Built-in Dynamic Optimization using Annotations¶

This part of the OM manual is a contribution by Massimo Ceraolo and Vitalij Ruge.

Foreword¶

Dynamic optimization using Annotations is the most user-friendly way to perform Dynamic Optimization(DO) in OpenModelica, since it allows the OMEdit graphical user interface to be used.

It is also more powerful that the Built-in Dynamic Optimization using Optimica language extensions, since it allows final constraints.

Formulation limitations and algorithm¶

We can formulate the optimization problem as follows:

$\begin{eqnarray} \min_{u(t)}{\ \left\lbrack M\left( \mathbf{x}\left( t_{f} \right), \mathbf{u}\left( t_{f} \right),t_{f} \right) + \int_{t_{0}}^{t_{f}}{L\left( \mathbf{x}(t), \mathbf{u}(t),t \right)\text{dt}} \right\rbrack} && \begin{array}[t]{l} \mbox{optimization purpose} \\ (M \mbox{ \textit{is the Mayer Term}} \\ L \mbox{ \textit{is the Lagrange Term}}) \end{array} \\ \mathbf{0} = \mathbf{f}\left( \mathbf{x}(t),\dot{\mathbf{x}}(t),\mathbf{u}(t),t \right) && \begin{array}[t]{l} \mbox{dynamical description} \\ \mbox{of the system} \end{array} \\ \mathbf{0} \leq \mathbf{g}\left( \mathbf{x}(t),\mathbf{u}(t),t \right) && \begin{array}[t]{l} \mbox{path constraints} \\ \mbox{e.g. physical limits of components} \end{array} \\ \mathbf{0} = \mathbf{r}_{0}\left( \mathbf{x}\left(t_{0} \right),t_{0} \right) && \mbox{Initial constraint} \\ \mathbf{0} = \mathbf{r}_{f}(\mathbf{x}\left( t_{f} \right)\mathbf{u}\left( t_{f} \right),t_{f}) && \mbox{Final constraint} \end{eqnarray}$

Where u(t) is the vector of input variables, on which DO works to try to get the desired minimum, and x(t) is the vector of state variables.

The equations above can be implemented in OpenModelica using the full power of Modelica language, and therefore there is a good freedom and flexibility, under the obvious constraint that the system must be regular enough for the convergence to take place.

However, there are limitations in the possibility operational limits can be set.

The formulation of operational limits in (3) is general, since it allows to use non-boxed constrains. Consider for instance a battery. The energy the battery can deliver is a function of the power we use to charge or discharge it. Therefore, the actual limit should be described as:

$E_{\min}(P) \leq E_{\text{bat}} \leq E_{\max}(P)$

Moreover, (3) is time-variant (the third argument of function g).

The OpenModelica optimization through annotations accepts as path constraints only time-invariant box constraints, so eq (3) is expressed in the simpler form:

$\mathbf{x}_{\min} \leq \mathbf{x}(t) \leq \mathbf{x}_{\max}$

$\mathbf{u}_{\min} \leq \mathbf{u}(t) \leq \mathbf{u}_{\max}$

$g_{\min} \leq g(\mathbf{x}(t),\mathbf{u}(t)) \leq g_{\max}$

OpenModelica uses the Radau IIA discretization scheme of order 1 or 5 depending on user input.

Using the first order the Radau IIA is equivalent to implicit Euler and to compute states, output and cost function only the values at the end of each interval are evaluated. E.g., if we have StopTime=1 and Interval=0.25, only values for t=0.25, t=0.5, t=0.75, t=1 will be considered. Therefore, the resulting value of the control variable at t=0 may be even totally wrong, since it has no influence on the result.

Syntax¶

OpenModelica provides specific annotations to allow the optimization problem to be described, as an alternative to the use of Optimica language. They are listed and commented below.

Request for an optimization. We must use two simulation flags and a command line option. These can conveniently be inserted into the code, to avoid selecting them manually all the time, as follows:
```
__OpenModelica_simulationFlags(s = "optimization", optimizerNP="1"),
__OpenModelica_commandLineOptions = "+g=Optimica",
```
OptimizerNP gives the number of colloction points and can be only 1 or 3. As already said the RadauIIA order is 2*OptimizerNP-1.

The user is recommended to use as a first attempt optimizerNP=1. In case of questionable results, they can try optimizerNP=3.

For the simulation, it is known that the stability ranges are different. At the same time, we lose stability with higher order (see 1).

Note that Optimica command-line option is added even if we do not use Optimica specific language constructs; this it is required for the use of optimization-related annotations.
Select optimization parameters. We must specify StartTime, StopTime, Interval, and Tolerance. The first two have the same meaning as in time simulations. Interval not only defines the output interval (as in time simulations), but has a more specific meaning: it defines the interval between two successive collocation points. I.e., optimization is done splitting the whole timespan in sparts having interval as length. Therefore this value may have a huge effects on the simulation output. For typical runs, number of intervals values from 100 to 1000-2000 could be adequate. These values are obviously set through the experiment annotation, e.g.:
```
experiment(StartTime = 0, StopTime = 20, Tolerance = 1e-07, Interval = 0.02),
```
the default tolerance is 1e-6. The user is warned that enlarging this value may affect the output quality adversely by large amounts (an example will be provided later). Going up to 1e-8 may be advisable in some cases.
Indicate the minimisation goal. We can indicate whether we must just minimise a quantity, or the integral of a quantity (see (1)., as follows:
```
Real totalCost = xxx annotation(isMayer = true); //minimize totalCost(tf)
Real specificCost = xxx annotation(isLagrange = true); //minimize integral of specificCost (tf)
```
Several isMayer and isLagrange goals can be set. The actual goal will be the sum of all goals (isMayer goals as they are, isLagrange goals first integrated between t₀ and t_f).

Obviously, it is possible in Modelica to use just isMayer=true also for Lagrange terms, integrating the Laplace integrand inside the model, but the internal numeric treatment will be different.
Describe the system. This is done in the usual Modelica way. Here we can exploit the huge power of modelica language and tools to automatically convert a system described in physical (and possibly graphical) terms into DAE equations
Define path constraints. As we said previously, they must be boxed and time-invariant. They are expressed using annotations as in the following example (taken from the full example described in the next section):
```
Real energyConstr(min = 0, max = energyMax) = storage.energy annotation(isConstraint = true); //timespan constraint on storage energy
```
Here, we see that the constraints are described through min and max values.
Define initial constraints. These are set using the existing modelica syntax to indicate initial values

Define final constraints. These are set using a specific annotation, as in the following example (taken from the full example described in the next section):

annotation(isFinalConstraint = true);

Some special care must be taken when dealing with final constraints. We must be sure that OM front-end does not do alias elimination of the constrained variable, since in that case it could pass on bounds from final constraints to the merged variable, and these final constraints would become path constraints. To avoid this potentially harmful alias elimination we must add to the final constrant an auxiliary parameter, as follows.

parameter Real p = 1 "Auxiliary parameter for energy final constraint";
Real energyConstr(min = 0, max = energyMax) = storage.energy annotation(isConstraint = true); //timespan constraint on storage energy
Real energyFinConstr(min = energyIni, max = energyIni) = p * storage.energy annotation(isFinalConstraint = true); //final time constraint on storage energy

The auxiliary parameter can also be used for scaling the constrained variable, so that it is roughly around one, so easing convergence. This could be done for instance as follows:

parameter Real p = 1e-3 "Auxiliary parameter for energy final constraint";
Real energyConstr(min = 0, max = energyMax) = storage.energy annotation(isConstraint = true); //timespan constraint on storage energy
Real energyFinConstr(min = p*energyIni, max = p*energyIni) = p * storage.energy annotation(isFinalConstraint = true); //final time constraint on storage energy

Preparing the system¶

To allow DO to operate in good conditions it is very important that the system has continuous derivatives.

Here we just give two examples:

In case of a combiTimeTable is used to describe non-linear algebraic functions, it is highly recommended to use Continuous derivative for the smoothness parameter
If we need to use the absolute value of a variable, we have derivative discontinuity around zero. This can be avoided, with negligible loss of precision, substituting $abs(x)$ with $sqrt(x^2+\varepsilon)$ , where eps is very low in comparison with the values $x$ usually assumes during the simulation.

Example 1: minimum time to destination¶

This example refers to a car, which is requested to cover the max possible distance using power from an engine which has a torque limitation and a power limitation.

The torque limitation is transformed in a maximum force that the wheels can transfer to the road to pus the car.

This is a very easy dynamic optimization problem, whose solution is the so-called bang-bang control: accelerate at the maximum possible degree, then, when half of the road is reached, decelerate with the maximum possible degree.

The code is very simple and it is as follows:

model BangBang2021 "Model to verify that optimization gives bang-bang optimal control"

   parameter Real m = 1;
   parameter Real p = 1 "needed for final constraints";

   Real a;
   Real v(start = 0);
   Real pos(start = 0);
   Real pow(min = -30, max = 30) = f * v annotation(isConstraint = true);

   input Real f(min = -10, max = 10);

   Real costPos(nominal = 1) = -pos "minimize -pos(tf)" annotation(isMayer=true);

   Real conSpeed(min = 0, max = 0) = p * v " 0<= p*v(tf) <=0" annotation(isFinalConstraint = true);

equation

   der(pos) = v;
   der(v) = a;
   f = m * a;

annotation(experiment(StartTime = 0, StopTime = 1, Tolerance = 1e-07, Interval = 0.01),
__OpenModelica_simulationFlags(s="optimization", optimizerNP="1"),
__OpenModelica_commandLineOptions="+g=Optimica");

end BangBang2021;

The constraint on power is especially worth considering. Above, we stated that path constraints are $0 \leq g(\mathbf{x}(t),\mathbf{u}(t))$ here we have box limits on pow, which is indeed a function of state variables as follows:

$- 30 \leq fv \leq 30\ = > \ - 30 \leq mav \leq 30$

And a and v are two state variables. So, these are two box constraints of the type $0 \leq \ g\left( \mathbf{x}(t),\mathbf{u}(t) \right)\$ as follows:

$mav = m\text{\ x}_{1}\text{\ x}_{2} \geq - 30\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mav = m\text{\ x}_{1}\text{\ x}_{2} \leq \ 30$ .

The results can be expressed in terms of force and power applied to the vehicle. They are as follows:

and they are as expected.

In this model we don’t use the Modelica capability to automatically determine the system equations from the graphical description of a system. In other words, the above general formulation $0 = f\left( \mathbf{x}(t),\dot{\mathbf{x}}(t),\mathbf{u}(t),t \right)$ is explicitly written as follows:

der(pos) = v;
der(v) = a;

In the following example, we will use this capability extensively.

Example 2: hybrid vehicle minimum consumption¶

This example refers to the electricity generation of a hybrid vehicle. These vehicles can choose at any time which amount of the propulsion power must be taken from a battery and which from the Internal combustion engine.

In this example the engine can be switched ON and off without penalty, so the DO can choose both when the ICE must be ON /OFF; and the power it must deliver when it is ON.

Objective of the control is to minimise the fuel consumption. This must be done keeping the energy inside the storage at the final time, equal to the one at t=0 (otherwise it is easy to have zero consumption: just keep the Internal Combustion Engine OFF all the time!)

For simplicity’s sake, the propulsion power, in this simple example is taken as being a sine wave plus an offset (needed to make the average positive). When the power is positive the wheels transfer power to the road, when negative they recover it (storing it into the battery).

To find the optimum, a fuel consumption curve is added, as follows:

The minimum is 210 g/kWh, and occurs when the ICE power is at the 76.8 % of the nominal power

The system diagram is as follows

The DO algorithm is required to determine the battery power outBatPower (positive when the battery delivers power) so that to minimise the fuel consumption toGrams. Block toGperkWh is normalised, so that it can be used for different engines, adapting the horizontal scale through gain, and the vertical’s through gain1.

This diagram defines the system, whose equations will be automatically determined by OpenModelica, through model flattening. However, some code must be manually written to perform DO.

Here the code is as follows:

//*** DO-related rows

input Real outBatPow;

//
Real totalCost = toGrams.y "minimize totalCost(tf)" annotation( isMayer=true);
//

Real energyConstr(min = 0, max = energyMax) = storage.energy annotation(isConstraint = true); //timespan constraint on storage energy

parameter Real fecp = 1 "final energy constraint parameter";

Real energyFinConstr(min = energyIni, max = energyIni) = fecp * storage.energy annotation(isFinalConstraint = true);//final time constraint on storage energy

Real icePowerConstr(min = 0) = itoIcePow.y annotation(isConstraint=true); //timespan constraint on Ice power

//*** End of DO-related rows

A few comments:

The choice isMayer=true on the objective function totalCost requires its final value to be actually minimised, not its integral (as would have been in case of the keyword isLaplace=true)
We have two different constraints on the storage energy: the storage energy must all the time be between 0 and the maximum allowed, and at the end of the simulation must be brought back to its initial value.
ICE can only deliver power, not absorb; so, we expect all the (regenerative) braking power and energy to be sent into the storage

Ideal storage¶

Here we consider the storage to be ideal: the flow of power in and out causes no losses to occur

In this case the solution of our optimization problem is trivial:

The Ice must supply the average power requested by the load, and when it does this it must do it at the optimal point which, as seen above is when its power is at the 76.8% of its nominal value
The battery supplies the load power minus ICE power.

Using iceNominalPower=60 kW we get the following output:

We see, as expected that the ICE is switched ON and OFF; and when it is ON it delivers at its 76.8 % of nominal power, i.e. at 46.1 kW. The battery delivers the difference, and when the load is negative absorbs all the power from it. The control is such that the energy at the end of the transient is the same as the one at t=0.

This result is good and confirms what we expected.

Effects of tolerance¶

We mentioned that reduction of tolerance may affect the result adversely by large, especially when the minimum, as in this case is very flat (since the specific fuel consumption curve used for our example is very flat).

In the following picture we see the result of the previous section as it appears when we release tolerance by changing it from 1e-7 to and 1e-6. Now the result is badly wrong (and the total cost has changed from to 25.6g 29.9g).

More realistic storage¶

To the DO to be useful, it must obviously go beyond what is exactly expected. Therefore, we repeat the simulation adding the simulation of some losses inside the battery. According to scientific literature, losses here are modelled through the following formula:

$L(t) = 0.03{|P}_{\text{batt}}(t)| + 0.04\frac{P_{\text{batt}}^{2}(t)}{P_{batt,nom}}$

Which reflects that they in part are proportional to the absolute value of battery current, partly to its square. The coefficients are typical for power electronic converters interfacing a battery.

Inside the code, however, the formula introduced is structurally different, since it has been transformed to avoid the derivative discontinuity of the absolute value of a quantity around zero, using a trick like the one reported in sect. 1.4.

We see that now the optimiser changes the Ice power when in ON state, to reduce the battery power at its peak, since the losses formula pushes towards lower powers.

Adding storage power limitation¶

As a last case for example 2, we ass tome limitation on the power that can be exchanged by the battery. This can be physically due to limitations of either the battery or the inverter connected to it.

To show better the effect, we first rise the ICE power to 100 kW, so that the interval of ICE operation is smaller:

Then we change the following row of code:

input Real outBatPow;

into:

input Real outBatPow(min = -maxBatPower, max = maxBatPower);

where

parameter Modelica.SIunits.Power maxBatPower = 30e3;

giving rise to the following results (with a much longer computation time than in the previous cases):

Example 3: Acausal vehicle¶

As a last example, we replicate the optimization of Example 2, but the power to be delivered directly deriving from simulation of a vehicle, modelled through its physical elements.

The considered diagram is as follows:

The upper part contains a vehicle model. The model follows a speed profile defined by the drive cycle driveCyc, through a simple proportional controller (simulating the driver). The power is applied to a mass; the drag force dragF sis the force against the movement due to friction (independent on speed) and air resistance (proportional to the square of vehicle speed).

The lower part contains the management of the storage, and the optimization algorithm, already discussed in example 2.

The results are shown in the following picture, where the obtained cost (red curve) is compared to what obtainable in case the ICE is continuously kept ON (blue curve), at a power (blue curve) that allows the battery energy at the end of the simulation to be equal to the one as t=0, as in the case of the optimised solution.

This example shows that the optimizer can find an ON/OFF strategy that more than halves the hybrid vehicle fuel consumption.

The following plot shows the ICE power in comparison with total power needed to cover the given trip profile mass.v. The rest is supplied by the battery.

1: Hairer, Ernst and Wanner, Gerhard, Radau Methods, 2015, pp 1213-1216, DOI 10.1007/978-3-540-70529-1_139.