Modeling, analysis and control of microgrids in dynamic and steady-state

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1 Thesis advance IV Modeling, analysis and control of microgrids in dynamic and steady-state M.I. Gibran David Agundis Tinajero 1, Dr. Juan Segundo Ramírez 2, Dra. Nancy Visairo Cruz 3 Abstract This document presents the fourth thesis advance of the period September February 217. A review of the thesis advance III is addressed in the Introduction. In Sections 3-5, an islanded microgrid with a two level hierarchical droop control and its implementation in the laboratory is presented. In Section 6, a shooting method for the computation of the steady-state solution of autonomous systems with unknown period is addressed. In Section 7, discussion and conclusions of the results obtained in the previous Sections are made. Finally, the activity schedule for the next semesters is shown in Section 8. 1 PhD student 2 Thesis advisor 3 Thesis co-advisor Contents 1 Introduction 1 2 Thesis advance in the period September February Hierarchical control for islanded microgrids using an OEPF Primary control: grid-forming droop control Extended power flow formulation Optimal extended power flow formulation DG units efficiency function PCCs voltages function Reactive power function Power restriction 4 Test system and laboratory implementation 5 5 Case studies Case I: Load profile Conventional droop control Hierarchical control scheme 5.2 Case II: Load and capacity profiles Shooting method for autonomous systems Autonomous systems [1] Discussion and conclusions 8 8 Activities schedule 8 References 9 1. Introduction In the previous thesis advance, an extended Newton-Raphson power flow formulation for islanded and grid-connected VSCbased microgrids with controlled distributed generation units was presented. Three different power management control strategies were addressed, and the detailed procedure for their inclusion in the conventional Newton-Raphson power flow problem was shown. This formulation was developed in order to compute the steady-state solution of controlled microgrids in an easy and straightforward manner overcoming the necessity of numerical integration. In this thesis advance, an optimal extended Newton - Raphson formulation is used to develop a two level hierarchical control for islanded microgrids. Furthermore, the proposed control is implemented and online tested in the microgrid laboratory of the Aalborg University [2]. On the other hand, due to the frequency of the droop controlled islanded microgrids is different to the nominal, a shooting method for autonomous systems with unknown period is studied in order to compute the steady state solution and stability of droop controlled microgrids. 2. Thesis advance in the period September February 217 In this period the following activities were made: 1. Review of the state of the art of control strategies and microgrids. 2. Development of an hierarchical control for islanded microgrids using an optimal extended power flow as secondary level control. 3. Laboratory implementation of an islanded microgrid with a hierarchical control. 4. Course An introduction to HVDC and MTDC transmission systems (approved. Course responsible: Professor Sanjay K. Chaudhary. 5. Course Transients in power systems (approved. Course responsible: Professor Filipe Faria da Silva. 6. Course Harmonics in power electronics and power systems (approved. Course responsible: Professor Claus Leth Bak.

2 Modeling, analysis and control of microgrids in dynamic and steady-state 2/9 7. Course Electricity Market and Power System Optimization (approved. Course responsible: Professor Zhe Chen. 8. Implementation of parallel Newton methods for the computation of the steady-state solutions using cloud computing (24 cores. 9. Study and implementation of a shooting method for the computation of the steady-state solution of systems with unknown period. 1. Revise and correction of the journal paper entitled An Extended Newton-Raphson Power Flow Formulation for Islanded and Grid-Connected VSC-Based Microgrids in collaboration with the external researchers Josep M. Guerrero and Mehdi Savaghebi, from the Aalborg University. 11. Writing and submission of the journal paper entitled Hierarchical Optimal Extended Power Flow Control for Islanded AC Microgrids in collaboration with the external researchers Josep M. Guerrero, Juan C. Vazquez, Nelson Diaz and Adriana Luna, from the Aalborg University. 12. Abstract submission to the 217 PES General Meeting of the accepted PES Transactions paper entitled Harmonic Issues Assessment on PWM VSC-Based Controlled Microgrids Using Newton Methods 13. Collaboration in the submitted conference paper entitled Monitoring Harmonic Distortion in Microgrids using Dynamic Mode Decomposition, in the 217 PES General Meeting. 3. Hierarchical control for islanded microgrids using an OEPF The proposed hierarchical control has the purpose of achieving specific operational goals of the islanded microgrid through of the optimized selection of the primary control references. The hierarchical control includes a primary conventional droop control and a centralized optimal extended power flow (OEPF control level as shown in Fig. 1. In this scheme, the OEPF control level is responsible for computing and sending the references K p, K q, and V to every distributed generation (DG unit primary control. In order to perform this computation, the information of the microgrid topology, loads and the DG units capacities are needed, furthermore, a set of constrained nonlinear functions representing the operational goals is required by the optimization method. On the other hand, using the reference values computed in the OEPF control level, the primary control regulates the voltage and frequency of the islanded microgrid, as well as the active and reactive power injected by each DG unit. DG units capacities PQ load [Kp, Kq, V * ] OEPF Droop Control Droop Control Droop Control DG2 DG1 DGn Figure 1. Hierarchical control scheme. Microgrid In regards to the aforementioned, a detailed explanation of each stage of the hierarchical control scheme is addressed below. 3.1 Primary control: grid-forming droop control Due to the islanded condition, the first control level has to regulate the frequency and the voltage of the microgrid. To achieve these objectives, the conventional grid-forming droop control is used [3, 4]. Fig. 2 shows the basic control structure in a three-phase grid-forming inverter. Observe that the control has two loops working on the dq reference frame; the external loop adjusts the grid voltage and the internal loop regulates the current supplied by the inverter. In this way, the controlled current charges the capacitor to keep the output voltage close to the reference provided by the voltage control loop [3]. On the other hand, notice that the frequency and voltage references used in the external and internal loops are given by the conventional droop approach, where, K p and K q are the slope values of the droop characteristic curves [5]. PWM v abc i d i q P Q LF1 i (t Signal processor v (t CF Signal processor c c LF2 v (t i (t PI PI PCC I Signal processor Microgrid PI PI K p * P Figure 2. Basic grid-forming control structure. The nominal parameters used for the primary control in the case studies are shown in Table 1. i d i q V * K q Q c c

3 Modeling, analysis and control of microgrids in dynamic and steady-state 3/9 Table 1. Parameters of the primary control Parameter Symbol Value P ω droop coefficient K p Q V droop coefficient K q Current loop proportional gain K pc 2 Current loop integral gain K ic 4 Voltage loop proportional gain K pv Voltage loop integral gain K iv 4.5 Commutation frequency f c 1 khz V = V K q Q (4 where ω is the nominal angular frequency of the system and V is the nominal voltage at the connection bus. Notice that from (3 and (4, the active and reactive power given by the n-th DG unit can be defined as, P n = ω ω K p (5 3.2 Extended power flow formulation In electric power systems, the conventional power flow formulation relies on the well-known power balance equations [6], P n = N m=1 Q n = V n V m Y nm cos(θ nm δ n + δ m N m=1 f or n = 1,...,N (1 V n V m Y nm sin(θ nm δ n + δ m f or n = 1,...,N (2 where V n, V m, δ n and δ m are the magnitudes and phase angles of the n-th and m-th bus voltages, respectively. Y nm and θ nm are the magnitude and phase angle of the admittance matrix, respectively. Depending on the topology of the system, the power flow can be formulated using three types of buses: voltagecontrolled (PV, load (PQ and slack [7]. In this way, the active power injected to the system by the PV buses is fixed and known, while the slack bus provides the missing active and reactive power needed by the system. In islanded droop controlled microgrids, this conventional power flow approach cannot be used in most of the cases because the following reasons [8, 9]: 1 The DG units used in a microgrid have a limited capacity, therefore, a slack bus can not be assigned. 2 The active and reactive power sharing among the DG units depends on the droop characteristics and cannot be pre-specified such as in the conventional power flow. 3 The frequency in a islanded microgrid is changing constantly within a range, while in the conventional power flow it is considered always fixed. Accordingly to the aforementioned, an extra bus classification has to be included in the power flow formulation to take into account the droop controlled DG units, hereafter called droop buses (DB [8]; these DB rely their formulation in the droop characteristic equations, ω = ω K p P (3 Q n = V V n K q (6 therefore, the mismatching equations P and Q of the power flow formulation becomes, [ Pn Q n ] [ Pn ( ω ω K = p Q n ( V V n K q Observe that, in (7 the angular frequency is still unknown, therefore, an extra equation related to it has to be included. To overcome this problem, one voltage angle of the DG units is fixed and its active power equation is used as the angular frequency equation as follows, P n Q n ω = P n ( ω ω K p Q n ( V V n K q P m ( ω ω K p ] (7 (8 where P m is the active power of the fixed angle bus. The set of mismatching equations given in (8 completes the formulation of the droop buses, which can be used together with the conventional PV and PQ buses. It should be mentioned that, in the case that one DG unit exceeds its active or reactive power capacity, the value is fixed at its specific limits and the computation of the power flows continues. It is important to notice that the control references K p, K q, and V that affect the steady-state behavior of the islanded microgrid appear explicitly in the DB formulation, giving the opportunity to use them as variables in the optimization stage. 3.3 Optimal extended power flow formulation In order to solve the optimization problem of the extended power flow, the fmincon MATLAB function is used. The formulation of the problem is specified as follows, min x f (x such that { c(x Ax b lb < x < ub (9

4 Modeling, analysis and control of microgrids in dynamic and steady-state 4/9 where x are the variables to optimize, f (x is the function to minimize, c(x is a nonlinear function, A is a matrix, b is a vector, and lb and ub the lower and upper bound restrictions for the x values, respectively. The equations proposed for the optimization formulation are based on four operational goals of the islanded microgrid: 1. The losses of each DG unit is minimized based on their efficiency curves, maximizing the global efficiency of the microgrid. 2. The voltages on the PCCs are maintained close to 1 p.u. 3. The reactive power shared among the DG units has to be equal in percent of their capacities. 4. The power injected by the DG units cannot exceed its maximum capacity (S n < S max n. The variables to optimize are the reference values of the primary control K p, K q, and V. The formulation of each operational goal as a function is explained in detail below DG units efficiency function The efficiency of the DG units is defined based on the relationship between the output power delivered to the microgrid and the input power provided by the primary source [1], η n = Pout n Pn in (1 this efficiency represents the losses due to conduction, switching of electronic devices, among others, additionally, the total losses depends on the characteristic of the technology used, the operation point and the switching frequency [1]. In this way, the efficiency of an inverter can be represented graphically for all the load range that it is able to handle. These efficiency curves can be obtained with simulation or experimentally and approximated by a second order function as follows [1], η n = P in2 n α 1 n P in n + α n + β 1 n P in n + β n (11 where αn 1, αn, βn 1 and βn are the coefficients obtained with the simulation or experimental results of the n-th DG unit. Hence, in a microgrid composed by n DG units, the global efficiency can be maximized with the reduction of the total losses in the DG units, taking into account that due to the short distances in the islanded microgrid the losses in the feeders can be neglected, otherwise, the conductor losses have to be taken into account. Notice that (11 computes the efficiency using the Pn in but in the power flow method the power computed is Pn out, therefore, substituting P in = P out /η (11 is reformulated and the following quadratic equation is obtained, η 2 n + η n ( β 1 n β n Pn out α n βn + Pout2 n β n α1 n βn Pn out = (12 finally, solving (12 the value of η n is obtained. Regarding the aforementioned, this operational goal can be formulated in the optimization problem as follows, F η = N (P in n=1 n Pn out 2 (13 where N is the number of DG units. Observe that, if all the DG units are working with a ideal efficiency (η = 1, the function F η will be equal to zero, in this way, the minimization of this function will maximize the global efficiency of the microgrid. In the case studies presented below, the efficiency parameters used for the DG units were extracted from [1] and are shown in Table 2. Table 2. Parameters of the efficiency curves Curve α 1 n α n β 1 n β n η η η PCCs voltages function Because of the power flow solution gives the voltage magnitudes in p.u. of the system buses, the function used to achieve this operational goal can be formulated as follows, F V = N n=1 (V PCCn 1 2 (14 notice that if the magnitudes of voltages V PCCn are equal to the reference of 1 p.u., the summation of the function will be equal to zero Reactive power function To formulate this operational goal, the reactive power obtained in the power flow and the specific power limits of each DG unit are used. For a microgrid with three DG units the function is as follows, F Q = N N k=1 n=1 ( Qn S max n Q k 2 Sk max k n (15 if the DG units share the reactive power equally in percent to their capacities, the function (15 will be zero.

5 Modeling, analysis and control of microgrids in dynamic and steady-state 5/ Power restriction The power restriction is included in the optimization problem as a inequality function in c(x as, S n S max n (16 while this inequality is true, the power injected by the n-th DG unit will not exceed its capacity. Regarding the aforementioned, the optimization problem is built as a sum of each operational goal function, subject to the capacity restriction as follows, to the dspace via ethernet using a User Datagram Protocol (UDP. The communication between the dspace and the CPC is performed through an interface using the professional software LabVIEW. When the OEPF stage is required, the dspace platform sends the information of the microgrid loads to the CPC which, using the LabVIEW interface, calls the MATLAB software to perform the optimization process. The CPC sends back the computed control parameters and the dspace uses this information in the DG units primary control. dspace min x f (x = F η + F V + F Q Sub ject to S n Sn max (17 Variable load Inverters finally, the solution of this problem gives the control references K p, K q, and V. 4. Test system and laboratory implementation Figure 3 shows the single line diagram of the microgrid used as test system. It includes three dispatchable DG units with LCL filters, two fixed R loads connected to PCC 1 and PCC 3, and a variable PQ load connected to PCC 2, additionally, the PCCs are connected through RL lines. RL lines Fixed loads Central computer PQ Load R L1 R Lin1 L Lin1 R Lin2 L Lin2 PCC 1 L 12 L 22 PCC 2 PCC 3 L 32 R L2 Figure 4. Laboratory implementation of the microgrid. R 12 R 13 C 11 R 22 R 23 C 21 L 11 R 11 DG1 R 21 R 31 DG2 R 32 L 21 L 31 DG3 Figure 3. Microgrid test system. R 33 C 31 The parameters of the test microgrid are shown in Table 3, they were obtained from the real values of the laboratory implementation. The implementation of the test system was made in the Microgrid Research Laboratory in Aalborg University with an online architecture as can be seen in Fig. 4. The primary control is modeled and included in a real time platform (dspace 16, while the OEPF stage is incorporated in a central computer (CPC which sends and receives information Table 3. Parameters of the microgrid Parameter Symbol Value Nominal voltage V RMS L L 4 V Nominal frequency f 5 Hz Nominal capacity S max 2.5 kva Filter resistance R 11, R 12, R 13, R 21, R 22, R 23, R 31, R 32, R 33.1 Ω Filter inductance L 11, L 12, L 13, L 21, L 22, L 23, L 31, L 32, L mh Filter capacitance C 11, C 21, C µf Line resistance R Lin1, R Lin2.45 Ω Line inductance L Lin1, L Lin mh Resistance Load R L1, R L2 115 Ω

6 Modeling, analysis and control of microgrids in dynamic and steady-state 6/9 P (W Q (VAr P load (W P DG1 P DG2 P DG P DG1 P DG2 P DG DG Q DG1-1 (a (b (c 2 1 P in % of the DG Capacity Q in % of the DG Capacity Efficiency (% 8 (d (e (f Q DG3 QDG2 1.1 V Q V DG3 V PCC3 PCC2 PCC1 Voltage (p.u (g (h (i 3 Q load (VAr Q DG1 Q DG2-1 Figure 5. Case I: Conventional droop control under load profile. Capacity (VA 2 1 DG2 DG3 DG1 DG2 DG3. 5. Case studies In order validate the proposed hierarchical scheme, in this section two case studies of the islanded MG under different operations conditions are presented. In the first case study, the load connected to the PCC 2 is changing following a load profile. The load profile have 24 changes made every 3 seconds, emulating a load variation every hour in a day. In the second case study, apart from the load profile, a power capacity profile is included in the DG units. In both cases, the control of the islanded microgrid is performed online using the hierarchical control; however, in the first case study, the experiment is also conducted using only the conventional droop control with the purpose to show the advantages of the proposed control scheme. 5.1 Case I: Load profile Conventional droop control The load profiles for active and reactive power can be seen in Figures 5(g and 5(h, respectively. Notice that, due to the control references K p, K q, and V are the same for all the DG units, the active and reactive power shared among the DG units is very similar (5(a and 5(d. Consequently, the efficiency of each DG unit is not taken into account in the operation of the microgrid as can be seen in 5(c, where in the worst case, droops up to 87 %. On the other hand, observe that the PCCs voltage, in the worst case, droops below.95 p.u. when the reactive power load is incremented Hierarchical control scheme The results obtained using the hierarchical control scheme are shown in Fig. 6. Observe how, in order to get a best efficiency, each DG unit injects a different amount of active power, having in the worst case an efficiency up to 95 %. Furthermore, in this case the PCC voltage is maintained close to 1 p.u., even with the abrupt changes of reactive power load. In this case, the reactive power is shared very similar as in the case with only the conventional droop control, and the differences are due the parameters in the laboratory are not exactly the same as in the power flow formulation, additionally, the small unbalances in the microgrid are neglected. 5.2 Case II: Load and capacity profiles In this case, the load and capacity profiles are shown in Figures 7(c and 7(f, respectively. Notice that the capacity of the DG units changes constantly and this causes a constantly variation of the active and reactive powers, because of the restriction of the capacity limits. Despite the changes of active and reactive power, the PCCs voltage is kept close to the nominal voltage value. On the other hand, observe in 7(b that the control tries to maximize the efficiency but the capacity restrictions limit the control optimization, i.e., if the n-th DG unit best efficiency is reached at 1.5 kw but the power capacity droops to 1 kw, it will not be able to reach its best efficiency. The results obtained in the case studies show that the proposed hierarchical control is a reliable online scheme to manage the operation of an islanded microgrid, achieving different operational goals. Additionally, the optimal power flow, which is based on the conventional formulation, has the advantage that any change in the microgrid topology can be

7 Modeling, analysis and control of microgrids in dynamic and steady-state 7/9 P (W Q (VAr P load (W P DG1 (a (b (c 2 1 P DG P P DG3 DG1 P DG2 9 P DG2 DG1 5 DG2 85 P in % of the DG Capacity -5 Efficiency (% 8 (d (e (f Q DG1 Q DG3 2 Q DG2 15 Q DG3 1.1 V PCC1 V PCC3 1 Q DG1 1. Q DG Q in % of the DG Capacity Voltage (p.u..8 (g (h (i 3 Q load (VAr -1 Figure 6. Case I: Hierarchical control scheme response under load profile. Capacity (VA 2 1 DG3 V PCC2 DG1 DG2 DG3 P (W Q (VAr (a (b (c P DG1 1 P DG3 P LOAD P DG2 8 DG1 DG2 DG3-5 6 Q LOAD Efficiency (% Voltage (p.u PQ load Capacity (VA (d (e (f 3 Q DG3 1.1 V PCC1 V PCC3 VPCC2 Q DG1 QDG DG1 DG2 DG3. Figure 7. Case II: Hierarchical control scheme response under load and capacity profiles.

8 Modeling, analysis and control of microgrids in dynamic and steady-state 8/9 updated in an easy and straightforward manner, besides, the operational goals can be also changed or improved depending on the operational needs of the islanded microgrid. 6. Shooting method for autonomous systems In general, the mathematical model of a controlled microgrid can be given by the following ordinary differential equation set, ẋ = f(t,x (18 where x is the state vector of n elements. A way to compute the steady-state solution of a controlled system represented by a set of ordinary differential equations such as (18, is to find a state vector x such that x = x T, where T is the fundamental period of the system, x is the state vector at t = t and x T =x(t + T, t ; x. This problem can be addressed through fast-time-domain methods [11, 12]; however, for droop controlled microgrids, this methods cannot be used because the fundamental period of the system is unknown. In order to overcome this problem, a shooting method for autonomous systems is addressed. 6.1 Autonomous systems [1] The formulation of the problem for autonomous systems, ẋ = f(x (19 can be represented as a two-point boundary-value, as shown below, x(t,ν ν = (2 where ν = x is an initial guess. In order to compute the values of ν and T, a correction is accomplished through the Newton-Raphson method. Let, δν = ν ν δt = T T (21 such that equation (2 becomes, x(t + δt,ν + δν (ν + δν = (22 Therefore, to compute the corrections, (22 is expanded in a Taylor series and only the linear terms are kept as follows, [ ] x ν (T,ν I δν + x T (T,ν δt = ν x(t,ν (23 where x/ ν is an n n matrix, I is the n n identity matrix, and x/ T is an n 1 vector. Observe that (23 constitutes a system of n equations but n + 1 unknowns, thus, an extra equation is needed. In this case, a phase condition equation is used to complete the system (23, that is, ( x Trans T (T,ν δν = (24 Using this condition equation, the following system of n + 1 equations is obtained: ( x ν (T,ν I x T (T,ν ] ( Trans [ δν x T (T,ν δt [ ] ν x(t =,ν (25 after determining the corrections, a convergence criterion is checked. If the criterion is not satisfied, the initial guess is updated as (T + δt,ν + δν and the procedure is repeated. The application of this method implemented in islanded microgrids is still in development, but it was proved in a small dynamic system. 7. Discussion and conclusions A new hierarchical control scheme with a primary droop control and a central optimal extended power flow for islanded microgrids was presented in this thesis advance. Two case studies were implemented in the laboratory and tested online in order to show the reliability of the proposed control. Additionally, the formulation of the shooting method for autonomous systems with unknown period was addressed. The results obtained reveal the advantages, in terms of reliability and efficiency, of the proposed control over the conventional droop control. Furthermore, it was shown that even with random load and capacity variations, the proposed control achieves, working online, the operational goals required. It is important to point out the disadvantages of this control which are, the need to know the topology and parameters of the microgrid, and overlook of the unbalances. Therefore, further improvements are needed, such as, to take into account the unbalances of the system in the power flow formulation and include a parameter estimator of the system. In this way, future research for the inclusion of the improvements might be addressed. 8. Activities schedule The activities schedule for the next period is shown in the Fig. 8, the activities are the following. 1. Review of the state of the art of microgrids.

9 Modeling, analysis and control of microgrids in dynamic and steady-state 9/9 2. Implementation with parallel programming techniques and cloud computing five Newton methods in the timedomain to compute the periodic steady-state solutions to perform the following studies: Computation of stability regions of a microgrid without harmonics. Computation and analysis of resonances in a microgrid with and without control. Stability analysis considering a frequency dependent model of the network equivalent and its interaction with the microgrid. Passive components design in the time-domain. 3. Academic stay. 4. Formulation and solution of the harmonic power flow problem. 5. Implementation and assessment of various control schemes proposed for microgrids using the techniques developed in the previous activities. ACTIVITY YEAR Figure 8. Activity schedule References [1] A. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, ser. Wiley Series in Nonlinear Science. Wiley, 28. [2] Intelligent microgrid laboratory. aalborg university. [Online]. Available: intelligent-microgrid-lab/. [3] J. Rocabert, A. Luna, F. Blaabjerg, and P. Rodríguez, Control of power converters in ac microgrids, IEEE Transactions on Power Electronics, vol. 27, no. 11, pp , Nov 212. [4] N. L. Díaz, E. A. Coelho, J. C. Vasquez, and J. M. Guerrero, Stability analysis for isolated ac microgrids based on pv-active generators, in 215 IEEE Energy Conversion Congress and Exposition (ECCE, Sept 215, pp [5] J. M. Guerrero, J. C. Vásquez, and R. Teodorescu, Hierarchical control of droop-controlled dc and ac microgrids: a general approach towards standardization, in 29 35th Annual Conference of IEEE Industrial Electronics, Nov 29, pp [6] W. Stevenson and J. Grainger, Power System Analysis, ser. McGraw-Hill series in electrical and computer engineering: Power and energy. McGraw-Hill Education (India Pvt Limited, 23. [7] H. Saadat, Power System Analysis, ser. McGraw-Hill series in electrical and computer engineering. McGraw- Hill, 22. [8] M. M. A. Abdelaziz, H. E. Farag, E. F. El-Saadany, and Y. A. R. I. Mohamed, A novel and generalized threephase power flow algorithm for islanded microgrids using a newton trust region method, IEEE Transactions on Power Systems, vol. 28, no. 1, pp , Feb 213. [9] C. Li, S. K. Chaudhary, J. C. Vasquez, and J. M. Guerrero, Power flow analysis algorithm for islanded lv microgrids including distributed generator units with droop control and virtual impedance loop, in 214 IEEE Applied Power Electronics Conference and Exposition - APEC 214, March 214, pp [1] F. H. Dupont, J. Zaragoza, C. Rech, and J. R. Pinheiro, A new method to improve the total efficiency of parallel converters, in 213 Brazilian Power Electronics Conference, Oct 213, pp [11] J. Aprille, T.J. and T. N. Trick, Steady-state analysis of nonlinear circuits with periodic inputs, Proceedings of the IEEE, vol. 6, no. 1, pp , Jan [12] A. Medina, J. Segundo, P. Ribeiro, W. Xu, K. Lian, G. Chang, V. Dinavahi, and N. Watson, Harmonic analysis in frequency and time domain, IEEE Transactions on Power Delivery, vol. 28, no. 3, pp , July 213.