DAB State-Space Average Model

The first step for designing controller for a system or analyzing its behavior is deriving the model of the system. Having the model of the system, many different design techniques can be utilized. For power converters there are mainly two different methods for deriving models.

One of them is circuit averaging [77] and the other is state space averaging. Basically, because of the utilization of switches in switching power converters, they are non-continuous non-linear systems since the topology of the circuit is changing during each cycle. Circuit averaging technique tries to represent the elements of the circuit with their average values such as average current or average voltage. State-space averaging mainly allows a switched system to be

(5.38)

Fig. 5.20 Power flow of DAB in p.u. versus phase shift in rad

116

approximated as a continuous non-linear system. Averaged models are not necessarily linear, so they can be linearized over an operating point. The result will be small-signal linearized model.

The main advantage of linear models is mainly their compatibility with control design methods based on linear systems theory and Laplace transform. For DAB modeling we use state-space averaging. In practice some criteria should be met so the state-space averaged model holds valid.

First, the natural frequencies of the circuit and all modulation or signal frequencies of interest must be sufficiently low compared to the switching frequency. This will provide validity of continuity approximation. Second, the criterion for linearity approximation is that the time variations or ac perturbations around the DC operating point should be sufficiently small.

Here we don’t want to mention the theory behind state-space averaging, but just as a reminder, assuming a power converter with just two switches, i.e. a simple buck converter including a controllable switch and a diode, if we call the configuration when the switch is on #1 and the configuration when the diode is on #2 and write the state space equations, we will have:

(5.39)

(5.40)

where is the state variable vector consisting of inductor current and capacitor voltage. It can be proved that this non-continuous system can be represented by a new state variable vector Y driven by new input vector V such that:

̅ ̅ (5.41)

̅ (5.42)

117

̅ (5.43)

D1 is the duty ratio of the controllable switch and D2 is the duty ratio of the diode. As it can be seen, instead of two sets of equations (5.39) and (5.40) for two different configurations, the system can be defined with just one state-space equation (5.41). In fact (5.41), (5.42) and (5.43) represent the average behavior of the same system. Average behavior means no switching is reflected in the model, so states do not exhibit any ripple. In other words, the new states y(t) in the new state-space equation (5.41) track the average behavior of x(t) in (5.39) and (5.40), provided that v(t) is defined to track the average behavior of u(t) and the switching frequency is high enough. In (5.41) without loss of generality, the name of state vector Y can be changed to X and input vector of V can be changed to U as was used in the original system, since X and U are more common. So averaged state-space model of the system can be rewritten as:

̅ ̅ (5.44)

where ̅ and ̅ are defined in (5.42) and (5.43) respectively. When the buck converter is operating in Continuous Conduction Mode (CCM) only two possible configurations are possible, so if the duty ratio of the controllable switch is shown with the parameter “d” we will have:

(5.45)

(5.46)

̅ ) (5.47)

̅ ) (5.48)

118

The example above involved only two different possible configurations, however, more configurations are possible. For example, in the case of a buck converter if the converter operates in Discontinuous Conduction Mode (DCM) there will be three different configurations and consequently three sets of matrices A1, B1 and A2,B2 and A3, B3 instead of two.

There are lots of state-space averaged models available in the literature for conventional DC-DC converters such as buck, boost, buck-boost and Cuk converter [78],[79],[80],[81].

Analysis of these converters is simple since they employ only two switches, one controllable switch and one diode. Maximum number of possible configurations in these converters is three while operating under DCM conditions. For converters with more switches, the possible number of configurations increases dramatically. For example for the case of DAB as shown in Fig. 5.18 the possible number of configurations increases to 6 in CCM while it is only 2 for a buck converter operating in CCM. Cuk in [81] derives a state-space averaged model which is extended to converters with multi-structural topological modes; however, multi-structural state-space averaged model presented in [81] is applicable only when the inductor current is DC with an AC component [82]. In the case of DAB as was shown in Fig. 5.19 the inductor current has a zero average due to the symmetry of the waveforms. If the state-space equations for all the configurations are written and time-averaged over one complete cycle period, lots of the components of the coefficient matrices will cancel out each other because of the symmetry relative to the time axis and the averaged state-space model will provide no useful information.

So in the case of DC-DC converters with zero inductor current average like DAB, state-space averaging cannot be used directly. Instead, an extension of state-space averaging which is called extended state-space averaging technique should be used. This technique is mainly based on the fact that the average power delivered during the positive cycle is equal to the averaged power

119

delivered during the negative cycle. So the state-space model can be time-averaged over only half of a cycle (positive or negative).

According to Fig. 5.18 and Fig. 5.19 the equivalent circuit and state-space equations of each time interval can be obtained:

120

To average the state-space model, (5.51), (5.54) and (5.57) should be multiplied by their corresponding duty ratios which are d1, d2 and d3 respectively and then added together. The duty ratios of each interval can be calculated as follows:

Fig. 5.23 DA-, DB1, DB4 conducting

121

The same as two previous time intervals, d3 can also be calculated, however, there is no need to that, since d3 is dependent on d1 and d2 and can be obtained as:

(5.63)

According to Fig. 5.18 and Fig. 5.19 during the time interval of ωt0 ≤ ωt < ωt2 the switch TA+ conducts, so which we call it d for simplicity can be calculated as:

(5.64)

There is an important point here to note. As shown in Fig. 5.18(b) and Fig. 5.19 during the time interval 0 ≤ ωt < ωt0 the gating signal is applied to switch TA+ however, this switch

122

doesn’t conduct until the current through DA+ reaches zero. This situation is different from conventional DC-DC converters like buck, boost, buck-boost or Cuk. In these converters the current commutates from the diode to the controllable switch almost instantly after applying the gating signal. In these converters the gating signal can be used to calculate the duty ratios,

123

( )

⁄ ) (5.69)

As it is calculated in (5.67) and (5.69), d1 and d2 and hence d=d1+d2 are non-linear functions of input (u) and state of the system (x2). Non-linear model (5.65) can be used directly using non-linear techniques or can be approximated as a linear system around its DC operating point.