Thermal circuit to state-space¶

This notebook is a go through complete example of obtaining a state-space representation from the bilding described by walls ans thermal circuits. The steps are:

in a folder, describe the walls (wall type and wall data), the thermal circuits and give the assembling matrix or lists;
from walls and thermal circuits obtain the set of disassembled circuits (function bldg2TCd());
assemble the set of disassembled thermal circuits (function assemble_TCd_matrix());
transform the thermal circuit into state-space representation (function tc2ss()).

In [1]:

import numpy as np
import pandas as pd
import pd_dm4bem

Assembling and converting the thermal circuits to state-space¶

Consider the disassembled thermal circuits shown in Figure 1 that we want to assemble as shown in Figure 2 and then obtain the state-space representation.

disassambled_TC

Figure 1. Four disassembled thermal circuits: wall_out, TC0, TC1, TC2, TC3.

disassambled_TC

Figure 2. The assembling of the four circuits from Figure 1.

The steps to obtain a state-space model from thermal circuits described in a folder are:

Obtain the disassambled set of thermal circuits from data given in a folder.

In [2]:

# Disassembled thermal circuits
folder_path = "bldg"
TCd = pd_dm4bem.bldg2TCd(folder_path,
                         TC_auto_number=True)

# For non auto-numbering of thermal circuits TC
# TCd = pd_dm4bem.bldg2TCd(folder_path, TC_auto_number=False)

Assemble the set of thermal circuits

In [3]:

# Assembled thermal circuit:

# from 'assembly_matrix.csv'
ass_mat = pd.read_csv(folder_path + '/assembly_matrix.csv')
TCm = pd_dm4bem.assemble_TCd_matrix(TCd, ass_mat)

# from 'assembly_lists.csv'
ass_lists = pd.read_csv(folder_path + '/assembly_lists.csv')
ass_mat = pd_dm4bem.assemble_lists2matrix(ass_lists)
TCl = pd_dm4bem.assemble_TCd_matrix(TCd, ass_mat)

Transform the thermal circuit into state-space representation.

In [4]:

# State-space from TC
[As, Bs, Cs, Ds, us] = pd_dm4bem.tc2ss(TCl)

The state-space representation is characterized by the matrices A (state), B (input), C (ouput) and D (feedthrough) and by the input vector u. Note the use of 's' (from state-space) in the names of As, Bs, Cs, Ds us in order to avoid confusion with the notation for differential-algebraic equations (A, G, C, b, f).

In [5]:

As

Out[5]:

	c1_θ0	c2_θ0	ow0_θ1	ow0_θ3
c1_θ0	-0.000240	0.000085	0.000000	0.000002
c2_θ0	0.002857	-0.003925	0.000000	0.000790
ow0_θ1	0.000000	0.000000	-0.000024	0.000002
ow0_θ3	0.000011	0.000107	0.000121	-0.000239

In [6]:

Bs

Out[6]:

	c1_q0	c2_q0	ow0_q0	c1_θ0	c2_θ0	ow0_θ0	ow0_θ4
c1_θ0	0.000152	0.000000	0.000000	9.182736e-07	0.000000	0.000000e+00	8.022399e-08
c2_θ0	0.000000	0.000278	0.000000	0.000000e+00	0.000031	0.000000e+00	2.600003e-05
ow0_θ1	0.000000	0.000000	0.000022	0.000000e+00	0.000000	1.970654e-08	0.000000e+00
ow0_θ3	0.000000	0.000000	0.000000	0.000000e+00	0.000000	0.000000e+00	2.931595e-07

In [7]:

Cs

Out[7]:

	c1_θ0	c2_θ0	ow0_θ1	ow0_θ3
c2_θ0	0.0	1.0	0.0	0.0

In [8]:

Ds

Out[8]:

	c1_q0	c2_q0	c3_q0	ow0_q0	c1_θ0	c2_θ0	ow0_θ0	ow0_θ4
c2_θ0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

In [9]:

pd.DataFrame({'u': us})

Out[9]:

	u
c1_q0	To
c2_q0	To
c3_q0	Ti_sp
ow0_q0	To
c1_θ0	Φa
c2_θ0	Qa
ow0_θ0	Φo
ow0_θ4	Φi

The state-space representation allows us to do the eigenvalues analysis and to find the maximum time step.

In [10]:

λ = np.linalg.eig(As)[0]    # eigenvalues of matrix As
λ = np.sort(λ)

print('Time constants:') 
print([f'{T:.2f} s' for T in -1 / λ])

print('\n2 x Time constants:') 
print([f'{T:.2f} s' for T in -2 / λ])

dtmax = 2 * min(-1. / λ)
print(f'\nMaximum time step: {dtmax:.2f} s = {dtmax / 60:.2f} min')

Time constants:
['249.30 s', '4093.21 s', '6729.13 s', '44033.06 s']

2 x Time constants:
['498.60 s', '8186.41 s', '13458.25 s', '88066.12 s']

Maximum time step: 498.60 s = 8.31 min

Comparing the results¶

The model used in this notebook is similar to that used in 03CubicBuilding.ipynb. The notations in the two models are different (see Figure 2 and Figure 3).

thermal_circuit

Figure 3. Thermal circuit for the cubic building used in 03CubicBuilding.ipynb

The correspondences between the notations in the two models (i.e., this notebook and 03CubicBuilding.ipynb) is shown in Tables 1, 2, and 3.

Table 1. Correspondence between states (i.e., temperature nodes with capacities) in Figures 2 and 3.

Fig. 2	ow0_θ1	ow0_θ3	c2_θ0	c1_θ0
Fig. 3	θ1	θ3	θ6	θ7

Table 2. Correspondence between temperature sources in Figures 2 and 3.

Fig. 2	ow0_q0	c1_q0	c2_q0	c3_q0
Fig. 3	q0	q8	q10	q11

Table 3. Correspondence between flow sources in Figures 2 and 3.

Fig. 2	ow0_θ0	ow0_θ4	c2_θ0	c1_θ0
Fig. 3	θ0	θ4	θ6	θ7

In [11]:

θ_order = ['ow0_θ1', 'ow0_θ3', 'c2_θ0', 'c1_θ0']
uT_order = ['ow0_q0', 'c1_q0', 'c2_q0', 'c3_q0']
uQ_order = ['ow0_θ0', 'ow0_θ4', 'c2_θ0', 'c1_θ0']
u_order = uT_order + uQ_order
y_order = ['c2_θ0'] 

By reindexing the matrices of the state-space model, it becomes easier to compare the results from this notebook and 03CubicBuilding.ipynb.

In [12]:

As.reindex(index=θ_order, columns=θ_order)

Out[12]:

	ow0_θ1	ow0_θ3	c2_θ0	c1_θ0
ow0_θ1	-0.000024	0.000002	0.000000	0.000000
ow0_θ3	0.000121	-0.000239	0.000107	0.000011
c2_θ0	0.000000	0.000790	-0.003925	0.002857
c1_θ0	0.000000	0.000002	0.000085	-0.000240

In [13]:

Bs.reindex(index=θ_order, columns=u_order)

Out[13]:

	ow0_q0	c1_q0	c2_q0	ow0_θ0	ow0_θ4	c2_θ0	c1_θ0
ow0_θ1	0.000022	0.000000	0.000000	1.970654e-08	0.000000e+00	0.000000	0.000000e+00
ow0_θ3	0.000000	0.000000	0.000000	0.000000e+00	2.931595e-07	0.000000	0.000000e+00
c2_θ0	0.000000	0.000000	0.000278	0.000000e+00	2.600003e-05	0.000031	0.000000e+00
c1_θ0	0.000000	0.000152	0.000000	0.000000e+00	8.022399e-08	0.000000	9.182736e-07

In [14]:

Cs.reindex(index=y_order, columns=θ_order)

Out[14]:

	ow0_θ1	ow0_θ3	c2_θ0	c1_θ0
c2_θ0	0.0	0.0	1.0	0.0

In [15]:

Ds.reindex(index=y_order, columns=u_order)

Out[15]:

	ow0_q0	c1_q0	c2_q0	c3_q0	ow0_θ0	ow0_θ4	c2_θ0	c1_θ0
c2_θ0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

Types of notation¶

The notation used for the nodes and flow can be symbols or numbers (in the case of autonumbering).

The data used in the next examples is extracted from the paper:

Ghiaus, C. (2013). Causality issue in the heat balance method for calculating the design heating and cooling load. Energy, 50, 292-301.

In [16]:

import pandas as pd
import pd_dm4bem

Thermal circuit file to state-space¶

File describing the thermal circuit with symbols¶

Let's consider a thermal circuit that uses symbols for numbering the temperature nodes and the oriented flow branches (Figure 1).

tc_sym

Figure 1. Thermal circuit with symbols.

The thermal circuit from Figure 1 is defined in the file ./TC_tc2ss/TC_tc2ss_sym.csv. In this case, symbols are used for the nodes and the branches.

In [17]:

pd.read_csv('./TC_tc2ss/TC_tc2ss_sym.csv')

Out[17]:

	A	θso	θw	θsi	θa	G	b
0	qw0	-1	1.0	NaN	NaN	2.9	NaN
1	qw1	NaN	-1.0	1	NaN	2.9	NaN
2	qv	NaN	NaN	NaN	1	38.3	Tov
3	qco	1	NaN	NaN	NaN	250.0	Tow
4	qci	NaN	NaN	-1	1	125.0	NaN
5	C	NaN	4000000.0	NaN	8.2E+04	NaN	NaN
6	f	Qo	NaN	Qi	Qg	NaN	NaN
7	y	NaN	NaN	1	1	NaN	NaN

Auto-numbering¶

The nodes and the oriented branches can be auto-numbered. The names of the nodes and branches are composed by the name of the circuit and θ0, θ1, ... , for nodes and q0, q1, ... for branches, e.g., a_θ1 stands for temperature node θ1 of circuit a.

In [18]:

TC_file = "TC_tc2ss/TC_tc2ss_sym.csv"
TC = pd_dm4bem.file2TC(TC_file, name="a",
                       auto_number=True)
[Asa, Bsa, Csa, Dsa, ua] = pd_dm4bem.tc2ss(TC)
Asa

Out[18]:

	a_θ1	a_θ3
a_θ1	-0.000001	7.085614e-07
a_θ3	0.000035	-5.016371e-04

Symbols¶

Alternatively, the nodes and the branches can keep the names given in the TC file (the default value of auto_numberis False).

In [19]:

TC_file = "TC_tc2ss/TC_tc2ss_sym.csv"
TC = pd_dm4bem.file2TC(TC_file, name="s")
[Ass, Bss, Css, Dss, us] = pd_dm4bem.tc2ss(TC)
Ass

Out[19]:

	s_θw	s_θa
s_θw	-0.000001	7.085614e-07
s_θa	0.000035	-5.016371e-04

Note that the order of symbols corespond to the order given in the description file ./TC_tc2ss/TC_tc2ss_num.csvnot to the alphabetical order.

File describing the thermal circuit with numbers¶

Instead of using symbols, the thermal circuit can be numbered (Figure 2).

tc_num

Figure 2. Numbered thermal circuit

The thermal circuit from Figure 1 is defined in the file ./TC_tc2ss/TC_tc2ss_num.csv. In this case, the nodes and the branches are numbered.

In [20]:

pd.read_csv('./TC_tc2ss/TC_tc2ss_num.csv')

Out[20]:

	A	θ0	θ1	θ2	θ3	G	b
0	q0	NaN	NaN	1	NaN	38.3	Tov
1	q1	1	NaN	NaN	NaN	250.0	Tow
2	q2	-1	NaN	NaN	1.0	2.9	NaN
3	q3	NaN	1	NaN	-1.0	2.9	NaN
4	q4	NaN	-1	1	NaN	125.0	NaN
5	C	NaN	NaN	8.2E+04	4000000.0	NaN	NaN
6	f	Qo	Qi	Qg	NaN	NaN	NaN
7	y	NaN	1	1	NaN	NaN	NaN

In [21]:

TC_file = "TC_tc2ss/TC_tc2ss_num.csv"
TC = pd_dm4bem.file2TC(TC_file, name="n")
[Asn, Bsn, Csn, Dsn, un] = pd_dm4bem.tc2ss(TC)
Asn

Out[21]:

	n_θ2	n_θ3
n_θ2	-5.016371e-04	0.000035
n_θ3	7.085614e-07	-0.000001

Note that the values of the state-space representations are the same but the order is different. The order is given by the .cvs files that define the thermal circuit.

In [ ]: