# User defined equations¶

User defined functions provide a powerful tool to the user as they enable the definition of generic and individual equations that can be applied to your TESPy model. In order to implement this functionality in your model you will use the tespy.tools.helpers.UserDefinedEquation. The API documentation provides you with an interesting example application, too.

## Getting started¶

For an easy start, let’s consider two different streams. The mass flow of both streams should be coupled within the model. There is already a possibility covering simple relations, i.e. applying value referencing with the tespy.connections.connection.Ref class. This class allows formulating simple linear relations:

$0 = \dot{m}_1 - \left(\dot{m}_2 \cdot a + b\right)$

Instead of this simple application, other relations could be useful. For example, the mass flow of our first stream should be quadratic to the mass flow of the second stream.

$0 = \dot{m}_1 - \dot{m}_2^2$

In order to apply this relation, we need to import the tespy.tools.helpers.UserDefinedEquation class into our model and create an instance with the respective data. First, we set up the TESPy model.

>>> from tespy.networks import Network
>>> from tespy.components import Sink, Source
>>> from tespy.connections import Connection
>>> from tespy.tools import UserDefinedEquation

>>> nw = Network(iterinfo=False)
>>> nw.set_attr(p_unit='bar', T_unit='C', h_unit='kJ / kg')

>>> so1 = Source('source 1')
>>> so2 = Source('source 2')
>>> si1 = Sink('sink 1')
>>> si2 = Sink('sink 2')

>>> c1 = Connection(so1, 'out1', si1, 'in1')
>>> c2 = Connection(so2, 'out1', si2, 'in1')

>>> c1.set_attr(fluid={'water': 1}, p=1, T=50)
>>> c2.set_attr(fluid={'water': 1}, p=5, T=250, v=4)


In the model both streams are well-defined regarding pressure, enthalpy and fluid composition. The second stream’s mass flow is defined through specification of the volumetric flow, we are missing the mass flow of the connection c1. As described, its value should be quadratic to the (still unknown) mass flow of c2. First, we now need to define the equation in a function which returns the residual value of the equation.

>>> def my_ude(ude):
...     return ude.conns[0].m.val_SI - ude.conns[1].m.val_SI ** 2


Note

The function must only take one parameter, i.e. the UserDefinedEquation class instance. The name of the parameter is arbitrary. We will use ude in this example. It serves to access some important parameters of the equation:

Note

It is only possible to use the SI-values of the connection variables as these values are updated in every iteration. The values in the network’s specified unit system are only updated after a simulation.

The second step is to define the derivatives with respect to all primary variables of the network, i.e. mass flow, pressure, enthalpy and fluid composition of every connection. The derivatives have to be passed to the Jacobian. In order to do this, we create a function that updates the values inside the Jacobian of the UserDefinedEquation and returns it:

• ude.jacobian is a dictionary containing numpy arrays for every connection required by the UserDefinedEquation.

• derivatives to mass flow are placed in the first element of the numpy array (index 0)

• derivatives to pressure are placed in the second element of the numpy array (index 1)

• derivatives to enthalpy are placed in the third element of the numpy array (index 2)

• derivatives to fluid composition are placed in the remaining elements beginning at the fourth element of the numpy array (indices 3:)

If we calculate the derivatives of our equation, it is easy to find, that only derivatives to mass flow are not zero.

• The derivative to mass flow of connection c1 is equal to $$1$$

• The derivative to mass flow of connection c2 is equal to $$-2 \cdot \dot{m}_2$$.

>>> def my_ude_deriv(ude):
...     c0 = ude.conns[0]
...     c1 = ude.conns[1]
...     if c0.m.is_var:
...         ude.jacobian[c0.m.J_col] = 1
...     if c1.m.is_var:
...         ude.jacobian[c1.m.J_col] = -2 * ude.conns[1].m.val_SI


Now we can create our instance of the UserDefinedEquation and add it to the network. The class requires four mandatory arguments to be passed:

• label of type String.

• func which is the function holding the equation to be applied.

• deriv which is the function holding the calculation of the Jacobian.

• conns which is a list of the connections required by the equation. The order of the connections specified in the list is equal to the accessing order in the equation and derivative calculation.

• params (optional keyword argument) which is a dictionary holding additional data required in the equation or derivative calculation.

>>> ude = UserDefinedEquation('my ude', my_ude, my_ude_deriv, [c1, c2])
>>> nw.solve('design')
>>> round(c2.m.val_SI ** 2, 2) == round(c1.m.val_SI, 2)
True
>>> nw.del_ude(ude)


## More examples¶

After warm-up let’s create some more complex examples, e.g. the square root of the temperature of the second stream should be equal to the logarithmic value of the pressure squared divided by the mass flow of the first stream.

$0 = \sqrt{T_2} - \ln\left(\frac{p_1^2}{\dot{m}_1}\right)$

In order to access the temperature within the iteration process, we need to calculate it with the respective method. We can import it from the tespy.tools.fluid_properties module. Additionally, import numpy for the logarithmic value.

>>> import numpy as np

>>> def my_ude(ude):
...     return (
...         ude.conns[1].calc_T() ** 0.5
...         - np.log(abs(ude.conns[0].p.val_SI ** 2 / ude.conns[0].m.val_SI))
...     )


Note

We use the absolute value inside the logarithm expression to avoid ValueErrors within the solution process as the mass flow is not restricted to positive values.

The derivatives can be determined analytically for the pressure and mass flow of the first stream easily. For the temperature value, you can use the predefined fluid property functions dT_mix_dph and dT_mix_pdh respectively to calculate the partial derivatives.

>>> from tespy.tools.fluid_properties import dT_mix_dph
>>> from tespy.tools.fluid_properties import dT_mix_pdh

>>> def my_ude_deriv(ude):
...     c0 = ude.conns[0]
...     c1 = ude.conns[1]
...     if c0.m.is_var:
...         ude.jacobian[c0.m.J_col] = 1 / ude.conns[0].m.val_SI
...     if c0.p.is_var:
...         ude.jacobian[c0.p.J_col] = - 2 / ude.conns[0].p.val_SI
...     T = c1.calc_T()
...     if c1.p.is_var:
...         ude.jacobian[c1.p.J_col] = (
...             dT_mix_dph(c1.p.val_SI, c1.h.val_SI, c1.fluid_data, c1.mixing_rule)
...             * 0.5 / (T ** 0.5)
...         )
...     if c1.h.is_var:
...         ude.jacobian[c1.h.J_col] = (
...             dT_mix_pdh(c1.p.val_SI, c1.h.val_SI, c1.fluid_data, c1.mixing_rule)
...             * 0.5 / (T ** 0.5)
...         )


But, what if the analytical derivative is not available? You can make use of generic numerical derivatives using the inbuilt method numeric_deriv. The methods expects the variable 'm', 'p', 'h' or 'fluid' (fluid composition) to derive the function to as well as the respective connection index from the list of connections. The “lazy” solution for the above derivatives would therefore look like this:

>>> def my_ude_deriv(ude):
...     c0 = ude.conns[0]
...     c1 = ude.conns[1]
...     if c0.m.is_var:
...         ude.jacobian[c0.m.J_col] = ude.numeric_deriv('m', c0)
...     if c0.p.is_var:
...         ude.jacobian[c0.p.J_col] = ude.numeric_deriv('p', c0)
...     if c1.p.is_var:
...         ude.jacobian[c1.p.J_col] = ude.numeric_deriv('p', c1)
...     if c1.h.is_var:
...         ude.jacobian[c1.h.J_col] = ude.numeric_deriv('h', c1)

>>> ude = UserDefinedEquation('ude numerical', my_ude, my_ude_deriv, [c1, c2])
>>> nw.set_attr(m_range=[.1, 100])  # stabilize algorithm
>>> nw.solve('design')
>>> round(c1.m.val, 2)
1.17

>>> c1.set_attr(p=None, m=1)
>>> nw.solve('design')
>>> round(c1.p.val, 3)
0.926

>>> c1.set_attr(p=1)
>>> c2.set_attr(T=None)
>>> nw.solve('design')
>>> round(c2.T.val, 1)
257.0


Obviously, the downside is a slower performance of the solver, as for every numeric_deriv call the function will be evaluated fully twice (central finite difference).

Last, we want to consider an example using additional parameters in the UserDefinedEquation, where $$a$$ might be a factor between 0 and 1 and $$b$$ is the steam mass fraction (also, between 0 and 1). The difference of the enthalpy between the two streams multiplied with factor a should be equal to the difference of the enthalpy of stream two and the enthalpy of saturated gas at the pressure of stream 1. The definition of the UserDefinedEquation instance must therefore be changed as below.

$0 = a \cdot \left(h_2 - h_1 \right) - \left(h_2 - h\left(p_1, x=b \right)\right)$
>>> from tespy.tools.fluid_properties import h_mix_pQ
>>> from tespy.tools.fluid_properties import dh_mix_dpQ

>>> def my_ude(ude):
...     a = ude.params['a']
...     b = ude.params['b']
...     c0 = ude.conns[0]
...     c1 = ude.conns[1]
...     return (
...         a * (c1.h.val_SI - c0.h.val_SI) -
...         (c1.h.val_SI - h_mix_pQ(c0.p.val_SI, b, c0.fluid_data))
...     )

>>> def my_ude_deriv(ude):
...     a = ude.params['a']
...     b = ude.params['b']
...     c0 = ude.conns[0]
...     c1 = ude.conns[1]
...     if c0.p.is_var:
...         ude.jacobian[c0.p.J_col] = dh_mix_dpQ(c0.p.val_SI, b, c0.fluid_data)
...     if c0.h.is_var:
...         ude.jacobian[c0.h.J_col] = -a
...     if c1.p.is_var:
...         ude.jacobian[c1.p.J_col] = a - 1

>>> ude = UserDefinedEquation(
...     'my ude', my_ude, my_ude_deriv, [c1, c2], params={'a': 0.5, 'b': 1}
... )


One more example (using a CharLine for data point interpolation) can be found in the API documentation of class tespy.tools.helpers.UserDefinedEquation.

## Document your equations¶

For the automatic documentation of your models just pass the latex keyword on creation of the UserDefinedEquation instance. It should contain the latex equation string. For example, the last equation from above:

latex = (
r'0 = a \cdot \left(h_2 - h_1 \right) - '
r'\left(h_2 - h\left(p_1, x=b \right)\right)'
)

ude = UserDefinedEquation(
'my ude', my_ude, my_ude_deriv, [c1, c2], params={'a': 0.5, 'b': 1},
latex={'equation': latex}
)


The documentation will also create figures of CharLine and CharMap objects provided. To add these, adjust the code like this. Provide the CharLine and CharMap objects within a list.

ude = UserDefinedEquation(
'my ude', my_ude, my_ude_deriv, [c1, c2], params={'a': 0.5, 'b': 1},
latex={
'equation': latex,
'lines': [charline1, charline2],
'maps': [map1]
}
)