We investigate the interactions of multiple solar PV inverters in voltage regulation on a different feeder of Southern California Edison. The Fontana feeder is a 12 kV distribution system about one mile from the local substation. The entire length of the interconnected distribution circuit is 7.8 miles including all mainline and branch circuits. Voltage regulation of the circuit is accomplished by switched capacitor banks placed along the length of the mainline of the distribution circuit. The capacitors are controlled using a time schedule with a voltage override. The voltage override operating set points are adjusted automatically if the ambient temperature, measured at the capacitor bank controller, is above 90 F. Voltage regulation at the substation is also accomplished using switched capacitors located at the substation. The capacitors at the substation are operated to both regulate the voltage at the substation bus bars and compensate var flows in the subtransmission system.
Figure 3.11: Schematic diagram of a distribution feeder with high penetration of Photovoltaics. Bus No. 1 is the substation bus and the 6 loads attached to it model other feeders on this substation.
Table 3.3: Network of Fig. 3.11: Line impedances, peak spot load KVA, capacitors, and PV generation’s nameplate ratings.
Network Data
Line Data Line Data Line Data Load Data Load Data PV Generators
From To R X From To R X From To R X Bus Peak Bus Peak Bus Nameplate
Bus. Bus. (Ω) (Ω) Bus. Bus. (Ω) (Ω) Bus. Bus. (Ω) (Ω) No. MVA No. MVAR No. Capacity 1 2 0.259 0.808 8 41 0.107 0.031 21 22 0.198 0.046 1 30 34 0.2
2 13 0 0 8 35 0.076 0.015 22 23 0 0 11 0.67 36 0.27 13 1.5MW
2 3 0.031 0.092 8 9 0.031 0.031 27 31 0.046 0.015 12 0.45 38 0.45 17 0.4MW 3 4 0.046 0.092 9 10 0.015 0.015 27 28 0.107 0.031 14 0.89 39 1.34 19 1.5 MW 3 14 0.092 0.031 9 42 0.153 0.046 28 29 0.107 0.031 16 0.07 40 0.13 23 1 MW 3 15 0.214 0.046 10 11 0.107 0.076 29 30 0.061 0.015 18 0.67 41 0.67 24 2 MW 4 20 0.336 0.061 10 46 0.229 0.122 32 33 0.046 0.015 21 0.45 42 0.13
4 5 0.107 0.183 11 47 0.031 0.015 33 34 0.031 0 22 2.23 44 0.45 Shunt Capacitors 5 26 0.061 0.015 11 12 0.076 0.046 35 36 0.076 0.015 25 0.45 45 0.2 Bus Nameplate 5 6 0.015 0.031 15 18 0.046 0.015 35 37 0.076 0.046 26 0.2 46 0.45 No. Capacity 6 27 0.168 0.061 15 16 0.107 0.015 35 38 0.107 0.015 28 0.13
6 7 0.031 0.046 16 17 0 0 42 43 0.061 0.015 29 0.13 Base Voltage (KV) = 12.35 1 6000 KVAR 7 32 0.076 0.015 18 19 0 0 43 44 0.061 0.015 30 0.2 Base KVA = 1000 3 1200 KVAR 7 8 0.015 0.015 20 21 0.122 0.092 43 45 0.061 0.015 31 0.07 Substation Voltage = 12.35 37 1800 KVAR
8 40 0.046 0.015 20 25 0.214 0.046 32 0.13 47 1800 KVAR
8 39 0.244 0.046 21 24 0 0 33 0.27
Figure 3.12: Joint distribution of the normalized solar output level and the normal- ized load level
3.5.1 Simulation setup
In this section we present an example to illustrate the effectiveness of our fast- timescale VVC control, compared with the current PV integration standards IEEE 1547 which require all the inverters to operate at unity power factor and not partici- pate in VAR control of the distribution circuit [67].
We use the 47-bus distribution feeder shown in Fig. 3.11 in our simulations. This circuit is a simplified model of an industrial distribution feeder of SCE with high penetration of renewables, integrated with 5 large PV plants. The network data, including line impedances, peak MVA demand of loads, and the nameplate capacity of the shunt capacitors and the PV generations, are listed in table 3.3. For all the loads in this industrial area, we assume a constant power factor of 0.8.
To focus on the fast-timescale control, we fix the trajectory of slow-timescale control (v0(T), c(T),
T = 1, . . . , M) to be the settings used in practice, and only solve problem (3.14)–
(3.15) for each time t. We use real data for load and solar generation over the course of a year. Using minute based data for one year, the empirical joint probability
Figure 3.13: Overall power savings in MW, for different load and solar output levels assuming a 3% voltage drop tolerance.
distribution of the load and solar output levels is shown in Figure 3.12. We have used a hot color scale, with hotter colors representing a higher probability for the system to spend time in. The axes are scaled to the peak power demand and total capacity of installed PV generation, respectively. We assume ni = 1 for all loads, corresponding to a constant current load model for this feeder. We further constrain the reactive power flow through the substation bus to be less than 2MVAR at all times. To compare the cost of the operation under the IEEE 1547 standard and our proposed algorithm, we choose a fixed voltage drop tolerance.
We implemented the proposed convex relaxation of the original VVC optimization problem and solved it using CVX optimization toolbox [69] in Matlab. In all our sim- ulations, we checked the inequality constraint in condition (3.28) for optimal solutions of the relaxed problem and confirmed that the inequality constraints were all active.
Figure 3.13 shows the difference in the objective function between current standard IEEE 1547 standard and our proposed solution for different load and solar output levels, given a 3% voltage drop tolerance. This can be interpreted as the overall power savings achieved by optimal control of inverter’s VAR dispatch. Interestingly, in typi-
Table 3.4: Simulation Results For Some Voltage Tolerance Thresholds Voltage Drop Annual Hours Saved Spending Average Power Tolerance Outside Feasibility Region Saving
3% 842.9h 3.93%
4% 160.7h 3.67%
5% 14.5h 3.62%
cal loading conditions, optimal inverter control can save more than 200KW in overall power consumption, even at night. The missing region in Figure 3.13 is the set of the infeasible settings of load and solar for the IEEE 1547 operation mode in the relaxed problem formulation, and is therefore also infeasible for the original problem. The feasible operation region for the proposed method is considerably a superset of that of IEEE 1547 operation mode. This demonstrates significant power quality benefits achieved by optimal inverter control. Table 3.4 summarizes the overall time that the feeder can save not spending outside the feasible region, and the overall operation cost benefits for the whole year, for different voltage drop tolerances. The results show more than 3% savings achieved by optimal reactive power dispatch for the inverters.