Study case

The electrical network in the case study is a section belonging to the network of the Electrical Engineering Building of Universidad Industrial de Santander. This building is located in Bucaramanga, Colombia, at 7°8’ North, 73°0’ West. During daylight hours (6 a.m. to 6 p.m.), the temperature ranges between 24 and 28 °C, while the solar irradiance varies between 2,0 and 7,6 kWh/m2 day, with an average of 4,9 kWh/m2 day ^{21) (22}. Fig. 2 presents the solar irradiance of one week for the month of June, 2018. The solar irradiance for that year was used for our case study.

Figure 2 Solar irradiance during a week for the month of June 

Fig. 3 shows a five-level building with an area of approximately 2.700 m2. The first four levels are for classrooms and student areas, and the fifth level is for the administrative area. Different strategies for the rational and efficient use of energy (REUE) have been implemented, such as natural lighting and ventilation, green roofs (580 m²), automation systems, and PV generation systems.

Figure 3 Electrical Engineering Building 

Fig. 4 shows the single-line diagram of the studied network (TP4), with each branch containing lighting loads, air extractors, and THWN 12AWG conductors. There is a TP4 distribution board for the electrical network on the fourth floor of the building. It consists of a lighting sub-board (TALU4) and a common coupling point (PCC) for the PV system installed on the terrace. The TALU4 contains ten single-phase circuits with dimmable luminaires, air extractors (classrooms), and on/off luminaires (bathrooms and corridors). The PV system has the following units: a solar tracker SS-1 (one panel), SFV1 (18 panels), SFV2 (18 panels), and SFV3 (six panels). Table I lists the general characteristics of the luminaires, air extractor, and PV systems.

Figure 4 Single-line diagram of the studied low-voltage network 

Table I General characteristics of the loads and PV system 

Modeling the elements

The studied distribution network is made up of a transformer, conductors, nonlinear loads, and a PV system. For these components (except for the transformer), it is necessary to establish a dependency on frequency changes in the frequency domain.

The transformer of the building has a capacity of 630 kVA and 13,2 kV/220 V, as well as a Δyn5 connection. This element is defined as the simulated network voltage source and, therefore, as an infinite power node or SLACK with a nominal phase voltage of 127 V.

The electrical conductors of the network circuits under study are of the THWN type and of different wire gauges. Their distance corresponds to the separation between the board and load. These elements are represented as frequency-dependent impedance Zh, as expressed in Eq. (1), where h indicates the harmonic order of the frequency under study, R is the resistance, and XL is the inductive reactance. Table II describes the characteristic of the electrical wires of each network circuit.

Table II Characteristics of the electric wires 

Finally, the nonlinear loads and the PV system are represented by Norton equivalent models of coupled and decoupled admittance matrices, as shown in Eqs. (2) and (3), respectively. This coupled approach allows the study of the harmonic interaction between the voltage and the current signals.

Here, represents the current vector of the device, frequency concatenated with the admittance matrix, is the vector of the voltage signal variations. The dimensions of the H and K models indicate the highest odd harmonic order used for the current and voltage signals, respectively.

Table III presents the coupled and decoupled Norton models for each nonlinear element (luminaires, extractors, and the PV system). The model for each single-phase load (luminaire or air extractor) was graphically represented by the real and imaginary parts of the admittance matrix for each operating condition. The PV system is a three-phase component, represented by six admittance matrices for each operating condition. The on/off luminaires and extractors have one operating condition, the dimmable luminaires have two operating conditions according to dimmer level (0 and 10 Vdc), and the PV system has three operating conditions according to solar irradiance (high and low) (23-26).

Table III Norton equivalent models for each element 

Adapted backward/forward (BF) iterative algorithm

Generally, the BF iterative sweep algorithm using Kirchhoff’s laws is employed to solve the power flows of distribution networks. It orders the network nodes to systematize the iterative process of calculating the currents that flow through the conductors from the currents injected into the nodes. Similarly, it helps to calculate the voltages in the lower nodes, starting from the voltage of the source node. Fig. 5 illustrates the nodal ordering applied to the scheme.

Figure 5 Example of nodal ordering scheme 

For this case study, adjusting the algorithm’s mathematical approach to the characteristics of the network allows considering unbalanced loads, harmonic distortion in the feeder, and the use of the Norton model to represent nonlinear loads and the PV system, as shown in Fig. 6.

Figure 6 Adapted backward-forward (BF) algorithm flowchart 

The dotted blocks in Fig. 6 highlight the process added or adjusted according to network characteristics, such as the input of the Norton models of nonlinear loads and the PV system and the initialization of the voltage signal for all k frequencies. Calculations within the iterative process are consecutively performed for each k frequency. The third, fifth, and seventh harmonic orders were the harmonic frequencies adopted for the study, as they were the most representative within the feeder signal. Therefore, the frequency value of k was 4.

There is a general stage involving the initialization of the conditions for the power flow, which includes the input of the data characteristics of the network, the input of the Norton models for each of the nonlinear loads and the PV system, the creation of arrays that represent the connections of the circuits in the network, and the input of the initial conditions of the voltage signal into each of the nodes of the network. Note that the initial voltage conditions were obtained from a prior review of the voltaje signals of the studied electrical network at different times of the day. In addition, the initial conditions of each node corresponded to the voltage of the SLACK bus.

The iterative process of this method consists of two sub-processes. First, the backward sweep calculates the injected currents in the load nodes and PV system according to the Norton model (BF-NC and BF-ND). The current IB flowing through branches is calculated using Eq. (4), where IL is the load or PV system current connected to the node, and Ii,N is the current of the branches i connected to the node N.

Next, the forward sweep is initialized and calculates the voltage drops of the branches, considering the I B currents and the specific impedances of each conductor. The node voltages were obtained progressively from the SLACK node to the farthest node. At the beginning of each forward sweep cycle, it is necessary to maintain the tension of the SLACK node in its initial conditions and apply Eq. (5), where VN−1 is the voltage of the upper node, VN is the voltage of the lower node, and IBZh represents the voltage drop in the branches.

The node voltage and branch current data are stored in arrays and exported to an Excel file. After finishing the iterative process, the percentage error in magnitude and the absolute error in phase angles between the current and previous iteration data are calculated to find the desired nodal voltage Branch current values.

Modeling the electric network in Simulink

The main objective of this study was to analyze the performance of the BF algorithm adjusted and programmed in MATLAB. To corroborate the results obtained by the BF-ND strategy, the electrical network was modeled in Simulink, considering the use of the decoupled Norton model to represent the loads and the PV system (SIM-ND strategy).

Simulink is a tool that solves power flows using the approach and solution of the state equations of the model in the frequency domain.

The modeling of the studied network considered its characteristics and the most representative odd harmonic orders (third, fifth, and seventh). The network was represented by four subsystems, one at each frequency (Fig. 7). In addition, there is a subsystem containing result export blocks from Simulink to MATLAB in order to facilitate the treatment of the data and compare them with the values estimated by the BF algorithm. The MATLAB Function and Controlled Current Source blocks allow modeling the loads and the PV system in each subsystem by applying only the Norton decoupled matrix model.

Figure 7 General structure of the Simulink model 

Operation scenarios

Table IV presents 18 scenarios for the operation of the electricity grid under study, which are based on two inputs. The first input corresponds to the operating conditions of the dimmable luminaires (0 and 10 Vdc), the on/off luminaires, the air extractors (on), and the PV system for two levels of solar irradiance (high-H and low-L). The second input is the representativeness of the number of devices in simultaneous operation.

Table IV Operation scenarios for the study case 

Data processing

At the end of each simulation, the magnitude and phase-angle data of the node voltages, Branch currents, and neutral currents were stored in an Excel file corresponding to each operation scenario. The magnitude and phase angle data were discriminated by each harmonic order under study (fundamental, third, fifth, and seventh).

After the simulation of the operation scenarios had been completed, the normalized root mean square errors (NRMSEs) were calculated by importing the data stored in Excel to the MATLAB Workspace. These errors were estimated from the waveforms obtained using the three analyzed strategies. Similarly, the RMS values of the voltage node and current in the branches were calculated. Finally, the power parameters were estimated to analyze the performance of the strategies in terms of the apparent, active, and non-active power, as well as the power factor.

Evolution of the performance of strategies

Fig. 8 shows the evolution of RMS errors in voltage for node TP4 (VTP4) and the current for Branch SLACK-TP4 (ISLACK-TP4), considering the operation scenarios. These errors were calculated for each scenario by selecting average values for each iteration. The results obtained by the BF-ND and BF-NC strategies are acceptable at the end of the second iteration, with average errors of 2,5e5% for the voltaje and 2,0e-2% for the current.

Figure 8 Evolution of the estimation error of the RMS voltage and current values at each iteration 

RMS values

Fig. 9 shows the single-line scheme of the studied electrical network that relates the RMS values of nodal voltage and branch current obtained by applying the BF algorithm with decoupled Norton modeling (Fig. 9a) and coupled Norton modeling (Fig. 9b). Specifically, it shows the results for the estimates of Scenario 22, where all the luminaires and air extractors work and the PV system operates under low solar irradiance, which implies high levels of THDi ²⁵.

Figure 9 Voltages and currents for operation scenario 22. The blue, green, and red values refer to voltaje or current values for phases A, B, and C, respectively 

There was a difference in the VRMS voltage drop values between the lower and SLACK nodes. The two strategies exhibit a similarity of VRMS values, with differences of less than 0,1 V in most load circuits and TP4 and TALU4 nodes.

In terms of the IRMS, there is a significant difference between the values of the two strategies because the coupled model correctly estimates the actual waveform of the current consumed by the load, unlike the decoupled model ^23,24,26. This difference is more noticeable in the PV system IRMS injected with differences of 2,7-3,6 A due to the low satisfaction in the estimation of currents by the decoupled model for low solar irradiance ²⁵.

Figs. 10 to 13 show the errors of the RMS values calculated for the three strategies. Fig. 10 shows that the VRMS errors in the TP4 node do not exceed 0,08% in scenarios where the PV system operates at low solar irradiance (e.g., E2, E4, E6, and E9). In comparison, the errors are less than 0,02% for scenarios in which the PV system operates at a high solar irradiance level (e.g., E1, E3, E4, and E7).

Figure 10 Voltage signal RMS value errors in node TP4 for the operation scenarios 

In contrast, Figs. 11 to 13 show the estimation errors regarding the IRMS. Fig. 11 compares the performance of IRMS error strategies for the SLACK-TP4 branch in scenarios where the PV system operates at low solar irradiance levels (E2, E4, E6, E9, E14, E16, E18, E20, and E22). The resulting errors are between 20 and 150 %. These differences between the strategies are caused by the power signal estimation capability of the Norton decoupled model (ND) compared to the Norton coupled model (NC), the latter being the most successful.

Figure 11 SLACK-TP4 branch current signal RMS values errors for operation scenarios 

This may be a consequence of that observed in Fig. 12, where the errors of the IRMS injected by the PV system in scenarios where it operates at low solar irradiance are greater than 20 %. However, in the specific scenarios E16, E18, E20, and E22, phase B and phase C errors exceed 150% when comparing SIM-ND strategies against BF-NC.

Figure 12 TP4-PV system branch current signal RMS value errors for the operation scenarios 

Fig. 13 shows IRMS errors greater than 1,0% in scenarios E1, E3, E15, E17, and E21, where the dimmable luminaires operate with the lowest dimmer level (0 Vdc), which provides greater harmonic distortion (THDi >18 %) than other levels of control (THDi <15 %) (²³, ²⁴.

Figure 13 TP4-TALU4 branch current signal RMS value errors for operation scenarios 

Waveforms and normalised root mean square errors (NRMSEs)

Fig. 14 presents the characteristics of the waveform of the SLACK bar and those estimated using the three strategies (SIM-ND, BF-ND, and BF-NC). There is a predominance of the flat-top waveform type in these waveforms, where the fifth harmonic order prevails over the third and seventh orders. Similarly, the voltage waveforms obtained using the three strategies are significantly similar. Fig. 15 shows a comparison of the NRMSEs of the voltage signals obtained via the three strategies for the TP4 node in each operation scenario. In general, the values were less than 0,05 %. However, in scenarios where the PV system injects power at a low solar irradiance level and the dimmable luminaires operate at their maximum dimmer level (10 Vdc - E14, E16, E20, and E22), phase A errors are between 0,05 and 0,1% when comparing SIM waveforms-ND with BF-NC, and BF-NC with BF-ND.

Figure 14 Voltage signal waveforms for all operation scenarios 

Figure 15 Voltage normalized root mean square errors (NRMSEs) at node TP4 for all operation scenarios 

This is the result of an overload in phase A luminaire circuits and the PV system. Similarly, the estimation of the current signal injected by the PV system with the coupled and decoupled Norton models shows differences when analyzing low solar irradiance levels. ²⁵ found that the coupled model estimates the waveform with an NRMSE of less than 2 %, in contrast to the decoupled model, with an error close to 20 %. It should be noted that, although there is prior knowledge of the current performance of Norton models in the modeling of loads and PV systems, this study aims to review the possible impacts of using these models on the voltage and power parameters.

The waveforms of the current signals (Figs. 16, 17, and 18) reveal the differences between the current signals estimated using each of the three strategies.

Fig. 16 clearly shows the differences between the waveforms of the current signals in the SLACK-TP4 branch, considering scenarios where the PV system operates at high solar irradiance (Fig. 16a) and scenarios with low solar irradiance levels (Fig. 16b). The reason for this is the impossibility of the decoupled Norton model to estimate the current signal successfully at this PV system operation level.

Figure 16 Current signal waveforms on the SLACK-TP4 branch for all operation scenarios 

Fig. 17 shows the current waveforms of the TP4-TALU4 branch, where the signals estimated by the three strategies are very similar. However, in the high solar irradiance scenarios (Fig. 17a), the máximum peak current value is 10 A. In the low-solar-irradiance scenarios (Figure 17b), the peak value is 20 A. This is because of the close relationship between the dimmer levels of the dimmable luminaires and the operation of the PV system.

Confirming what is shown in Fig. 16, Fig. 18 reveals the differences between the waveforms of the currents injected by the PV system operating at the two levels of solar irradiance: high (Fig. 18a) and low (Fig. 18b), demonstrating the inability of the decoupled Norton model to estimate the injected current signal at low solar irradiance.

In general, the current waveform injected by the PV system at high solar irradiance levels (Fig. 18a) predominantly affects the current of the SLACK-TP4 branch above the current waveforms from the TALU4 board (Fig. 18a). In contrast, in scenarios with low solar irradiance, the waveform of the current from the TALU4 board (Fig. 18b) is predominant in the SLACK-TP4 branch current above the PV system waveform (Fig. 18b) because the luminaires and air extractors operate at their máximum power, whereas the PV system does not.

Figure 17 Current signal waveforms on the SLACK-TP4 branch for all operation scenarios 

Figure 18 Current signal waveforms on the SLACK-TP4 branch for all operation scenarios 

Figs. 19and 20 present the NRMSEs of the waveforms of the current signals in the TP4-PV and TP4-TALU4 branches. Fig. 19 shows the scenarios where the low solar irradiance levels (E2, E4, E6, E9, E14, E16, E18, E20, and E22) work; there is a difference between the waveforms estimated by BF-NC and those estimated by SIM-ND and BF-ND, obtaining NRMSEs greater than 20 %.

Figure 19 Current NRMSEs in the TP4-PV branch for all operation scenarios 

In contrast, in scenarios E1, E3, E7, E8, E15, and E17, the NRMSEs for the estimated current signals were less than 1,0% since the PV system operated at a high irradiance level.

Fig. 20 shows the NRMSEs of the current waveforms of the TP4-TALU4 branch. The NRMSEs obtained are greater than 3,9% for the scenarios where the dimmable luminaire operates at a low dimmer level (0 Vdc - E1, E3, E15, E17, E19, and E21), contrary to the scenarios in which the luminaire operates at a high dimmer level (10 Vdc - E2, E4, E16, E18, E20, and E22), with NRMSEs of less tan 3,3 %.

Figure 20 Current NRMSEs in the TP4-TALU4 branch for all operation scenarios 

The reason for these errors is that the dimmable luminaires, at the low dimmer level (0 Vdc), have a THDi index higher than 19 %, and, at a high dimmer level (10 Vdc), the THDi index is less than 13% ²⁴.

Factors of influence

The results of the waveform errors of the current signals allow analyzing the potential impact of operating the extractors (minimum and maximum use) and the luminaires (minimum and máximum use and high 10 Vdc dimmer level) in scenarios where the PV system operates at low solar irradiance (E2, E4, E6, E9, E14, E16, E18, E20, and E22). Fig. 21 presents the matrix representations of the impact on the waveform of the current signal of the TP4-PV and TP4-TALU4 branches of the network under study.

Figure 21 NRMSE load influence matrix for the TP4-TALU4 branch 

According to Figs. 21a and 21b, the number of luminaires or air extractors does not influence the waveform errors of the current signals of the branch feeding the TALU4 sub-lighting board. Similarly, NRMSEs close to 3,0 and 4,0% indicate the impact of using the coupled Norton model to represent the luminaires and air extractors, respectively.

Figs. 21c and 21d allow stating that the presence of the luminaires and air extractors does not affect the waveform of the current signal of the PV system, confirming that this estimation depends on the load model used. In this regard, the coupled Norton model more accurately estimates the current signal of the PV system when operating at low solar irradiance levels ²⁵.

Analysis of the power parameters

Considering the results obtained for the waveforms in Section 3.3, an analysis of the power parameters (apparent power, active power, non-active power, and power factor) was performed for all the operation scenarios. Similarly, the errors in these parameters were calculated from the results of the BF-NC strategy.

Fig. 22 shows the power parameter values of the main network branches (SLACK-TP4, TP4-TALU4, and TP4-PV) for scenario 21. This figure reveals the imbalance between the phases of the PV system and the load circuits. In addition, the power factor has values below 0,97 due to the capacitive nature of the loads, which produces an increase in non-active power. It should be noted that the differences in the apparent and active power values and, consequently, in the non-active power values are due to the coupled and decoupled Norton models’ estimation capacity regarding the current signals.

Figure 22 Estimated power parameters for scenario 21 

As for the errors in the estimated power parameters when comparing the two BF strategies, Fig. 23 shows the relationship between these variables.

Figure 23 Percentage errors of power parameters (BF-NC vs BF-ND) 

Fig. 23a presents the apparent power errors for all the operation scenarios. The errors for the TP4-TALU4 branch are higher than 0,7% in the scenarios where the dimmable luminaire works at its low dimmer level (0 Vdc), contrary to those where the luminaire works at maximum dimmer level (10 Vdc), where the errors are less than 0,8 %. Similarly, the errors in scenarios where the PV system operates at low solar irradiance are above 20 %. In comparison, the errors do not exceed 5% at high solar irradiance.

Fig. 23b presents the errors regarding active power. For the TALU4 lighting sub-board, values between 0,1 and 0,4% can be observed in all operation scenarios. However, the active power errors in the branch from the feeder to the general floor panel (SLACK-TP4) are influenced by the active power errors injected by the PV system. Low solar irradiance scenario errors are between 20 and 90 %, which is reflected in the SLACK-TP4 branch errors, mainly when the PV system works with air extractors.

For the non-active power errors in Fig. 23c, the values are higher than 10% in scenarios where the dimmable luminaires operate at a low dimmer level (0 Vdc). These errors are in the order of 60 and 80 %, respectively, when operating at a high dimmer level (10 Vdc). In addition, the PV system operating at low solar irradiance affects the non-active power errors of the feeder branch (SLACK-TP4).

Following the non-active power errors, the same behavior can be observed for the power factor errors of the TP4-PV branch. Therefore, the error values of the feeder branch to the main floor panel (TP4) are between 30 and 90% for these drivers.

Performance analysis of the strategies

Tables V and VI present a summary of the performance analysis of the studied strategies in terms of the waveforms and RMS values of the voltage, current, and power parameters. The following symbol conventions is used in the Tables, ✓: poor (Error >10), ✓✓: ordinary (6 <Error <10), ✓✓✓: acceptable (1 <Error <5), ✓✓✓✓: good (0,1 <Error <0,9), and ✓✓✓✓✓: excellent (Error <0,1).

The BF-NC strategy was established as a reference for the performance analysis because, within the load models in the frequency domain studied in this research, the Norton model equivalent to the coupled admittance matrix presented in the literature accurately estimates the current of a nonlinear load ^23,25,27-30.

According to the results in Table V, the three strategies used exhibit excellent performance (0,03-1,3 %) in estimating the waveforms and RMS values of the voltage signals of the nodes of the electrical network. In terms of current, Table V shows the excellent performance of the BF-NC strategy, as the harmonic interaction is considered in the modeling of the PV system and the loads. However, the BF-ND and SIM-ND strategies, when compared against BF-NC, showed an acceptable performance (5,0 %) in scenarios of high solar irradiance, as well as a poor performance (>10 %) in scenarios of low solar irradiance.

Similarly, Table VI shows the performance analysis of the apparent power (S), active power (P), non-active power (Q), and power factor (fp) parameters. The performance of the BF-ND and SIM-ND strategies regarding non-active power for high and low solar irradiance scenarios is poor (>10 %) because of the behavior of dimmable luminaires working together at the lowest dimmer level (0 Vdc) and the operation of the PV system at low solar irradiance levels.

Table V Performance of the studied strategies in terms of waveforms and RMS voltage values 

Note: ✓- Poor, ✓✓- Ordinary, ✓✓✓- Acceptable, ✓✓✓✓- Good, ✓✓✓✓✓- Excellent.

Table VI Analysis of performance of strategies in terms of power parameters 

Note: ✓- Poor, ✓✓- Ordinary, ✓✓✓- Acceptable, ✓✓✓✓- Good, ✓✓✓✓✓- Excellent.