Effects of Load-Based Frequency Regulation on Distribution Network Operation

The illustration in Fig. 1 demonstrates how the actions of aggregated DERs could affect network operation. For instance, voltages may vary more when loads provide regulation, which may increase the prevalence of voltage violations. If EVs are also on the network, their actions (charging or discharging) may improve some constraint violations and worsen others.

We use GridLAB-D [14] to run power flow simulations of distribution networks with time-varying loads. GridLAB-D performs quasi-steady state analysis using a Newton-Raphson algorithm at each time step to solve for a network’s three-phase, unbalanced power flow solution. We also use GridLAB-D’s dynamic, physics-based models of heating, ventilation, and air conditioning (HVAC) systems. We test networks using a 1 hour simulation with 2 second time steps. To ensure appropriate initialization of dynamic states, we simulate the 24 hours that precede the test hour (using a 30 second time step for the first 23.5 hours and a 2 second time step for the last 0.5 hours).

We use network models from the Pacific Northwest National Lab’s (PNNL’s) prototypical feeder database [14]. These models are of actual networks and are prototypical in that they represent common network types in the U.S. [15]. The networks include fuses, voltage-regulators, capacitor banks, and distribution transformers. Each network is paired with the typical meteorological year (TMY) hourly weather data for its region, which provides the HVAC models with realistic ambient temperature inputs.

The original feeder models have only one planning load (ZIP model) per distribution transformer; we require higher resolution load models. We disaggregate the transformer-level loads using a method provided by PNNL [16] that estimates the number of houses represented by each planning load and then constructs a model for each house. The house model comprises individual load models (HVAC, water heaters, and pool pumps) and ZIP models that aggregate all other loads in the house. Other modifications made to improve realism are described in Appendix A.

GridLAB-D’s HVAC model includes a thermal model of a house, as well as a model of the space-conditioning devices. The disaggregation method [16] selects randomized parameter values (e.g., wall insulation, house footprint size, temperature setpoints), such that each HVAC model is unique. Throughout this paper we use the term “AC model” to refer to the thermal model of the house, as well as the air conditioner device itself. GridLAB-D’s AC model is based on [17] and has three dynamic states: indoor air temperature $\theta$, mass temperature $\theta_{\text{m}}$, and on/off status $S$ [18]. These variables evolve according to

where $\theta_{\text{amb}}$, $Q_{\text{g}}$, and $Q_{\text{ac}}$ are time varying inputs that represent outdoor air temperature, internal heat gain, and cooling capacity of the air conditioner, respectively. The constant parameters are defined as: $C_{\text{a}}$ the thermal mass of indoor air, $C_{\text{m}}$ the thermal mass of indoor mass, $U_{\text{a}}$ the overall heat transfer coefficient, $H_{\text{m}}$ the coefficient of heat transfer between indoor mass and indoor air, $r$ the ratio of the internal gains absorbed by air to the total internal gains, and $\theta_{\text{low}}$ and $\theta_{\text{high}}$ the lower and upper limits of the AC’s temperature deadband.

We non-disruptively [9] control ACs with on/off commands that maintain indoor temperature within the user-set deadband. We use the simplest form of the probabilistic dispatch method (see [19, 20, 10]) in which a switching probability is broadcast to all ACs and, based on this probability, each AC individually decides whether to switch. Specifically, when the switching probability $u$ is broadcast at the $k$th time step, the $i$th AC will switch if it is available for external control and $p_{i}(k)<u(k)$, where $p_{i}(k)$ is drawn from the uniform distribution $\mathcal{U}(0,1)$. An AC is “available” for external control if three conditions are satisfied: 1) its indoor air temperature is within the temperature deadband; 2) the AC has not switched within the last two minutes; and 3) if the AC were switched, it would not reach its temperature limit in under two minutes, as predicted by a model. The first condition ensures the control will be non-disruptive to the user and the last two conditions protect the unit’s compressor by preventing excessive switching.

We use the simple proportional control scheme from [10] to calculate the switching probability $u$. For a desired power level $P_{\text{des}}$, the switching probability is calculated as

where the parameter $K_{\text{P}}$ is a proportional gain, $P_{\text{meas}}$ is the measured power of the AC population, $\bar{P}_{\text{ON}}$ is the average power of ACs that are on in steady-state, and $N_{\text{AC}}$ is the number of ACs that are available to switch. We assume that $N_{\text{AC}}$ and $P_{\text{meas}}$ are measured perfectly.

The primary aging mechanism for distribution transformers is the deterioration of coil insulation due to heat from resistive losses [21]. To estimate transformer aging, we use GridLAB-D’s built-in model, which is based on sections 5-7 of IEEE Standard C57.91-1995 [21]. Figure 2 shows three key variables within the model: the transformer’s load, winding temperature, and estimated minutes aged.

The model has two dynamic states: $\tilde{\theta}_{\text{oil}}$ the difference between the transformer’s top-oil temperature and ambient temperature (i.e., $\theta_{\text{oil}}-\theta_{\text{amb}}$); and $\tilde{\theta}_{\text{w}}$ the difference between the hot-spot winding temperature and the top-oil temperature (i.e., $\theta_{\text{w}}-\theta_{\text{oil}}$). Given these states, a transformer’s thermal dynamics are described by

where $\tilde{\theta}_{\text{oil,u}}$ and $\tilde{\theta}_{\text{w,u}}$ represent the ultimate temperatures that would be reached if the present load were sustained indefinitely [21]. The ultimate temperatures are computed as $\tilde{\theta}_{\text{w,u}}=\tilde{\theta}_{\text{w,R}}L^{1.6}$ and $\tilde{\theta}_{\text{oil,u}}=\tilde{\theta}_{\text{oil,R}}\big{(}(L^{2}l_{\text{fl}}/l_{\text{nl}}+1)/(l_{\text{fl}}/l_{\text{nl}}+1)\big{)}^{0.8}$, where $L$ is the time-varying load (per unit). The oil time constant $\tau_{\text{oil}}$ is computed according to equations (14) and (15) in [21]. Values for the thermal parameters in the above equations are derived from data found in [21, 22, 23, 24] (see Table 1).

The final stage of the model calculates a transformer’s aging rate $F_{\text{AA}}$ according to the empirically derived formula [21]

where $\theta_{\text{w}}$ is the winding temperature and $\theta_{\text{w}}=\theta_{\text{amb}}+\tilde{\theta}_{\text{oil}}+\tilde{\theta}_{\text{w}}$. The nominal aging rate is equal to one and occurs when $\theta_{\text{w}}=110^{\circ}$C. Note that, in this model, the aging rate is not dependent on the transformer’s age.

We test feeders with a scenario designed to reveal the negative effects of load-based regulation. In this scenario, we assume 100% of residential ACs are controllable, we test networks during the peak-load hour of the year, and we maximize the amplitude of the regulation signal. We find each feeder’s peak-load hour by running a simulation over the summer months without regulation. We compare this simulated peak-load to the feeder’s original planning load. If the simulated peak is less than 90% of the planning load, we use [16] to iteratively adjust the simulated peak by repopulating the feeder with more houses until the peak is between 90% and 100% of the planning load. For the regulation signal, we select a segment of the PJM Reg-D signal [25] in which the energy consumed during the regulation hour is equal to that of the base case. We scale the signal such that its maximum magnitude is 0.4 p.u. – the largest magnitude the AC population can track before performance begins to deteriorate.

We assess the effects of regulation by identifying changes in network variables between the base and regulation cases. Variables of interest and their corresponding constraints are listed in Table 2. First in the table are constraints on the voltage magnitude at service nodes, which are where service lines connect to distribution transformers. The continuous voltage limits are based off of ANSI standard C84.1 [26] and are only violated if surpassed for over two minutes. The emergency limits are from [27] and are similar to those in [26] and [28]. Second in the table is voltage unbalance of 3-phase nodes: unbalance should be kept below 3% to keep 3-phase motors from overheating [26].

Third in the table are constraints to prevent transformers from aging too rapidly. Many utilities use apparent power as a proxy for aging rate. We use a limit of 200% of a transformer’s rating as in [28]. Unlike utilities, we have access to our simulated transformers’ aging rates. We constrain a transformer’s aging rate such that its average rate over the simulation hour must be less than one. If a transformer surpasses this limit, its lifetime will be shorter than nominal (20 years). Fourth in the table is current flow, which is monitored for overhead and underground lines on the primary side of distribution transformers. We set the over-current constraint to 100% of the line’s rating. We also monitor the status of all fuses.

We conduct a survey across five networks with the goal of determining how sensitive the results are to different feeder parameters and topologies. Each feeder is from one of five climate regions in the U.S. and varies in terms of voltage level, topology, geographical density of buildings, etc. [15]. Table 3 lists the feeders’ key characteristics and conditions during the peak-hour. The first two parts of a feeder’s name indicates (climate region number)-(voltage level). For brevity, we will refer to the feeders only by region number. In this study, we expect the effects of regulation to be larger for feeders with higher percentages of AC load (i.e., R2, R4, and R5). We also anticipate that regulation could cause voltage issues on feeders with long lines, which are typically in rural areas (i.e., R1 and R4).

In this study, we examine the effects of load-based regulation when EVs are also active on the network. We assume 20% of residential houses have EVs. All EVs are identical and are able to both charge and provide power to the grid by discharging. We consider two scenarios: “EV+” in which all EVs charge for an hour, and “EV-” in which all EVs discharge for an hour, where the latter could occur if the EVs were coordinated by an aggregator. We model EVs as constant power loads that charge/discharge at 3.3kW with unity power factor. We conduct this study on feeder R1.

This study is designed to investigate one of the results from the main study – that some transformers have an increased aging rate due to load-based regulation. Our goal is to determine whether some transformers consistently have an increased aging rate across multiple trials. We run six randomized trials, all on feeder R1, with different random instantiations of the initial on/off status of the ACs as well as the ACs’ probabilistic responses; all other parameters are kept constant.

We find that – when EVs are charging – voltage violations due to load-based regulation worsen. Voltage results are presented in Table 5. We restrict our attention to the 1.05 p.u. continuous limit because it is the only constraint violated. In the base case, the results for the EV+ and No-EV trials are similar: 5.69% and 5.35% of nodes exceed the 1.05 p.u. limit for at least one time step, respectively. However, in the regulation case, over-limit nodes increase to 20.07% in the EV+ trial and decrease to 4.85% in the No-EV trial. A similar divergence occurs in the EV+ and No-EV trials when examining nodes with continuous voltage violations (i.e., over-limit for more than 2 minutes). In contrast to the other trials, the EV- trial has only 1.34% of its nodes over-limit for any duration, and this percentage remains almost constant across cases and durations.

The divergence in the No-EV and EV+ results is due, in part, to network-voltages’ increased sensitivity to load variation (due to load-based regulation), at higher levels of base load. Empirically, we see that as base loading increases (from EV- to EV+) the change in voltage variation due to load-based regulation increases (see Fig. 7). Analytically, we find that the sensitivity of voltage to changes in power injections due to regulation increases as loading levels increase. To derive this sensitivity, we use the DistFlow voltage equation [29], as reformulated in [30]:

where $V_{i}$ and $V_{k}$ are the voltage magnitudes at buses $i$ and $k$, $P_{ik}$ and $Q_{ik}$ are the real and reactive power flows received by bus $k$, and $r$ and $x$ are the resistance and reactance of the line connecting the buses. Solving the quadratic for $V_{k}^{2}$ and then taking the square root, we obtain an expression for $V_{k}$. Given this expression, the sensitivity of $V_{k}$ to changes in power injection at bus $k$ due to regulation is

where $\mathbb{Q}$, $\mathbb{P}$, and $\mathbb{V}$ are the three sensitivity terms. To evaluate (6), we must find expressions for each derivative. We assume $V_{i}$ is an infinite bus; thus $\partial V_{i}/\partial P_{\text{reg}}=0$ and $\mathbb{V}=0$. We solve for the partials of $V_{k}$ in the $\mathbb{Q}$ and $\mathbb{P}$ terms using the previously derived expression for $V_{k}$. To solve for the remaining derivatives, we assume that changes in flow are only due to changes in regulation power, i.e., $\partial P_{ik}/\partial P_{\text{reg}}=1$ and $\partial Q_{ik}/\partial Q_{\text{reg}}=1$. Finally, we find $\partial Q_{ik}/\partial P_{\text{reg}}$ given that ACs have a constant power factor of 0.97.

Using (6), we find an expression for the voltage sensitivity at the downstream node of a representative line in R1. We evaluate the node’s voltage sensitivity at all operating points from the base case simulation. For each trial, the computed voltage sensitivity, averaged across time, increases as base loading increases (see Fig. 7). Note that $\mathbb{P}$ contributes more to the increase in sensitivity than $\mathbb{Q}$.

The increase in voltage sensitivity with increased loading may explain why there is a large change in over-limit nodes in the regulation case of the EV+ trial, but only small changes in the other trials (see Table 5). It should be noted that the EV- trial results deviate from what might be expected (i.e., an increase in voltages) because a capacitor bank switches off due to an initial rise in voltage; the end result is a net decrease in voltage for most nodes.

We find that some transformers are more likely to age faster due to load-based regulation and some are more likely to age slower. Figure 8 shows the observed distribution of the number of trials transformers experience an increased aging rate due to regulation. We compare this distribution to the “expected distribution”, which represents the hypothesis that all transformers are equally likely to have an increased aging rate. Specifically, we model the transformers’ aging outcomes as i.i.d. Bernoulli random variables with probability 0.49 of increased aging. (This value is the observed prevalence of increased aging in the randomized trials.) The expected distribution, then, is the number of increased aging outcomes that occur over six Bernoulli trials, i.e., a Binomial distribution. The difference between the two distributions in Fig. 8 indicates that the aging outcomes cannot be modeled as i.i.d. random variables. Instead, the edges of the observed distribution indicate that a large portion of transformers is very likely to experience an increased aging rate, and another large portion is very likely to experience a decreased aging rate.

Load-based regulation does not have a consistent effect on transformers’ aging rates, in part, because of an unanticipated effect of probabilistic dispatch. In the following discussion, we will refer to switching in response to probabilistic dispatch as “dispatch-switching”. We will also refer to a cycle through an AC’s deadband without dispatch-switching as a “natural cycle”. When all ACs receive the same $u$, an AC with a low natural duty-cycle is more likely to be dispatch-switched on than off, and an AC with a high natural duty-cycle is more likely to be dispatch-switched off than on. See Appendix B for a derivation of this result. If an AC is dispatch-switched on more than off, it will “cool-cycle”, i.e., cycle in the cooler part of its deadband. This cycling behavior reduces the average temperature of the house, thus increasing its average power consumption and, consequently, the transformer’s aging rate. A similar argument can be made for why warm-cycling reduces transformer aging rates. As shown in Fig. 9, there is a negative correlation between the percent change in aging rate of a transformer and the average natural duty cycle of the ACs the transformer supplies. Transformers supplying ACs with a lower than average duty cycle ($<0.503$) frequently have an increased aging rate, and transformers supplying ACs with a higher than average duty cycle frequently have a decreased aging rate.

Our first step towards designing a control strategy for load-based regulation that is both network-protecting and computationally efficient is to identify a reduced-set of network constraints. Although this set will be network dependent, we expect that voltage unbalance and line current constraints could be safely omitted from the reduced-set for most networks. Transformer aging and power constraints could also be omitted, since load-based regulation did not cause any additional violations in our results. However, some utilities may choose to enforce stricter constraints that protect transformers from increased aging rates. Finally, we expect that in most networks we will need to include continuous voltage constraints in the reduced-set because of the increase in voltage variation due to regulation and the increased risk of violations in high loading scenarios.

We propose a few strategies for providing load-based regulation while protecting distribution networks. First, to prevent cool-cycling and thereby the likelihood of increased aging rates for transformers, ACs closer to naturally switching should be dispatched first (i.e., priority-stack control [31]). Second, to prevent voltage violations, the amount of participation by loads in voltage-weak areas could be reduced, or separate resources could be dispatched for voltage support. Reduction in participation could be implemented in one of two ways: 1) by allowing only a portion of loads in the area to participate in regulation, or 2) by dispatching loads in that area less frequently. Both of these strategies require some coordination between DER aggregators and the distribution operator.