Inner and outer walls

Windows

Room air load

The thermal zone model is based on the simplified control engineering model which is described in the German VDI 6007 standard. The basic idea behind the previously mentioned model is the equivalence between the differential equation of the thermal conduction in a wall and the processes in an idealized electrical cable, as shown in the following equation:

$\frac{\partial T\left(t,x\right)}{\partial t}=\frac{1}{R\cdot C}\cdot \frac{\partial T^2\left(t,x\right)}{\partial x^2}\text{,}$

where T is the temperature, R is the thermal resistance of the wall layer, and C is the heat capacity of the wall layer.

According to the norms 6007 and 6020 individual building components can be divided into three basic groups:

The different wall types can be modeled using the equivalence explained earlier, leading to a 2-transfer functions model of a thermal zone, depicted in Fig. 4.5. The resistances and capacities of the walls are calculated depending on the wall’s structure and its geometry.

The model representation of a thermal zone in Modelica according to VDI 6007 is shown in Fig. 4.6 (Lauster, Streblow, & Müller, 2012).

The heating medium which flows into the component via the enthalpy connector is heated up to the prescribed set temperature by an ideal heating element. The thermal output ${\dot{Q}}_{\text{th}}$ of this heating element is determined by a PID controller which controls the flow temperature within the component. By using the efficiency table of the boiler the current fuel consumption is calculated.

${\dot{E}}_{\text{fuel}}=\frac{{\dot{Q}}_{\text{th}}}{\eta }\text{.}$

The externally prescribed target compressor speed of the heat pump determines the load state. With the source and sink temperatures as environmental status the maximum condenser heat flow and the corresponding power consumption are delivered by two separate tables. Based on these values the actual COP and thermal output ${\dot{Q}}_{\text{th}}$ can be calculated according to the following equations (Huchtemann & Müller, 2009):

$\text{COP}=\frac{{\dot{Q}}_{\text{th}}}{P_{\text{el},\text{max}}}\text{,}$

$P_{\text{el}}=P_{\text{el,}\text{max}}\cdot {\left(\frac{1}{N_{\text{target}}}\right)}^3\text{,}$

${\dot{Q}}_{\text{co}}={\dot{Q}}_{\text{th}}={\dot{Q}}_{\text{co,}\text{max}}\cdot {\left(\frac{1}{N_{\text{target}}}\right)}^3\text{.}$

At first the electrical and thermal efficiencies η_el and η_th are calculated depending on the coefficients of the polynomial, the maximum electrical power output, medium mass flow rate $\dot{m}$ , and return temperature T_R:

${\eta }_{\text{el}}=a_0+a_1\cdot {P_{\text{el,}\text{max}}}^2+a_2\cdot P_{\text{el,}\text{max}}+a_3\cdot {\dot{m}}^2+a_4*\dot{m}+a_5\cdot {T_{\text{R}}}^2+a_6\cdot T_{\text{R}}\text{,}$

${\eta }_{\text{th}}=b_0+b_1\cdot {P_{\text{el,}\text{max}}}^2+b_2\cdot P_{\text{el,}\text{max}}+b_3\cdot {\dot{m}}^2+b_4*\dot{m}+b_5\cdot {T_{\text{R}}}^2+b_6\cdot T_{\text{R}}\text{.}$

Depending on the generation mode (heat or power driven) the current thermal and electrical output power is calculated using the power/heat demand input signal and the efficiencies. Based on the output flows and the efficiency, the gas consumption ${\dot{E}}_{\text{fuel}}$ is determined by (Rosato & Sibilio, 2012):

${\dot{E}}_{\text{fuel}}=\frac{P_{\text{el}}}{{\eta }_{\text{el}}}\text{.}$

Using the input radiation, orientation, and location data, the total radiation on the tilted panel surface G can be calculated. The thermal efficiency η can be calculated depending on the difference between the mean panel temperature T_m and the environmental air temperature with polynomial coefficients a_i which are either given by the manufacturer or determined empirically. Hereby the total thermal output of the panel ${\dot{Q}}_{\text{th}}$ is determined:

$\eta =\frac{1}{G}[{\eta }_0-a_1\cdot \left(T_{\text{m}}-T_a\right)-a_2\cdot {\left(T_{\text{m}}-T_{\text{a}}\right)}^2],$

${\dot{Q}}_{\text{th}}=G\cdot A_{\text{panel}}.$

The enthalpy connectors for loading and unloading the storage tank are, depending on their position, connected to a particular discrete layer of the tank or a coil inside the tank. For the case of tank loading the function of the tank can be described as follows.

The enthalpy of a passing medium is to a certain extent transferred from the medium to the layers through which it flows or to the layers which are connected to the corresponding heating coil. After passing the predefined layers of the tank or the heating coil, the medium leaves the storage tank with a certain enthalpy difference which is transferred to the internal storage medium. The layers itself are also connected to each other. The exchange of enthalpy between these layers occurs on the one hand by simple heat conduction ${\dot{Q}}_{\text{cond}}$ in both directions (to the upper and lower neighboring layer) and on the other hand based on buoyancy induced convection. Those two effects result in the effective heat conduction coefficient λ_eff that is used to calculate the effective heat transfer by conduction ${\dot{Q}}_{\text{cond}}$ . Additionally each layer is thermally connected with the environment to which heat losses ${\dot{Q}}_{\text{loss}}$ can occur depending on the temperature difference of the medium T_layer,n and the outside air T_env.

These different heat flows also depend on the height of each discrete volume h_layer and their lateral and front surface area A_layer,front and A_{layer,lateral} respectively:

${\dot{Q}}_{\text{cond}}=\frac{{\lambda }_{\text{eff}}}{h_{\text{layer}}}\cdot A_{\text{layer,}\text{front}}\cdot \left(T_{\text{layer},n}-T_{\text{layer,}n+1}\right)\text{,}$

${\dot{Q}}_{\text{loss}}=k_{\text{tank,}\text{lateral}}\cdot A_{\text{layer,}\text{lateral}}\cdot \left(T_{\text{layer},n}-T_{\text{env}}\right)\text{.}$

The unloading process of the storage tank is vice versa.

The discrete volumes of the heat exchanger are arranged in an alternating manner. Thus, each volume containing the hot medium is neighboring a volume containing the cold medium. Those neighboring discrete volumes are thermally connected to each other by a heat transfer component. Based on the heat flow between the individual volumes ${\dot{Q}}_i$ , which is influenced by the temperatures of hot and cold volumes T_h,i and T_c,i and the heat transmission coefficient k, the total heat flow within the heat exchanger ${\dot{Q}}_{\text{th}}$ can be calculated:

${\dot{Q}}_i=k\cdot A_i\cdot \left(T_{\text{h},n+1-i}-T_{\text{c},i}\right)+k\cdot A_i\cdot \left(T_{\text{h},n-i}-T_{\text{c},i}\right)\text{,}$

$=k\cdot A_i\cdot \left(T_{\text{h},n+1-i}+T_{\text{h},n-i}-2T_{\text{c},i}\right)\text{,}$

${\dot{Q}}_{\text{th}}={\displaystyle \sum }_{i=0}^n{\dot{Q}}_i\text{.}$

The heat ${\dot{Q}}_{\text{th}}$ flowing into the component is transferred to a thermal capacity element representing the mass of the radiator. A radiator wall element calculates the partial energy flows through convection and radiation that are transferred to the surrounding environment. The convective part ${\dot{Q}}_{\text{conv}}$ is determined by the heat transmission coefficient k, the effective radiator area A, and the temperature difference between the radiator wall T_rad and the environment T_env. The radiative part of the energy transfer ${\dot{Q}}_{\text{rad}}$ is determined by Stefan Boltzmann’s law, which uses the emissivity ɛ of the radiator and the Stefan Boltzmann constant σ. This energy transport is carried out by a conventional thermal connector and a radiation port which will be connected to the thermal zone model (Section 3.5.1.1).

${\dot{Q}}_{\text{th}}={\dot{Q}}_{\text{conv}}+{\dot{Q}}_{\text{rad}}$

${\dot{Q}}_{\text{conv}}=k\cdot A\cdot \left(T_{\text{rad}}-T_{\text{env}}\right)$

${\dot{Q}}_{\text{rad}}=\sigma \cdot \varepsilon \cdot A\cdot \left({T_{\text{rad}}}^4-{T_{\text{env}}}^4\right)$

The pump component simply prescribes the mass flow $\dot{m}$ of the heating medium in the system. The enthalpy and temperature of the medium stay unchanged:

$\dot{m}={\dot{m}}_{\text{prescribed}}\text{,}$

$h_{\text{out}}=h_{\text{in}}\text{,}$

$T_{\text{out}}=T_{\text{in}}\text{.}$

The three-way valve model is able to represent either a distribution or a mixing valve. As a distribution valve it splits a mass flow ${\dot{m}}_{AB}$ into two separate mass flows ${\dot{m}}_A$ and ${\dot{m}}_B$ according to the valve opening position.

Used as a mixing valve two separate mass flows ${\dot{m}}_A$ and ${\dot{m}}_B$ are mixed to one mass flow ${\dot{m}}_{AB}$ depending on the valve position.

Since the pump (Section 3.5.1.9) can ideally prescribe a mass flow rate, conventional throttle valves are not necessary for a system setup.

The mass flows and enthalpies of each enthalpy connector for the distribution/mixing valve are calculated as follows:

Distribution valve

${\dot{m}}_A={\dot{m}}_{AB}\cdot y\text{,}$

${\dot{m}}_B={\dot{m}}_{AB}\cdot \left(1-y\right)\text{,}$

$h_A=h_A\text{.}$

Mixing valve

${\dot{m}}_{AB}=y\cdot {\dot{m}}_A+\left(1-y\right)\cdot {\dot{m}}_{B,}$

$h_{AB}=\frac{{\dot{m}}_A\cdot h_A+{\dot{m}}_B\cdot h_B}{{\dot{m}}_{AB}}\text{.}$

The line model is a lumped π line model with a voltage reference pin (Fig. 4.9).

The resistance R and the inductance L are the total resistance and inductance of the line, calculated from the parameters, whereas the capacitances C₁ and C₂ share the total line capacitance:

$R=r\cdot x\text{,}$

$L=l\cdot x\text{,}$

$C=c\cdot x\text{,}$

$C_1=C_2=\frac{1}{2}C\text{.}$

The effect of the line capacitance can be neglected in low voltage lines by setting C = 0, leading to a purely inductive model.

The photovoltaic module is a simplified model with two pins which converts solar irradiance and ambient temperature data into electrical active power depending on the surface area of the module. This power value is then used to compute the relation between voltage and current. When information on reactive power is not available, its value is set to zero as a default. The nominal operating conditions, such as ambient temperature, cell temperature, and radiation are parameters which may be chosen according to the specific PV panel in use. The efficiency is modeled to decrease at higher ambient temperatures than the nominal ambient temperature, depending on a temperature coefficient which depends on the type of PV panel.

This model is not a component-based model like a diode PV model. Therefore, it is not useful for a detailed analysis of PV panel properties; it should rather be used to represent the behavior of a PV module as a source within an electric system (Fig. 4.10).

If a specific output is required, electrical quantities can be measured by connecting the corresponding sensor model.

At first, the cell temperature T_cell is calculated from the ambient temperature input and the solar irradiance input, depending on the nominal operating conditions:

$T_{\text{cell}}=T_{\text{ambient}}+\left(T_{\text{NOCT}}-T_{\text{ambient,NOCT}}\right)\cdot \frac{\text{Irr}}{\text{Ir}{\text{r}}_{\text{NOCT}}}\text{.}$

Now, the efficiency η is updated according to the difference between T_cell and T_NOCT, weighted with the temperature coefficient for efficiency.

Finally, according to Skoplaki and Palyvos (2009) the active power output is given as:

P = Irr. area. η,

which specifies the relationship between the dynamic phasor current and voltage across the two pins of the model with the equation:

$〈i〉=\frac{{\left(P+jQ\right)}^{*}}{〈v^{*}〉}\text{.}$

The battery storage system is a simple model of a generic battery. It can be charged or discharged up to a maximum capacity with a certain power and efficiency, while updating its state of charge. A charging or discharging logic can be connected to the battery via the charge/discharge status input. When information on the end-of-discharge or end-of-charge voltage are unknown, the default values can be used (Fig. 4.11).

This model does not specify the battery technology and side effects which may occur due to the specific chemical processes some battery types use.

The battery is modeled as a voltage source with an internal resistance, whose voltage level depends on the state of charge (Wagner, 2010).

The battery storage is considered to operate in DC, a logic which translates between the dynamic phasor voltage and current in the rest of the system and their root mean square values was included within the battery model (Fig. 4.12).

Assuming that the voltage source depends on the state of charge linearly for simplicity:

$v_{\text{DCsource}}=v_{\text{min}}+(v_{\text{max}}-v_{\text{min}})\cdot \frac{\text{SOC}}{\text{SO}{\text{C}}_{\text{max}}}\text{.}$

When the maximal storage capacity is reached, the voltage is limited by the end-of-charge voltage.

In the first time step, SOC = SOC₀. Through the power parameter, the DC current is computed from:

$i_{\text{DC}}=\frac{P}{v_{\text{source}}}\text{.}$

The state of charge is then updated through the equation:

$\text{SOC}=\text{SO}{\text{C}}_0+\int \eta \cdot a\cdot i_{\text{DC}}dt\text{,}$

while SOC ≤ SOC_max, and a = ±1 depending on whether the battery is charging or discharging.

The current and voltage of the DC voltage source are converted back to AC as ideal sinusoidal signals with an adjustable default frequency of 50 Hz, where the amplitude is given by $\sqrt{2}$ times the DC voltage or current.

The relation between the dynamic phasor current and voltage is given by the complex power:

$〈i〉=\frac{{\left(P+jQ\right)}^{*}}{〈v^{*}〉}\text{.}$

For the transformer, an ideal model as well as a standard model is available for both single-phase and multiphase applications. The ideal transformer changes the voltage according to the turns ratio with an optional magnetization, whereas the standard transformer is modeled with two inductors and their mutual inductance (Fig. 4.15).

Additionally, a prototype of a YY and YD transformer (YD5, YD11) have been modeled.

Ideal transformer

Standard transformer:

Ideal transformer

The relation between the primary voltage v₁ and the secondary voltage v₂ is given by:

$〈v_1〉=n\cdot 〈v_2〉\text{.}$

If the user chooses to consider the magnetization, the behavior is modified to include the magnetization current i_m (as a function of the primary current i₁ and secondary current i₂) and the magnetic flux ψ_m with respect to the primary side of the transformer:

$〈i_{\text{m}}〉=〈i_1〉+\frac{〈i_2〉}{n}\text{,}$

$〈{\psi }_{\text{m}}〉=L_{\text{m}}\cdot 〈i_{\text{m}}〉\text{.}$

Standard transformer

The standard transformer also includes the primary and secondary inductance as well as their influence on each other. The equations then are:

$〈v_1〉=L_1\cdot \frac{d〈i_1〉}{dt}+M\cdot \frac{d〈i_2〉}{dt}\text{,}$

$〈v_2〉=L_2\cdot \frac{d〈i_2〉}{dt}+M\cdot \frac{d〈i_1〉}{dt}\text{.}$

Three-phase transformers

The three-phase transformers are obtained by connecting single-phase transformers with a wye and/or delta connector.

In case the ideal transformer is used, a resistor and inductor can be added in order to approximate the real behavior. The YY and YD transformers are models comprising the connection logic (wye or delta) and the transformer model (ideal or standard) as well as additional components if the ideal transformer model is used instead of the standard transformer. For example, the connection logic for a YD transformer is obtained by connecting components in series in the following order:

If a specific output is required, electrical quantities can be measured by connecting the corresponding sensor model.

According to Ghosh and Prelas (2011), the active power curve of a wind turbine is described by the equation:

$P=c_{\text{p}}{\varepsilon }_{\text{g}}{\varepsilon }_{\text{b}}\frac{1}{2}\rho A{v}^3,$

where $\frac{1}{2}\rho Av³$ is the power of the wind acting on a surface A, and the factors c_pɛ_gɛ_b are limiting factors on the actual power exploitation which varies depending on the specific type of wind turbine and is limited by 0.593 according to Betz’s law.

From the power output, the dynamic phasor voltage and current can be calculated with the equation:

$〈i〉=\frac{{\left(P+jQ\right)}^{*}}{〈v^{*}〉}\text{.}$