Automatic thermal model identification and distributed optimisation for load shifting in city quarters

Figures & data

Table 1. Symbols used for the building model identification methodology.

Symbol	Description
$\alpha$	solar irradiance model parameter
$\text{β}$	(constant) internal gain model parameter
$h$	MRI slices enumerator
$I_{\mathrm{g},o}$	façade irradiance, where $oϵ\left\{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t},\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{h},\mathrm{w}\mathrm{e}\mathrm{s}\mathrm{t}\right\}$
$j$	zone enumerator
$k$	discrete time index
$l$	number of MRI slices
$\lambda$	lumped RC model parameter
$n$	number of model states (length of $\mathit{x}$)
$N_{\mathrm{p}}$	prediction horizon
$N_{\mathrm{z}}$	number of zones
${\dot{Q}}_{\mathrm{c}\mathrm{o}\mathrm{m}}$	heating power of commercial zones
${\dot{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	maximum heating power (of zone)
${\dot{Q}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$	total heating power
$T_{\mathrm{e}}$	envelope (or wall) temperature
$T_{\mathrm{f}}$	feed temperature of the heating circuit
$T_{\mathrm{h}}$	heating (or floor) temperature
$\overline{T_{\mathrm{h}}}$	optimal linearisation point of $T_{\mathrm{h}}$
$T_{\mathrm{i}}$	inner (or zone) temperature
$T_{\mathrm{r}\mathrm{e}\mathrm{f}}$	zone reference temperature (setpoint)
$t_{\mathrm{s}}$	sampling time
$u_{\mathrm{h}}$	relative heating control action
$\mathit{x}$	state vector (consisting of $T_{\mathrm{i}}$, $T_{\mathrm{e}}$ and $T_{\mathrm{f}}$)

Figure 1. Data slicing for the model relevant identification (MRI) approach using a Kalman Filter (KF) for the generation of the required start values ${\mathit{x}}_{0,h}$ for each data slice $h=1\dots l$. In order to obtain meaningful start values for the first MRI slice ${\mathit{x}}_{0,1}$, a small part at the beginning is required to ensure the filter’s convergence.

Table 2. Symbols used for the distributed optimisation methodology.

Symbol	Description
${\mathit{\xi }}_j$	vector of local optimisation variables
$f_j$	local objective functions
$J_i$	objective function of block $i$ in ADMM
${\mathit{z}}_i$	optimisation variables of block $i$ in ADMM
${\mathit{h}}_j\left({\mathit{\xi }}_j\right)$	vector of local equality constraint functions
${\overline{\mathit{h}}}_i\left({\mathit{z}}_i\right)$	vector of block equality constraints in ADMM
${\mathit{g}}_j\left({\mathit{\xi }}_j\right)$	vector of local inequality constraint functions
${\overline{\mathit{g}}}_i\left({\mathit{z}}_i\right)$	Vector of block inequality constraints in ADMM
$\mathit{H}$	vector of coupling constraint functions
$\rho$	penalty parameter
$\nu$	vector of Lagrange multipliers for the coupling constraints
${\hat{\mathit{\xi }}}_j$	local copy of the vector of optimisation variables

Note: The index $j$ denotes the zone.

Table 3. Overview of the basic data and use of the buildings of quarter Q1 of Reininghausgründe in Graz, Austria.

	Floors	Gross floor area	Office / Commercial	Residential	Heating	Cooling
Building	–	m²	m²	m²	kW	kW
BT01 (Greentower)	20	14590	2412	12178	620	410
BT02	16	14941	2412	12529	636	418
Malthouse	3	2181	2181	–	130	144
BT04	7	6859	1131	5728	292	192
BT05	10	6051	1436	4615	261	190
Main office	2	1267	1267	–	district heat	84
Brewhouse	4	1500	1500	–	district heat	99
BT08	6	6166	2466	3700	275	239
BT09	10	7000	1000	6000	297	190
BT10	6	4246	4246	–	210	280
sum		64801	20051	44750	2721	2246

Figure 2. Exemplary result of the identification of the linearised zone model for one of the simulated zones.

Figure 3. Evaluation of the mean absolute error (MAE) over the look ahead time for each zone (blue). The mean over all zones is indicated in orange.

Figure 4. Comparison of the duration lines of the total thermal load of the apartment and commercial zones without EMS (woEMS) and with EMS (wEMS).

Figure 5. Co-simulation results without EMS (woEMS) and with EMS (wEMS). The upper plot shows the total heat flow from the energy hub. The lower plot shows the mean zone temperature of the apartment zones, weighted by the respective floor area, and the chosen comfort band of 22 $\pm$ 1 °C.

Figure 6. Results of the economic evaluation. The overall costs are about 9% lower when using the distributed EMS.