Electronic – How to implement control of the bidirectional buck-boost dc/dc converter in software

The continuous mode boost and buck-boost converters exhibit control-to-output transfer functions \$G_{vd}(s)=\hat{v}(s)/\hat{d}(s)\$ containing two poles and one RHS (right half-plane) zero, called zero of nonminimum phase.

Your original transfer funtion is:

$$G_{vd}(s)=\frac{1}{V_{RAMP}}\times \frac{V_{IN}}{\left ( 1-D \right )^2}\times \frac{1/\underline{L}C\times\left ( 1-s\left ( \underline{L}/R \right ) \right )}{s^2+s\left ( 1/RC \right )+1/\underline{L}C}$$

Starting from a simpler function (without the RHP zero), named \$ G_{vd}'\$:

$$G_{vd}'=\frac{1}{V_{RAMP}}\times \frac{V_{IN}}{\left ( 1-D \right )^2}\times \frac{1/\underline{L}C}{s^2+s\left ( 1/RC \right )+1/\underline{L}C}$$

Or, placed as a standard second order t.f:

$$G_{vd}'=K_{DC}\times \frac{1/\underline{L}C}{s^2+s\left ( 1/RC \right )+1/\underline{L}C}$$

where the DC gain is \$K_{DC} =\frac{1}{V_{RAMP}}\times \frac{V_{IN}}{\left ( 1-D \right )^2}\$.

The equation can be rewritten as:

$$G_{vd}'=K_{DC}\times \frac{1}{\left ( \frac{s}{\omega_0} \right )^2 + \frac{s}{\omega_0Q}+1 }$$

With \$\omega_0=1/\sqrt{\underline{L} C}\$ and \$\omega_0Q=R/\underline{L}\$

Similarly, the original transfer function can be expressed as (RHP zero included):

$$G_{vd}=K_{DC}\times \frac{\left (1-\frac{s}{\omega_{RHP}} \right )}{\left ( \frac{s}{\omega_0} \right )^2 + \frac{s}{\omega_0Q}+1 }$$

The response will be:

$$\hat{v}(s)= \frac{K_{DC}}{\left ( \frac{s}{\omega_0} \right )^2 + \frac{s}{\omega_0Q}+1}\times\hat{d}(s) - \frac{K_{DC}/\omega_{RHP}}{\left ( \frac{s}{\omega_0} \right )^2 + \frac{s}{\omega_0Q}+1 }\times\hat{d}(s)\times s$$

The response of the original system is the sum of two components: The first is equivalent to the modified system response (without the zero) and the second is the derivative (scaled) of that one. For the case of a stable system with a step input in \$t = 0\$, this last component will have substantial influence at the beginning and then will vanishes when \$t\rightarrow\infty \$. Note the negative sign leads to a momentary opposite effect in output (nonminimum phase).

UPDATE:

The presence of zero RHP in the model is explained as follows: For the output voltage to increase, the duty cycle must be increased in such a way that the inductor will be disconnected from the load for a long time, causing the output voltage to drop (i.e. in the opposite direction to the one desired). The controller must be designed to meet the project requirements and avoid oscillations while maintaining duty cycle below an undesirable 100% - being limited by the PWM integrated circuit itself.

Ideally, during nighttime there are no loads and the electricity from the AC grid is cheaper than during daytime, so the aim is to store that cheap energy into the batteries during the night.

A battery will cost $300-$1000 per stored kWh. If it can do 3000 cycles, then one stored kWh costs $0.1-$0.3 in battery amortization costs. Thus your scheme is profitable only if the difference between day/night kWh prices from the grid is more than this, which is unlikely, or if the grid is unreliable or intermittent.

Now, back to modeling:

You are modeling your two DC-DC converters as voltage sources. Since their outputs are connected together by a low resistance (ie, wire) any difference in output voltage will result in large currents.

The correct way is to use a current-mode approach.

Both DC-DC converters are now current sources. They will adjust their duty cycle to keep average inductor current constant. In reality, inductor current is a sawtooth with ripple centered on its average, but the result is the same. There are many ways to implement this, for example check LED drivers.

So, you split the problem into several smaller problems. First, design a DC-DC converter which takes a "desired current" as input variable, and regulates its duty cycle to deliver this current independent of voltage variations. Let's call CM the current source on the left (mains-powered) and CB the one connected to batteries. CB's current can be positive or negative since it can both charge and discharge the batteries.

Now you have current sources, so you can parallel them (not so with voltage sources).

Then you implement a control loop, which inputs the load voltage at point V3 on your schematic, and decides the total current I that must be delivered by CM and CB together. This synthetizes an impedance.

Then an algorithm will need to decide how to split this current between CM and CB depending on grid kWh price, battery state of charge, user settings, etc.

FYI I did a similar thing for a bicycle dynamo light: the incoming power source is intermittent. It charges a capacitor. This is then used to power the LEDs. A microcontroller looks at the capacitor voltage, and controls a bidirectional buck-boost current source/sink which charges/discharges a LiIon battery from/to the capacitor. It also adjusts LED current depending on available power.

Differences are that in my case, high ripple on the capacitor is desirable, since it allows to store energy from the generator. In your case ripple is undesirable, so the algorithm would be different.

I chose the battery voltage to be always below the main capacitor rail voltage, so I did not need a 4-switch buckboost, just a standard halfbridge. Current regulation is done by hysteretic control, so frequency varies, but it is dead simple.

The key is to have a current regulation loop which is fast (say your DC-DC runs at 200-500 kHz, then the current loop will react in less than 10 µs), and a capacitor that is large enough (nothing huge though) that the outer control loop does not need to account for the phase shift of the inner current control loop.

None of this needs complicated state space analysis...