# Mathematica NDSolve for DAE and System Modeling

I'm a student for HVAC engineering and I've been using Mathematica for my study (mostly differential-algebraic equation system or Finite-Element-Analysis) and my hobby project for years. I really enjoy Mathematica, but I always have been struggling with NDSolve and its DAE solver, because it seems to be not very robust (see this post).

I have this feeling that NDSolve (or IDA) does not particularly favor discontinuous functions such as Clip, Min, Max. For most engineering problems and modeling method, I think that these functions are not avoidable. I've been browsing this forum for solutions quite a long time and there appeared some solutions for it, but they are mostly only for some specific case and not applicable for my case (e.g. a simple P-controller with a minimum of a close-to-zero value such as \$MachineEpsilon will very likely trigger the NDSolve::ndsz error for unknown reasons, not sure if it's a bug or not).

Till recent, I found that since v11 Mathematica allows connection to Modelica with CreateSystemModel. I've heard that Modelica has a strong DAE solver and I would really like to test this Mathematica-Modelica feature. But the documentation for System Modeling seems to be not very detailed. For the first trial I would like to realize a simple house heating model from this post.

(* effective heat capacity of building *)
Cwirk = 50 25 3;
(* import outdoor temperature *)
li = Import[
"http://rredc.nrel.gov/solar/old_data/nsrdb/1991-2005/data/tmy3/\
725958TYA.CSV"];
(* interpolate outdoor temperature *)
tae = Transpose[{Range, Drop[Drop[li, 1][[All, 32]], 1]}]
eq = {
(* equation for building *)
Q[tau] - 200 (tt[tau] - ta[tau]) == Cwirk tt'[tau],
(* heating capacity of floor heating system *)
Q[tau] - 100 (28 - tr[tau]) vF[tau] 7/6 == 0,
(* the water outlet temperature of floor heating *)
tr[tau] - tt[tau] - (28 - tt[tau]) Exp[-0.9/(7/6 vF[tau] 0.22)] == 0,
(* a simple P-controller for the flow rate *)
vF[tau] - Max[Min[(20 + 20 (20 - tt[tau])), 100], 10^(-10)] == 0}


Here is what I've got so far.

model = CreateSystemModel[eq, tau]
data = CreateDataSystemModel[tae]
wholeModel =
ConnectSystemModelComponents[{model, data}, {"ta" \[Element] data}]


The connection seems to be unsuccessful and SystemModelSimulate returns nothing. It would be really appreciated if a simple example can be provided for the new System Model feature.

EDIT

Here is a simplified version of the above problem. It is about the thermal behavior of a house. The thermal resistance is R (e.g. $$1/200 \frac{K}{W}$$) and it has a heat capacity of Cwirk (e.g. $$50 \cdot 25 \cdot 3 \frac{W h}{K}$$). The variable t is the room temperature and ta is the outdoor ambient temperature. Q stands for the heating output of the in-house heating device (we can set Q to a constant such as $$1000 W$$). So the conservation equation is

$$C_{Wirk} \cdot \frac{dt}{d\tau} = - \frac{t - t_a}{R} + \dot{Q}$$

The t_a is obviously an input to this model. We can use the weather data from the list tae.

li = Import[
"http://rredc.nrel.gov/solar/old_data/nsrdb/1991-2005/data/tmy3/\
725958TYA.CSV"];
(* interpolate outdoor temperature *)
tae = Transpose[{Range, Drop[Drop[li, 1][[All, 32]], 1]}]


It's unclear for me how to connect the weather data to the equation/model. It would be very appreciated if any explanation or maybe an example can be provided.

• The question is very interesting. However, the variables (such as vF, tt, li, tae) and the equations are very misleading to me. If you are new to Mathematica's system model, I suggest to use a simple enough system. Dec 4, 2019 at 10:51
• ConnectSystemModelComponents works with components that have connectors. You probably need to specify in CreateSystemModel which variables are connectors, and then use those connectors to connect it with the data model. I suggest reading the documentation page of ConnectSystemModelComponents. Dec 4, 2019 at 10:56
• @xinxinguo I've added a simplified example. Dec 4, 2019 at 12:12
• @MalteLenz thank you for the link. Actually, this is exactly what I'm confused about. After reading this documentation, I still have no idea how to connect an external list to the defined model. Dec 4, 2019 at 12:14
• Maybe this example is useful. Dec 4, 2019 at 12:28

Based on your simplified version, and using hard-coded data, create the model with the data source:

data = {{0, 6}, {1, 9}, {2, 1}, {3, 9}, {4, 7}, {5, 6}, {6, 1}, {7,
8}, {8, 0}, {9, 2}, {10, 9}};
datamodel = CreateDataSystemModel[data]


cwirk = 50 25 3;
q = 1000;
r = 1/200;


Create the model containing the equations. Note that we are defining the ta as a Real input:

eqmodel =
CreateSystemModel[{cwirk t'[tau] == (t[tau] - ta[tau])/r + q},
tau, {ta \[Element] "RealInput"}]


Connect them. Note how we connect the output from the data model, data.y, to the input we created in the equation model, eqs.ta:

simmodel = ConnectSystemModelComponents[
{"data" \[Element] datamodel, "eqs" \[Element] eqmodel},
{"data.y" -> "eqs.ta"}
]


And then you can simulate and plot:

sim = SystemModelSimulate[simmodel, 10]
SystemModelPlot[sim, {"eqs.t"}]

• Thank you, it is exactly what I need! Dec 4, 2019 at 12:49
• I tried your code in Mathematica. It returns Notification: The initialization problem is underdetermined, the \ following variables where chosen to be initialized from their start \ values: eqs.t. error and Error: Compilation of generated code failed (exit code 1). Build log: "/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.\ xctoolc\[Ellipsis] ~~~ 17 errors generated. *** Error code 1 Stop in /private/var/folders/8b/yyk12r657g3cmld0fs36r_4m0000gn/T/\ WolframSystemModeler-12.0. Does it mean that I have to install SystemModeler? Dec 4, 2019 at 13:24
• You don't need SystemModeler, but you need to install a C++ compiler. In the case of macOS, you should be able to follow this guide: wolfram.com/system-modeler/compiler-mac If you are using Catalina and some versions of XCode you may need to apply a workaround: support.wolfram.com/kb/48717 Since you don't have SystemModeler you will need to set the "PlatformSDKPath" using the tools from reference.wolfram.com/language/WSMLink/tutorial/… instead. Dec 4, 2019 at 13:52
• I got it working! I have Xcode but I had to manually set the PlatformSDKPath with SystemModelConfigurationSetConfiguration[]`. Thank you! Dec 4, 2019 at 14:39