CHAPTER 10

Averaging for Modeling and Simulation

The trend of power converters leads to the utilization of high-frequency switching and shows nonlinear characteristics due to the on/off operation. When a power supply system, e.g., microgrid, is formed by significant numbers of switching-mode converters, simulation can be challenging, especially for long-term study of daily power generation from PV and wind, slow variation of the state of charge in the energy storage system (ESS), etc. The high switching frequency demands very high sampling rate in the simulation setting to maintain high switching resolution, which eventually results in slow simulation speed. Both academy and industry demand an efficient and generalized simulation model to evaluate long-term system operation of distributed renewable power generation and energy storage. Modeling and simulation are increasingly important for dynamic analysis and controller synthesis. Modeling is a general term that constructs mathematical models for the simulation study, dynamic analysis, or both. The averaging technique has been widely applied for modeling power converters since the low-frequency component critically dominates system dynamics. Thus, the averaging technique can neglect high-frequency switching effects but reveal key system dynamics for simulation and analysis.

This chapter focuses on the averaging technique for numerical simulation and dynamic analysis. The method should support fast simulation without losing any critical dynamics in transient states. The modeling approach is based on the mathematical representation of system dynamics rather than any specific simulation tool. The approach is generalized for the whole operating status, including light load conditions and the DCM of power converters.

10.1  Switching Dynamics

The design and operation of high-efficiency power converters are mainly based on the fast on/off switching technology to manage and regulate power flow. High switching frequency, up to megahertz, becomes desirable, since the size and capacity of passive components, e.g., inductors and capacitors, can be significantly reduced to improve power density, reduce costs, and enhance system dynamics. For numerical simulation, the high switching frequency generally leads to slow numerical simulation to capture the fast switching dynamics. The sampling frequency for simulating switching-mode converters in discrete time is usually sized to be at least 100 times the switching frequency for accurate representation. For example, the sampling time should be 100 ns, assigned for simulation if a converter switching frequency is 100 kHz, which results in the 1% resolution of the switching duty ratio. When the sampling time is reduced to 10 ns, the resolution of the PWM is improved to 0.1%, which is more accurate and representative, but takes a longer time to fulfill the simulation. The majority focuses on special hardware and software to meet the demand for intensive simulation. However, the solution is very costly but is not general enough for wide implementation. The widely used simulation platforms have been listed in Sec. 1.4.2.

The dynamics of high-frequency switching can be of interest for short-term analysis to reveal the dynamics in each switching cycle, which has been widely used in the steady-state analysis of inductor current and capacitor voltage for DC/DC conversion. However, the periodic oscillation effect is mostly unimportant for long-term analysis and simulation. The averaging technique for power electronics has been developed since the 1970s to analyze system dynamics. The averaging approach is based on the fact that the switching frequency is much higher than the critical dynamics that are commonly represented by LCR circuits.

10.2  Continuous Conduction Mode

In CCM, the dynamics of both inductor current and capacitor voltage are as considered in the discussions that follow regarding the dynamic modeling and analysis.

10.2.1  Buck Converter

A standard buck converter can be divided into two regions, namely, switching and linear, as illustrated in Fig. 10.1. The linear section is formed by the passive components of L, C_O, and R, which can be modeled by the circuit theory. The switching mechanism is formed by the semiconductors, switched for the power modulation and voltage conversion. The interlink between the linear region and nonlinear section is the voltage at the switching node, v_sw, which is pulsating and discontinuous. The averaging computation focuses on the pulsating voltage, v_sw, in which the averaged value is derived to be when non-ideal factors are neglected. The voltage conversion is based on the proportion of the on-state duty ratio, d_on, in CCM. After the averaging, the equivalent circuit can be plotted as shown in Fig. 10.2. The linear circuit analysis can be applied and expressed by

FIGURE 10.1 Buck converter for modeling by averaging based in CCM.

FIGURE 10.2 Equivalent circuit of buck converter for modeling by averaging.

where and represent the averaged values of the inductor current and output voltage, respectively. The model can be transformed to the state-space format in (10.3), where the two states are represented by and , while the input is the on-state duty ratio of the active switch, d_on. The state-space model represents a linear physical system when V_in is constant for the modeling. The model becomes available for the state-space-based analysis and controller design. The state feedback is a useful control technique to regulate the state variables, and , and determine the control action, d_on.

Besides the state-space representation, the system dynamics can be represented by the single-input-single-output (SISO) system using differential equations. When the averaged value of the output voltage is the controlling target, a differential equation can be derived from the two first-order equations in (10.1) and (10.2) into the second-order format of

When the averaged value of i_L is of interest, the differential equation can be derived to represent the SISO dynamic relation between the averaged current, ¯i_L, and the on-state duty ratio, d_on, as

The differential equations in (10.4) and (10.5) can be transferred to the frequency-domain representation using Laplace transform. The transfer function in s-domain is a typical way to show the dynamic correspondence of a SISO system. For the buck converter, the transfer functions become as in (10.6) and (10.7) with the consideration of the controlled outputs, and , respectively. The transfer functions in (10.6) and (10.7) are commonly represented by the generalized formats, as expressed in (10.8) and (10.9) for the following analysis.

where

The DC gains in the generalized transfer functions are symbolized as K_0v and K_0i that represent the ratios between the outputs and inputs in the steady state. The denominator of G_i(s) is the same as that of G_v(s), which indicates two important parameters: the system damping ratio, ξ, and undamped natural frequency, ω_n. The transfer functions show the difference in the numerators by comparing (10.8) with (10.9).

10.2.2  Dynamic Analysis of Second-Order Systems

The dynamic model of buck converters in the CCM has been derived by the averaging technique as a standard second-order transfer function, as in (10.8). The mathematical model indicates two poles without zero. The poles can be determined and demonstrated on a complex s-plane or pole-zero map. It is well known by the control theory that all poles must be in the left-half plane (LHP) to ensure a system stability. It should generally not be concerned since the dynamic model of a buck converter always shows the stability. Thus, the study should focus on the dynamic features regarding the values of the undamped natural frequency, ω_n, and damping ratio, ξ. The important measure includes the settling time and percentage of overshoot in response to a step change. The undamped oscillation scenario in buck converters commonly refers to the non-load condition where R = ∞. The dynamic analysis indicates that non-load condition leads to the zero value of the damping factor, ξ = 0, according to (10.10). The analysis is based on the assumption that the Equivalent Series Resistance (ESR) in the circuit is zero. The switching operation triggers endless oscillation of the LC circuit and follows the resonant frequency of ω_n. When 0 < R < ∞, the nonzero value of ξ indicates a damping effect in the circuit, which makes the self-oscillation disappear in the steady state. The value of ω_n becomes the representative of the system dynamic speed. A fast response can be expected for the high value of ω_n. According to (10.10), the high speed results from the low values of L × C_O.

The value of ξ represents how much damping or oscillation the system presents. According to (10.10), the damping factor for the buck converter depends on the parameters of the passive components, L, C_O, and R. When the CCM is operated, a low value of R indicates light damping. The ratio of L and C_O is also critical and related to the damping ratio, which should be considered in the design stage. Thus, a converter circuit design should be comprehensive to consider not only the steady-state ripples but also the dynamic performance regarding the response speed and damping factor. However, an over-damped system design should be avoided, e.g., ξ > 2, since it shows a sluggish dynamic response. Figure 10.3 demonstrates the effect of ω_n and ξ on the step response regarding the response speed and damping performance, where V_in = 1.

FIGURE 10.3 Step response of the second-order system showing effect of (a) ξ; (b) ω_n.

Significant oscillation and overshoot are visible in Fig. 10.3a when ξ < 0.3, which prolongs the settling time. The response speed is fast when ω_n is high in the system dynamics, as demonstrated in Fig. 10.3b. The settling time of a step response regarding the system dynamics as in (10.8) can be estimated to be for the standard second-order transfer function. Table 10.1 summarizes the value of the percentage of overshoot (P.O.) in step response of the transfer function in (10.8), which is influenced by the value of ξ. When ξ = 0.7, the step response shows the value of P.O. less than 5% and fast response speed, as shown in Fig. 10.3a. The information can be treated as a reference for dynamic analysis and control system design.

TABLE 10.1 Percentage of Overshoot in Step Response of Standard Second-Order Systems

When the inductor current in a buck converter is the controlled objective, the dynamic analysis follows the transfer function in (10.9), where R = 1. It shows the same denominator as the transfer function in (10.8) with the same values of ω_n and ξ. The difference lies in that the numerator in (10.9) presents a dynamic term of β_is + 1, which indicates a zero with the value of . The negative value of the zero indicates a minimal-phase (MP) system. Figure 10.4 demonstrates the effect of minimal-phase zero during the step response. When the absolute value of is high, its effect is insignificant to the system response. If β_i is high in value, such as β_i = 10⁻³, the step response shows fast speed in the initial stage; however, it causes higher overshoot in comparison with the case of β_i = 0 and β_i = 10⁻⁴. Following (10.10), the β_i value is affected by the design of C_O and changed with the load condition, R. For buck converters, a high overshoot is expected in the step response of light load conditions since R is high in value.

FIGURE 10.4 Step response of second-order system with one minimal-phase zero,

10.2.3  Boost Converter

The boost converter shows the separation of the inductor, L, and the output capacitor, C_O, as illustrated in Fig. 3.22a. Different from the buck converter, a linear region is no longer clearly distinguishable. When the active switch is on-state, the system dynamics has been derived and expressed in (3.32) and (3.33). When it is off-state, the differential equation is derived as (3.34) and (3.35) to illustrate the switching dynamics. The system dynamics should be identified by the state variables of i_L and v_o, which link to the energy storage components, L and C_O. The control variable is the on-state duty cycle of d_on or the off-state duty ratio of d_off for the PWM to switch the active switch, Q.

In CCM, the state equations of (3.32) and (3.34) can be averaged within one switching cycle, T_SW, and expressed as (10.11).