is also the need to indicate which of the system component is under the threat of series resonance [3]. The papers [4 5] describes the use of frequency scan technique to detect the possible resonant frequencies in the electrical network while [5, 6] presented a harmonic resonance mode assessment based on the analysis of eigenvalues and eigenvectors of the system bus impedance matrix. Further details of modal analysis in the study of resonance are treated in the technical literature [9-11]. Felix O. Kalunta is currently pursuing his Ph.D degree with the Department of Electrical/Electronic Engineering, University of Lagos, Nigeria. He is on study leave from Federal Institute of industrial Research Lagos, Nigeria (e-mail: felka3@yahoo.co.uk). Osita U. Omeje is currently pursuing his Ph.D degree with the Department of Electrical/Electronic Engineering, University of Lagos, Nigeria (e-mail: ositaomeje@gmail.com) Frank N. Okafor is a professor with the Electrical/Electronic Engineering Department, University of Lagos, Nigeria. (e-mail: cfrankok@yahoo.com). These methods are primarily suitable for parallel resonance problem with little adaptation to the analysis of series resonance. The reason lies in the close relationship between loop impedance and the occurrence of series resonance. An attempt to apply the modal analysis to series resonance problem was also made [8] but yielded an incomplete solution. A dummy branch method was later incorporated into the modal analysis but the approach seems more like a short circuit study which does not reflect the actual series resonance scenario. The loop admittance scan technique is an adaptation of frequency scan for locating the resonance peaks in a series resonance problem. Calculations are performed to determine the loop admittance matrix of the concerned network, while the driving point admittances are the determinants of the resonant frequencies. The complexities involved in the calculation of the loop admittance matrix for large power networks especially when cable capacitance is involved have necessitated the need for a modified admittance scan. In such cases, matrix of network elements R, L and C are easily assembled by computer programming and later synthesized to form a loop impedance matrix. This is the approach adopted in this paper. The formation of dummy loops in the calculation of the network matrices is applied in order to account for the shunt capacitances without altering the network topology. Other factors that affect resonance characteristics like skin effect of cables are also investigated in this paper. This study will account for skin effect by calculating the resistance R of the cable at various discrete frequencies using an equation that varies according to cable type [11]. R = R 1 (.187 +.532 h ), where h 2.35 ----------- (1) where R 1 is the resistance of the cable at the fundamental frequency and h is the harmonic order. II. MODAL ANALYSIS APPLIED TO SERIES RESONANCE The determination of series resonant frequencies is based on mesh analysis at each harmonic frequency h in per unit. Resonance mode analysis in this case is based on the fact that the loop impedance matrix of a power network becomes singular at resonant frequencies. This requires the calculation of the eigenvalues of the system as well as their sensitivities to changes in system parameters. Imagine that a system experiences resonance at frequency h according to the frequency scan analysis. It implies that some elements of the loop current vector have large values at h. This in turn implies that the inverse of the [Z h ] matrix has large elements. This phenomenon is primarily caused by the fact that one of the eigen-values of the Z matrix is close to zero. In fact, if the system had no damping, the Z matrix would become singular due to one of its eigenvalues becoming zero. The above reasoning leads us to believe that the characteristics of the smallest eigenvalue of the [Z h ] matrix could contain useful information about the cause of the harmonic

resonance. The above analysis can be formally stated as follows: [V] h = [Z loop ] h [I loop ] h The current vector for each harmonic order is as follows [I loop ] h = [Z loop ] h [V] h, h = 1, 2, 3 n ---------- (2) Where[Z loop ] h, is the loop impedance matrix [V] h is the harmonic loop voltage vector. As usual, the loop impedance matrix can be decomposed into the following form: [Z] = [L][D][T] is the eigen-decomposition of the [Z] matrix at frequency h. [L] and [T] are the left and right eigenvector matrices of [Z] [D] - diagonal eigenvalue matrix of Z. T= L is due to the fact that Y is symmetric. Equation (2) now becomes [I]=[L][D] [T] [V] ------------------------ (3) If [T][V] and [T][I] are defined as the modal voltage and current vectors respectively. It can be seen that admittance scan equation has been transformed into the following form: J 1 λ 1 V m1 J [ 2 λ ] = 2 V [ m2 ] ------------- (4) J n [ λ n ] V mn This can be abbreviated as, [J k ] h = [λ k ] h [V mk ] h -------------------- (5) The inverse of the eigen value, λ k, has the unit of admittance and is named modal admittance. One can easily see that if λ 1 = or is very small, a small applied mode 1 voltage will lead to a large mode 1 current. On the other hand, the other modal currents will not be affected since they have no 'coupling' with mode 1 voltage. In other words, one can easily identify the 'locations' of resonance frequencies in the modal domain. After identifying the critical mode of resonance, it is possible to find the 'participation' of each loop in the resonance. This can be done using the well-known participation factor theory described in [1]. III. FORMULATION OF LOOP ADMITTANCE SCAN The admittance scan is a series resonance counterpart of the bus impedance scan. The scan is performed in one network mesh at a time. A sinusoidal voltage of unit amplitude V = 1, and of certain harmonic frequency is inserted into this mesh and the corresponding loop current is calculated. The process is repeated for other harmonic frequencies in per unit. For the purpose of illustration, consider a radial network in fig. 1 containing capacitor banks with a non-linear load connected to bus J through a transformer and a long cable. The network is partitioned along the point of common coupling between the consumer distribution network and public utility supply as in fig. 2. The consumer side of the network is modeled at each harmonic frequency h, and the system supply side is reduced to its Thevenin equivalent also at each harmonic frequency. There are two buses where voltage amplification could be excited by parallel resonance and also three meshes or loops where current amplification can occur by series resonance. Fig. 1: A simple consumer premises supplied from the public utility network Fig. 2: The equivalent circuit model of the 2 bus consumer network The application of loop analysis to fig. 2 produces the following equation, [V] h = [Z loop ] h [I loop ] h The current vector at each harmonic frequency, h is as follows [I loop ] h = [Z loop ] h [V] h, h = 1, 2, 3 n ------------- (6) The matrix [Z loop ] h is known as the loop impedance matrix. The inverse of the loop impedance matrix is known as loop admittance matrix, Y loop Y loop = [Z loop ] ---------------------------- (7) This matrix is the counterpart of bus impedance matrix and is therefore useful in the determination of the frequencies at which harmonic series resonance occurs. Equation (6) can now be written as loop I 1 [ I 2 ] I 3 loop Y 11 Y 12 Y 13 = [ Y 21 Y 22 Y 23 ] Y 31 Y 32 Y 33 V 1 [ V 2 ] ------------------ (8) V 3 For the purpose of resonance analysis, only the driving point admittances Y l kk are required. l Therefore Y kk = [ I k ], k j V k V j = [I loop ] h k = [Y loop ] h h kk [V loop ] k, V j = k j ---- (9) Suppose V k = 1p. u for all h =1, 2, 3 n [I loop ] h h k = [Y loop ] kk ------------------------ (1) A graph of [I loop ] k against h is equivalent to a graph of [Y loop ] kk against h. Therefore, driving point admittance versus frequency plot is obtained for each mesh in the network. Therefore, driving point admittance versus frequency plot is obtained for each mesh in the network. This technique when applied for the determination of series resonant frequencies in power networks can be referred to as admittance scan. The off diagonal or transfer admittances could be considered in the admittance scan when dealing

with many harmonic sources applied simultaneously at different network meshes. The current change in the k-th mesh due to the insertion of unit harmonic voltages in meshes k, i and j is stated as C p = the inverse shunt capacitance at p-th dummy loop. I k = Y kk + Y ki + Y kj ------------------ (11) IV. MODIFICATION OF NETWORK MATRICES TO ACCOUNT FOR THE SHUNT CAPACITANCES Procedure: 1. The three (n n) network matrices are assembled as usual in the absence of the shunt capacitances. These are designated as R old, L old and G old respectively. Fig 3: Equivalent circuit diagram showing two dummy loops for shunt capacitors C 1 and C 2 2. All parallel connection of capacitances can be combined together by summation. 3. Dummy loops m in number are created at each node where the shunt capacitances exist such that the total number of loops becomes(n + m), see fig 3. The dummy loops are assigned loop numbers n + 1, n + 2, n + m 4. Extra rows and columns corresponding to the number of the created dummy loops are added to each of the three network matrices to form a partitioned matrix as shown, R new = [ R old A ] ------------------- (12) A Q L new = [ L old B ] ---------------------- (13) B H G new = [ G old U ] ----------------------- (14) U V 5. The dummy loops are eliminated using the kron reduction formular, R new = R old AQA T, L new = L old BHB T and G new = G old UVU T 6. The matrix of network elements R, L and G are finally synthesized according to equation (15) to obtain the loop impedance matrix. This is repeated for each harmonic frequency, h. [Z loop ] = [R new ] + jω 1 h[l new ] j[g new ]/ω 1 h - (15) Where L n n loop inductance matrix R n n loop resistance matrix G n n loop inverse capacitance matrix ω 1 fundamental frequency in rad/s For a dummy loop p created between two actual loops i and j A ip = A pi = R sp, A jp = A pj = R sp, Q pp = R sp, elsewhere the entries are zero. Similarly, B ip = B pi = L sp, B jp = B pj = L sp, H pp = L sp and V pp = C p, elsewhere the entries are zero. Where R sp = the total resistance in the branch common to loops i, j and p L sp = the total inductance in the branch common to loops i, j and p Fig. 4: The sequential process of carrying out an admittance scan based on the creation of dummy loops The updated matrices obtained from application of the above procedure to fig. 3 are, R Loop = L Loop = G Loop = [ [ R 11 R 21 R 31 R s1 L 11 L 21 L 31 G 11 G 21 G 31 [ R 12 R 22 R 32 R s1 L 12 L 22 L 32 G 12 G 22 G 32 R 11 R 23 R 33 G 11 G 23 G 33 L 11 L 23 L 33 R s1 R s1 R s1 C 1 ] C 2 ] --------------- (16) L s2 L s2 L s2 ] --------- (17) ------------ (18)

Modal Impedance (pu) Impedance (pu) Admittance (pu) Modal Admittance (pu) Equation (15) is then applied after the process of eliminating the dummy loops, [Z loop ] = [R new ] + jω 1 h[l new ] j[g new ]/ω 1 h 12 1 Loop1 Loop2 Loop3 Y loop = [Z loop ] new ------------------ (19) 8 6 The entire process of carrying out an admittance scan based on the creation of dummy loops is described in the flow chart in fig. 4. The modified admittance scan and modal loop analysis has been applied to the equivalent circuit shown in fig.2, and the results are displayed in Fig. 7 and 8 respectively. With the circuit, series resonance result could be compared to that of parallel resonance solved by frequency scan and Resonance Mode Analysis (see fig. 5 and 6). 4 2 1 2 3 4 5 6 Fig. 7: Results of modal Loop analysis on test system 45 4 35 bus1 bus2 bus3 4.5 4 3.5 Loop1 Loop2 Loop3 3 3 25 2 15 2.5 2 1 1.5 5 1 1 2 3 4 5 6 Fig 5: Bus Impedance Scan Results of the test system.5 1 2 3 4 5 6 Fig. 8: Loop Admittance Scan results of the test system. 7 6 5 4 3 2 1 bus1 bus2 bus3 V. APPLICATION TO A DISTRIBUTION NETWORK The network diagram in fig.9 represents an 11kV underground radial distribution network which feeds nine load centers containing two major harmonic sources and four capacitor banks. It is an expansion of the distribution network used as case study in Reference [7]. The load centers are connected by 35mm 2 x 3, 11kV underground cables whose parameters are as follows, 1 2 3 4 5 6 Fig. 6: Results of bus Resonance mode analysis (RMA) The results indicate that the modified loop admittance scan compares favourably to its counterparts: modal-loop analysis and bus impedance scan. The key resonance modes according to the results displayed in Fig. 5 8 occur at frequencies (pu) = 4, 17 and 27. The modal admittance scan (fig.7) produces only one resonance peak for each mesh whereas about three peaks are captured in each mesh in the results of modified loop admittance scan. This achievement is because the network topological structure was not altered in the process of applying the proposed technique in localizing the series resonant frequencies. Resistance/ph/km =.243Ω, Inductance/ph/km = 5.23 x 1-4 H Capacitance/ph/km = 2.456 x 1-8 F The frequency response characteristics resulting from application of modified loop admittance scan across the entire distribution network are shown in fig. 1 13.

Fig. 12: Frequency response considering the effect of cable capacitance Fig. 9: Diagram showing the 11kV radial distribution network in a consumer premises Fig.13: Frequency response considering cable capacitance and skin effect The results indicate the occurrence of series resonance peaks at the various meshes M1 M9 as indicated. Fig 1: Frequency response neglecting the effect of cable capacitance and skin effect In figure 1 where the effects of cable capacitance and skin effect are neglected, the resonant frequencies in per unit are h = 6, 7, 14 and 15. In figure 11 where the effect of aggregate harmonic sources is considered, the resonant frequencies are the same as in fig. 1. In figure 12 where only the effect of cable capacitance is considered, the resonance frequencies in per unit are h = 6, 7 and 14. In figure 13 where the effects of cable capacitance and skin effect are considered, the resonance frequencies in per unit are h = 6, 7 and 14. Fig 11: Frequency response for aggregate sources neglecting the effect of cable capacitance and skin effect Comparison between figures 1-13 indicates that cable capacitance contributes immensely to the shifting of the resonant frequencies in M3 M6 while the skin effect only reduces the magnitude of loop admittance and widens the frequency curve. Since resistance instead of inductance or capacitance (that is the main contributors to system resonance) varies according to the skin effect, consequently, considering the skin effect in the calculations does not shift the resonant frequencies. However, the increased resistance could dampen the admittance peaks at resonance points thereby decreasing the branch currents. Skin effect also

increases the bandwidth of resonance. This implies the greater chance of current amplification occurring within the neighbourhood of the resonant peaks. Comparison between fig. 1 and 11 indicates that the application of many harmonic frequency sources only impacts on the magnitude of the driving point admittances and therefore is of no consequence to this study which is focused on the resonant frequencies. VI. CONCLUSION The modified loop admittance scan proves to be an acceptable method for capturing all the dominant frequencies involved in series resonance without any restriction to the utilized frequency step. The application of a single unit voltage is sufficient in the definition of admittance scan. Specific achievements are recorded in this study: the determination of series resonant frequencies by the application of loop admittance matrix, the use of matrix partitioning to treat independently the connection of shunt capacitors and the contribution of skin effect as well as cable capacitance on these results. It is noted that skin effect does not contribute to the shifting of resonant frequencies but only cable capacitance. Hence, in the calculation of the resonance peaks, the contribution of skin effect can be neglected. However, skin effect makes a significant contribution to the value of branch currents which necessitates its consideration. Conference on Innovations in Engineering and Technology (IET 211), Faculty of Engineering, University of Lagos, Nigeria, 8th 1th August 211, pp 568 577. [8] H. Zhou, Y. Wu, S. Lou and X. Xiong, Power System Series Harmonic Resonance Assessment based on Improved Modal Analysis, Journal Of Electrical & Electronics Engineering, Istanbul University, 27, vol.7, No.2, pp 423 43. [9] C. Yang, K. Liu and Q. Zhang, An Improved Modal Analysis Method for Harmonic Resonance Analysis, IEEE International Conference on Industrial Technology (ICIT 28), ISBN: 978-1-4244-175-6, Chengdu, China, 28, pp.1 5. [1] K. Md Hasan, K. Rauma et al, Harmonic Resonance Study for Wind Power Plant, International Conference on Renewable Energies and Power Quality (ICREPQ 12), Santiago de Compostela, Spain, 8th to 3th March 212. [11] K. Nisak, K. Rauma et al, An Overview of Harmonic Analysis and Resonances of a Large Wind Power Plant, Annual Conference of the IEEE Industrial Electronics Society (IECON 211), 7 1 November, 211. VII. REFERENCES [1] Z. Huang, Y. Cui and W. Xu, "Application of Modal Sensitivity for Power System Harmonic Resonance Analysis", IEEE Transactions on Power Systems, vol. 22, 27, pp. 222 231. [2] J. Li, N. Samaan and S. Williams, "Modeling of Large Wind Farm Systems for Dynamic and Harmonics Analysis", Transmission and Distribution Conference and Exposition, T&D, IEEE/PES, 28, pp. 1 7. [3] C. Chien and R. Bucknall, "Harmonic Calculations of Proximity Effect on Impedance Characteristics in Subsea Power Transmission Cables", IEEE Transactions on Power Delivery, ISSN: 885-8977, vol. 24, October 29, pp. 215 2158. [4] H. K. Lukasz, J. Hjerrild and C. L. Bak, Wind Farm Structures: Impact on Harmonic Emission and Grid Interaction, European Wind Energy Conference 21, Warsaw, Poland, 21. [5] R. Zheng and M. Bollen, Harmonic Resonances Due to a Grid-Connected Wind Farm, In Proceedings of the 14th International Conference on Harmonics and Quality of Power (ICHQP 21), Bergamo, Italy, 26-29 September 21, pp. 1 7. [6] J. Arrillaga and W. Neville, Power System Harmonics, ISBN: 47851295, West Sussex: Wiley & Sons, 23. [7] F. O. Kalunta and F. N. Okafor, Harmonic Analysis of Power Networks Supplying Nonlinear Loads, International