Importancia de la información financiera ambiental que el Contador revela.

In order to develop the spectrum density model and improve the accuracy of system identification model, exciting signals were generated and injected into the system during the training period. These exciting signals include zone temperature setpoints, internal equipment and occupancy schedules. Therefore, certain constraints added into our system excitation process, for example, the boundary of temperature setpoints, the minimum temperature setpoint or equipment schedule updating time span, and so on. Different system excitation strategies, such as Pseudo- Random Binary signal, Pseudo-Radom Sequences, Multi-sine signal, have been discussed and applied in on-linear process systems in different areas [150, 158, 159]. However, there are few publications on building energy system excitation found in existing literature. Pseudo-random Binary Sequence (PRBS) excitation signals (Eq. 3.3) for building temperature setpoints was generated and applied in [76].

𝑇𝑠𝑝,𝑖(𝑘) { 21, 25, 25, 𝑒𝑥𝑐𝑖𝑡𝑒𝑑 𝑧𝑜𝑛𝑒, 𝑃𝑅𝐵𝑆 = 0 𝑒𝑥𝑐𝑖𝑡𝑒𝑑 𝑧𝑜𝑛𝑒, 𝑃𝑅𝐵𝑆 = 1 𝑛𝑜𝑛 − 𝑒𝑥𝑐𝑖𝑡𝑒𝑑 𝑧𝑜𝑛𝑒𝑠 Eq. 3.3

3.3.1 Excitation Signal Generation Function Selection

Sum of sinusoids (SINE) model is used to generate the exciting signals (Eq. 3.4), because sinusoids signals are versatile periodic and can adjust signal shape and character of the power spectrum by adjusting their parameters.

𝑈𝜏+1= 𝑈𝜏+ √2𝑎𝜏𝑠𝑖𝑛(𝜔𝜏𝑡𝑇 + 𝜑𝜏) Eq. 3.4

Where 𝑈𝜏+1 is the excitation signal; √2𝑎𝜏 is a magnitude scale parameter from 0 to 1; 𝜔 is periodic frequency parameter from 0 to 2𝜋 ; 𝑇 is the sampling time, and 𝜑 is the phase lag parameter from 0 to 2𝜋, which do not affect the signal spectrum. Lowering T will result in a higher frequency bandwidth [158]. Another benefit of using sum of sinusoids input signals is that they enable the user to directly specify the shape and character of the power spectrum. The guidelines of excitation signal generation function parameter determining from Rivera et al. [175] is applied to ensure the signals contains necessary frequency information:

𝟏

𝜷𝒔𝝉𝒅𝒐𝒎𝑯 ≤ 𝝎𝝉≤

𝜶𝒔

𝝉_𝒅𝒐𝒎𝑳 Eq. 3.5

where, 𝜏𝑑𝑜𝑚𝐻 and 𝜏𝑑𝑜𝑚𝐿 correspond to the high and low estimates of the dominant time constant of the system (denote the slowest and the fastest systems time constants) [158]. 𝛼𝑠 and 𝛽𝑠 are user- decisions on high and low frequency content based on identification requirement. Typically, 𝛼𝑠 is 2 and 𝛽𝑠 is 3, corresponding to 95% of settling time [175]. In additional to function magnitude scale parameter, periodic frequency parameter and phase lag parameter, harmonic suppression is another important parameter which can decompose the output signal to obtain a more accurate estimate of the linear and nonlinear components.

3.3.2 Excitation Function Parameter Specification and Data Generation

The procedure of excitation signals design is summarized in Figure 3-1. The response time constant of a dynamic system is a measure of how quickly the system responds to an input change. It is usually determined by experiments. For example, the impulse response of a dynamic system can be expressed as:

𝑥(𝑡) = (𝛼/𝑇)𝑒−𝑡/𝑇 _{Eq. 3.6}

where, T is the response time constant, 𝛼 is a state parameter. The response time for the system output, x(t), to reach 95% of its final steady state value after an input change, is defined as 𝑇.95For the building in this studied project, 𝜏𝑑𝑜𝑚𝐻 = 360 minutes, and 𝜏𝑑𝑜𝑚𝐿 = 30 minutes; 𝛼𝑠 determines the high frequency content in the excitation signal and represents the response speed. 𝛽𝑠 specifies low frequency information corresponding to the system settling time. 𝛼𝑠 and 𝛽𝑠 are chosen to be 2 and 3 for the 95% of the system settling time, respectively. For the temperature setpoint excitation signals in this project, 𝑇𝑚𝑎𝑥 = 32𝑜𝐶 (90𝑜𝐹) and 𝑇𝑚𝑖𝑛 = 10𝑜𝐶 (50𝑜𝐹); while for the schedule ratio excitation signals, 𝑅𝑚𝑎𝑥 = 1 and 𝑅𝑚𝑖𝑛= 0.

3.3.3 Excitation Signal Injection Interval and Data Sampling Length Determination

According to Braun’s guidelines, the excitation injection frequency cannot be too high. The injection time step should be larger than the system response time. The building response time is related to building thermal mass, which can only be found through experiment test. The building studied in this project is a small light building. It system response time is relatively short. A parametric experiment test has been conducted to find out the best injection frequency. However, once the system excitation frequency changed, the building dynamics would change, and the power density model sapling length should also be updated accordingly. Table 3-2 presents the excitation frequency and sampling length testing results. Finally, 30 minutes with 6 hours and 30 minutes with 4 hours have been chosen for excitation frequency and sampling length for core zone model and perimeter zone model, respectively.

Table 3-2. Excitation frequency and sampling length testing summary

R2 Excitation Frequency Sampling Length Excitation Frequency Sampling Length Excitation Frequency Sampling Length

15 Min 3 Hour 30 Min 4 Hour 60 Min 6 Hour Training

Period 94.65% 93.70% 92.76%

Forecasting

Period 97.30% 95.30% 91.91%

3.3.4 Excitation Signal Evaluation

A primary goal of the system excitation is to lead to produce enough data which can discriminate between any two different models in the data set. The requirement for the informative experiments for open loop operation means that the input should be persistently exciting, and contains sufficiently many distinct frequencies. The covariance matrix is typically inversely proportional to the input power. The desired property of the waveform is defined in

terms of the crest factor (𝐶𝑟), which shows the ratio of peak values to the average value of the

signal (𝑢). The crest factor is defined as:

𝐶𝑟2= 𝑚𝑎𝑥 𝑢2_(𝑡) 𝑙𝑖𝑚 𝑁→∞ 1 𝑁∑𝑁𝑡=1𝑢2(𝑡) Eq. 3.7

A good signal waveform is consequently one that has a small crest factor. Therefore, a constrained minimum crest factor optimization problem is created to determine the parameters in excitation signal generation function:

𝐽(𝑎𝜏, 𝜔𝜏, 𝜑𝜏) = min 𝑎𝜏,𝜔𝜏,𝜑𝜏

𝐶𝑟, 𝜏 = 1, 2, ⋯ 𝑁 Eq. 3.8

subject to Eq. 3.9 and other hard boundaries (Eq. 3.10) for building temperature setpoints and equipment schedule:

10℃ = 𝑈𝑇,𝜏𝑚𝑖𝑛< 𝑈𝑇,𝜏< 𝑈𝑇,𝜏𝑚𝑎𝑥 = 32 ℃ Eq. 3.9

0 = 𝑈𝐸,𝜏𝑚𝑖𝑛< 𝑈𝐸,𝜏 < 𝑈𝐸,𝜏𝑚𝑎𝑥= 1 Eq. 3.10

Following the procedure discussed above, the exciting signals for temperature setpoints and equipment schedules used in this study are generated and shown in Figure 3-2, where all the excitation signals are injected into the building model every 30 minutes.

Figure 3-2. Building operation excitement

3.3.5 Training and Validation Data Generation

The system excitation signals discussed in section 2.3 was modeled and generated in Matlab. In order to apply these excitements into EnergyPlus model, BCVTB is used to exchange data between Matlab and EnergyPlus. Here BCVTB plays a master role in data exchange between Matlab and EnergyPlus through run-time coupling, as shown in Figure 3-3. During the entire study, typical meteorological year (TMY) weather data for Philadelphia is used. During the training and validation period, excited and unexcited building control signals will be sent to EnergyPlus model following the procedure in Figure 3-1, respectively. Simulation results and control signals will be sent back and stored in Matlab for system identification model training and validation. The length of training time is changeable according to the forecasting accuracy requirement.

Figure 3-3. Building operation data for on-line model training and validation