Improved Energy Balance Control for Boost Converters Without Estimating Circuit Energy Losses

The previously developed control methods based on the conservation of energy in circuits require the accurate estimation of energy losses, which is difficult to measure and calculate for boost converters. Consequently, there always exist steady-state errors in the output voltage if neglecting such circuit energy losses. To address this issue, an improved energy balance control (IEBC) method is proposed in this paper by integrating a simplified energy balance controller (SEBC) with a PI controller. The proposed IEBC can reduce the steady-state output voltage errors without requiring the estimation of circuit energy losses. Furthermore, the proposed IEBC can operate in both the continuous current mode (CCM) and the discontinuous current mode (DCM), thus accurate static and fast dynamic performances are achieved over the entire load operation range. Moreover, the stability of the IEBC is proved using the Lyapunov stability criterion. Compared with that of the SEBC, both simulations and experiments validate the feasibility and robustness of the proposed IEBC method.


I. INTRODUCTION
With the expanding of power converters applications, the requirement for the performance of power converters has become increasingly high, for instance, small size, light weight, high efficiency and so on. In this case, the conventional PID controller, although having the advantages of simple control principle and strong robustness, cannot fulfil the rising requirements of converters, e.g., fast dynamic response [1], [2]. Then an anti-windup PI controller has been proposed. By switching between the saturation and linear regions, improved dynamic response was implemented to the load variation. However, a switch condition first have to determine and there is no significant improvement on the input variations or disturbances [3], [4]. Recently, to get superior static and dynamic performances, various control methods were developed, such as hysteresis control, sliding-mode control, fuzzy control, etc [5]- [10]. As an attempt to obtain excellent performances, the law of energy conservation, which states that the total amount of energy in The associate editor coordinating the review of this manuscript and approving it for publication was Fuhui Zhou . a system is constant, has been introduced into the control field. It was firstly introduced to control the rigid body of active magnetic bearings, which provided a new view on the closed loop control [11]. Up to now, the application areas based on such a principle covered stability control of power systems, hamiltonian system, photovoltaic systems, active power filters, converters and so on [12]- [16].
As for converter applications, the conservation of energy was introduced to obtain the reference current and voltage of converters in a hybrid control scheme [17], [18], then improved dynamic performances were achieved. However, this scheme is only suitable for the critical discontinuous current mode (CDCM) and the discontinuous current mode (DCM). Under the continuous current mode (CCM), it is difficult to obtain a reference current, since not all the energy absorbed by an inductor can be released in full at the end of one switching cycle. A controller covering entire load operating ranges was presented in [19], [20] based on the law of energy conservation. Simulation results of a buck converter illustrate that it is not vulnerable to changes of converter operation modes (CCM, CDCM, DCM) and can offer fast dynamic performances. A switching control scheme (SCS) using the law of energy conservation was presented for controlling buck converters [21]. By considering the energy losses of a circuit when maintaining the energy conservation in the circuit, the SCS achieved accurate static and fast dynamic performances. Similarly, by considering the energy losses of the circuit, accurate static and fast dynamic performances were achieved based on the energy balance in boost converters [22]. However, the mathematical calculation of energy losses in a circuit is complex. Moreover, parasitic parameters of circuit components are usually unknown and difficult to be measured in actual circuits, which leads to the inaccurate estimation of such energy losses. Thus the energy losses in a circuit are usually neglected, as a result, leading to degraded performance, mainly the steady-state errors of the output voltage. To reflect this phenomenon, according to the previously developed control methods using the circuit energy conservation, a simplified energy balance control (SEBC) method, which neglect energy losses of circuits, is firstly derived in this paper. Then combined with a voltage PI controller, an IEBC method is proposed to tackle the degraded performance caused by neglecting circuit energy losses.
The main contents of the paper are as follows: Using the SEBC method, Section II analyses the steady-state errors to the output voltage caused by neglecting circuit energy losses; Section III designs and implements the proposed IEBC method; The stability of the IEBC boost converter is analyzed in Section IV; Section V discusses simulation and experimental results; Conclusion is given in Section VI.   [18], [19]. Defining T s as a switching cycle duration, at the beginning of the n th switching cycle, S 1 is turned on by a clock pulse with a fixed frequency, and the DC source injects energy into the circuit. As time goes on, the energy that the DC source injects into the circuit increases by integration and is compared with the sum of the consumed energy of load (u ref − u in )i o T s , the stored energy of the inductor (n−1)T s +T s (n−1)T s u i dt and the circuit energy losses, mainly due to the parasitic DC resistance R of the inductor

II. THE ANALYSIS OF THE SEBC
At the instant when the output of the integrator reaches the control reference, S 1 is turned off. S 1 remains off until the next clock pulse arrives. Thus the control methods based on the conservation of energy in the circuit keeps the output voltage to a desired value.
It should be noted that, the variables u in , i , i o of (1) are not the functions of time t but sampled and updated at the beginning of every sampling period T c , which is the time point kT c . And these variables are regarded as constants, since the duration of a sampling period is short. Due to its complicated measurement and computation, the circuit energy losses, mainly (n−1)T s +T s (n−1)T s i 2 R dt of (1) produced by the parasitic DC resistance R of the inductor, is usually neglected. As a result, (2) represents the control equation of SEBC.

III. THE DERIVATION OF CONTROL EQUATION AND IMPLEMENTATION OF THE PROPOSED IEBC A. THE CONTROL EQUATION DERIVATION OF THE IEBC
It is generally known that a PI controller attempts to minimize the error over time by adjusting the control variable u(t) so as to force a measured process variable y(t) to follow a desired value r(t). It means that the merit of the PI controller is to eliminate the errors between a control objective and a controlled object, and as a result, the PI controller relies only on the response of the measured process variable, not on the exact mathematical model. In consideration of such features of PI controller, a voltage PI controller is introduced to modify the SEBC method for eliminating the steady-state errors of the output voltage u steady due to neglecting the energy losses of the circuit. To implement the PI voltage controller, the following changes are made to equation (2) of SEBC.
Firstly, the converter is emulated by an equivalent resistance R e shown in Fig. 2, thus i is expressed as (n−1)T s +t on (n) Based on (3), the following equation can be deduced, Multiplying both sides of (2) by R e R s , (5) is derived from (2). Based on (3) and (4) R s , which is obtained from the PI voltage controller [23], as shown in FIGURE 2. The proportional term K p of the PI voltage controller is designed to be high enough, so that the dynamic response is not slowed down. Thus, the control equation of the proposed IEBC method is derived as (6). With the high robustness of the PI controller, the steady-state errors of the output voltage due to the neglection of circuit energy losses can be eliminated.

B. THE IMPLEMENTION OF THE IEBC
As shown in FIGURE 2, the implementation of the IEBC method is described as below. The control reference (u m − calculated at the beginning of every sampling period and kept as the same during the entire sampling period. After getting the control reference, the proposed IEBC method is implemented by integration and comparison. The implementation of a CCM boost converter using the proposed IEBC method is described as below: the integral begins in the time instant when S 1 is turned on by a clock pulse with a fixed frequency. Over time, the integral W int keeps increasing from its initial value as follows: and W int is constanly compared with the control reference. At the moment when W int (t) reaches to the control reference, a reset pulse is generated by the comparator in FIGURE 2 to reset the RS flip-flop as Q = 0, which turns off S 1 . In the meantime, the integral is reset. S 1 keeps as the off-state until the next clock pulse arrives, then the (n + 1) th switching cycle starts. Since a DCM boost converter has the similar implementation procedures with that of CCM, it is not presented in detail here.

IV. STABILITY ANALYSIS OF THE IEBC BOOST CONVERTER
Under the steady-state conditions, the inductor absorbs and releases equal energy, which means W (t) = 0. Thus, neglecting W , (6) is rewritten as follows: Then the average model of (8) is obtained as: From (9), d can be derived as: And such a boost converter can be described as: where s = 1 represents the on-state of switch S 1 , s = 0 represents the off-state of switch S 1 . Substituting s in (11) with the value of d derived above, the state-space averaged model of the energy balance controlled (n−1)T s +t on (n) boost converter is obtained as follows.
Let the values of di dt and du dt in (12) equal to 0, then the equilibrium point is obtained as follows: The Jacobian matrix evaluated at this equilibrium point is derived as follows Substituting (13) into the jacobian matrix gets By solving the characteristic equation det[λI − J ] = 0, the following characteristic quasi-polynomial equation is obtained: By performing a second-order Pad approximation, (16) can be written as follows: and a 0 > 0 are obtained, which satisfies the Routh−Hurwitz criterion. Hence, the IEBC is stable. 2 At t = 0.1 s, the load is stepped to 27 abruptly. The converter changes its operation mode from CCM to CDCM. u o jumps to 30.8 V due to such a sudden load step but soon settles to the pre-set value 30 V. The system does not become unstable.

Simulation and experimental results
3 At t = 0.15 s, the converter enters the perfect DCM when the load is further changed to 47 . As shown in FIGURES 3, the converter still operates stably and u o keeps at the pre-set value.
To demonstrate the superior performances of the IEBC method, compared with the SEBC method and a  current-mode PID controller, the responses to load and input voltage disturbances of the converter are discussed as follows. By using the SISO tool box in MATLAB, the PID controller is established with the 382 Hz cross-over frequency and the 54.5 degrees phase margin.
Step changes in loads, which is 10 → 27 → 47 , are applied. At t = 0.22 s, the load is changed back to 10 . Then step changes in the input voltage as 15 V → 18 V → 15 V are applied. FIGURES 4 and 5 show the responses of the converter to such load and input voltage step changes. The results demonstrate that, compared with the PID, the voltage peak overshoot u o is reduced from 2.2 V (using the PID) to 0.8 V (using the IEBC) under the case that the load steps from 10 to 27 . And the settling time t settling is shortened to 4 ms (using the IEBC) from 13 ms (using the PID). These comparison results of the responses are summarized in TABLE 1, which shows that, compared with that of the PID, u o of the IEBC is significantly reduced and t settling of the IEBC are significantly shortened.
Meanwhile, from FIGURES 4 and 5, it reveal that, under various operation conditions, u o using the SEBC cannot settle to the pre-set value but has certain steady-state errors u steady after reaching steady states. For example, under operation conditions of R = 10 , u o has the steady-state errors as 1.6 V to the pre-set value 30 V. TABLE 2 summarizes the steady-state value of u o and the steady-state errors u steady using the SEBC under various operation conditions. In contrast, the results in FIGURES 4 and 5 demonstrate that, there are no steady-state errors in u o using the IEBC because of the function of the PI voltage controller.

B. EXPERIMENTAL COMPARISON RESULTS
Based on a dSPACE DS1104, an experimental boost converter prototype is constructed. The current and voltage are VOLUME 8, 2020    show that u o using the IEBC is significantly reduced and t settling using the IEBC is significantly shorten compared with that of the PID controller. And from FIGURES 8 and 9, it is observed that although u o using the SEBC is reduced and

VI. CONCLUSION
In this research, the IEBC method has been proposed and implemented effectively for controlling boost converters. On the basis of the control principle of the previously developed control methods based on the conservation of energy in circuits, in the proposed control method, a PI voltage controller is added, as a result, the output voltage steady-state errors due to the neglection of circuit energy losses is removed. Furthermore, the stability of the converter controlled by the IEBC has been proved using Routh-Hurwitz criterion.
Simulation and experimental results demonstrate the proposed IEBC method can settle the output voltage to a pre-set value though circuit energy losses are not considered. Meanwhile, the comparison to the PID controller reveals that the IEBC keeps the advantage of the previously developed control methods based on the conservation of energy in circuits, which is superior dynamic performance, in terms of smaller voltage shoots and shorter settling times under the step changes of input voltage and load current. These results demonstrate that, using the IEBC, accurate static and fast dynamic performances are achieved even though circuit energy losses are neglected due to the complexity in their measurement and calculation.