Netload Range Cost Curves for a Transmission-Aware Distribution System Planning under DER Growth Uncertainty

Several research efforts have been dedicated in the last years to integrate DER uncertainty into distribution system planning. These efforts have resulted in advanced planning models embedding relevant short-term uncertainties that stem from DERs, including those associated with renewable generation [14], EV user behavior [15] and demand response availability [16] among others (see [17, 5] for a thorough literature review). However, less attention has been given to the incorporation of the long-term DER growth uncertainty into distribution system planning, with only a handful of works directly addressing this issue in the context of RPV and EV growth [6, 18, 19]. Furthermore, existing distribution system planning models that embed DER uncertainty [14, 15, 16, 6, 18, 19] have neglected the impact of distribution network investments on transmission-level reinforcement needs, which leaves a research gap in this regard.

A number of centralized planning models considering T&D coordination have been proposed in the last years. These models have integrated relevant aspects such as the uncertainty related to load growth and variable renewable generation [20], the impact of distribution investment decisions on the transmission-level energy prices [21], the installation of electric vehicle charging stations [22], and the use of open-source data and tools [23]. In general, these works are based on large-scale mathematical programs that can model the operation of assets at both T&D levels, therefore allowing an overall assessment of the impact of distribution-side investments on the transmission level. However, as a major drawback, this centralized approach directly conflicts with the current T&D independent planning workflows and data exchange limitation imposed by the regulatory framework. While decentralized T&D planning models have recently been proposed to address the above practical issue through the use of iterative hierarchical optimization methods [24, 25], they do not provide a clear design of incentives that would encourage T&D systems to collaborate, while maintaining their independence in the planning process. Furthermore, existing T&D planning models [20, 21, 22, 23, 24, 25] have an additional pending research gap in terms of including long-term DER growth uncertainty in their decision-making process.

In power systems operations, different T&D coordination schemes have also been proposed in the literature [26, 27]. In this context, the idea of using flexibility-vs-cost curves to describe the aggregate distribution-side flexibility seen from the transmission-side has gained considerable attention in the last years [28, 29, 30], as this approach allows T&D operators to interact with each other in a direct yet secure fashion. However, to the best of our knowledge, this framework has only been applied to operations so far, thus leaving an untapped potential for T&D planning applications.

Hence, there is a need for practical network planning strategies that create a link between distribution and transmission systems planners to deal with long-term DER growth uncertainty and its impact on grid infrastructure needs at both T&D levels.

To bridge the gaps discussed above, we propose a methodology for distribution system planning that considers DER growth uncertainty and provides transmission system planners with different options to control substation peak netload conditions via distribution investments. These options are presented in a form of Netload Range Cost Curves (NRCCs) that can be easily integrated into transmission planning processes. The contributions of this paper are threefold:

1. 1.

   The introduction of NRCCs as a form of expressing distribution system investment options to the transmission planners, while ensuring independence in the T&D planning process.

2. 2.

   A novel distribution system planning methodology that provides: (i) a simple and compact representation of the distribution-side’s planning flexibility, which can foster a collaboration between T&D system expansion planners; (ii) a “menu” of options of distribution systems plans with associated long-term netloads that can be used as an input for transmission studies.

3. 3.

   Detailed computational experiments based on a realistic distribution network. Our results demonstrate the applicability and benefits of the proposed transmission-aware distribution planning approach in the context of uncertain netload growth and adoption of RPV.

The rest of the paper is organized as follows: Section II describes conventional approaches to address the distribution system planning problem. Section III presents our proposed transmission-aware distribution planning strategy embedding DER growth uncertainty. Section IV reports computational experiments and results, demonstrating the relevance and advantages of our proposed distribution planning approach. Finally, we draw our main conclusions in Section V.

Under an approach based on deterministic optimization, network upgrade plans are obtained considering a single user-defined load and DER growth scenario. Candidate investment options can include the reinforcement of existing network corridors as well as the installation of new voltage regulators and/or BESSs [4, 5]. The resulting deterministic planning model includes investment-related decision variables $\bm{x}=\left[\bm{x}^{B},\bm{x}^{L},\bm{x}^{R}\right]$, operation-related decision variables $\bm{y}=\left[\bm{\rho}^{P},\bm{\rho}^{Q},\bm{p}^{B,D}\bm{p}^{B,C},\bm{q}^{B},\bm{f}^{P},\bm{f}^{Q},\bm{e}^{B},\bm{v}^{\dagger},\bm{\tilde{v}}^{R}\right]$, and sets describing the hours of the day $h\in\mathcal{H}$, representative days $d\in\mathcal{D}$, distribution lines $l\in\mathcal{L}$, candidate BESSs $b\in\mathcal{B}$, and network buses $n\in\mathcal{N}$, as described next:

Thus, (1) is a mixed-integer linear programming model, whose objective function (1a) aims to minimize the total investments costs resulting from line reinforcements and new voltage regulators and BESSs. Active and reactive nodal power balances are enforced through (1b)-(1c) for the substation that connects the distribution system to the main transmission grid, and via (1d)-(1e) for the rest of the buses. Apparent power capacity limits for lines are enforced through (1f), in which an inner piece-wise linear representation is used based on approximating hyperplanes $j\in\mathcal{J}$ with coefficients $\Upsilon^{P}_{j}$ and $\Upsilon^{Q}_{j}$ following [31]. The operation of BESSs is represented through constraints that describe the evolution of the stored energy over time (1g), the maximum stored energy limit (1h) and the apparent power capacity limit (1i). Voltage drops resulting from line powerflows are described through (1j)-(1k), where $\mathcal{L}^{0}$ is the subset of lines whose receiving end buses do not have (neither existing nor candidate) voltage regulation, and $\mathcal{L}^{R}$ is the subset of lines whose receiving end buses have (either existing or candidate) voltage regulation. Bus voltage limits and voltage regulation capabilities and are enforced through (1l) and (1m)-(1n), respectively, where $\mathcal{N}^{R}$ is the subset of buses with either existing or candidate voltage regulators and $\mathcal{N}^{CR}$ is a more restrict subset, which comprises the buses with non-existing but candidate voltage regulators. Constraint (1o) ensures that one of the line reinforcement options (including a ’no-upgrade’ option) is chosen for each network corridor, where $\mathcal{L}^{C}(n,n^{\prime})$ indicates the set of candidate line reinforcements for each pair of connected buses $(n,n^{\prime})$. Non-negative and binary decisions variables are described by (1p)-(1q).

The main drawback of the deterministic planning model is the utilization of a single deterministic growth scenario, which leaves all other possible realizations of load and DER growth neglected in the decision-making process for new investments. A conventional alternative to address this issue is to formulate the investment problem within a scenario-based framework considering a set of load and DER growth scenarios as follows:

The proposed distribution system planning model, given an available budget, aims to optimize the investment plan so as to avoid peak netloads, under the realization of different DER load growth scenarios, that largely deviate from the peak netload that would have been a result of an optimization exercise that determines investments while considering a single expected long-term DER load growth scenario. This model is written as follows:

where $\bm{\lambda}=\left[\lambda^{D},\lambda^{R}\right]$, in the optimal solution, represent maximum direct (D) and reverse (R) peak netloads among all considered scenarios, which can indicate transmission-side’s reinforcement needs resulting from considering multiple possible DER growth scenarios. Moreover, $\Lambda^{D}$ and $\Lambda^{R}$ are the originally ‘expected’ direct and reverse peak netloads resulting from a deterministic DER growth projection; $\left[\,y\,\right]^{+}=\max\{y,0\}$ is the positive-part function; $W\in[0,1]$ is a weighting factor for prioritizing the penalization of direct or reverse peak netloads compared to the expected scenario; and, finally, $\Gamma$ is a tunable parameter that indicates the available budget to invest in the distribution grid.

Thus, (2) is a mixed-integer linear programming model. The objective function to be minimized in (2a) penalizes the largest deviation in terms of peak netload among all considered DER load growth scenarios compared to the expected scenario, therefore aiming at minimizing the ‘unexpected’ transmission-side’s reinforcement needs resulting from DER growth uncertainty. Constraints (2b)-(2c) link the active power drawn from the transmission system and the largest values of direct and reverse power flows. Constraint (2d) limits the total investment cost based on the available budget and the last two sets of constraints ensure a secure system operation in all considered DER growth scenarios.

The distribution planning model (2) allows for the simultaneous consideration of multiple possible network upgrade options based on the budget parameter $\Gamma$. In particular, by adjusting the value of $\Gamma$, different possible network upgrade plans with corresponding associated peak netload intervals and budget requirements can be obtained. This information can then be used to build the proposed NRCCs for facilitating the dialog between distribution and transmission system planners, as described next.

Based on (2), NRCCs for characterizing the distribution-side planning flexibility can be built using the procedure illustrated in Fig. 1. For this purpose, first, our methodology requires, as an input, a set of projected DER long-term growth scenarios $k\in\mathcal{K}$ with corresponding bus netloads $\bm{LD}^{P}(\xi_{k})$. Although we describe a way to obtain this input in Section IV, proposing an ideal methodology to generate scenarios is out of the scope of this paper and our proposed approach assumes the required set of scenarios is readily available. Given the bus netload scenarios, the distribution planning model (2) is solved iteratively for different budget values $\Gamma$, resulting in a range of possible direct and reverse peak netloads $\lambda^{D}$ and $\lambda^{R}$. Finally, using the above values, the NRCCs are obtained. These curves can then be shared with the transmission planning entity for its independent network studies.

One of the main advantages of the proposed NRCCs lies in their compact representation of the distribution-side’s planning flexibility in a way that can be treated by the transmission counterpart. An example of the integration of NRCCs in transmission planning is illustrated in Fig. 2, in which it can be observed that instead of having a single peak netload value for Bus 2, now a range of possible peak netload values for this bus is available, each of which with an associated cost. Therefore, distribution grid investments are presented as “cost vs netload range guarantees” at the substation.

From a transmission planner perspective, these curves describe the marginal cost of decreasing the netload range in the substation. Therefore, they can be treated equivalently to any other nodal capacity investment (such as generator or storage asset) in the expansion planning process. From a procedural perspective, the NRCCs are a non-invasive form of coordinating T&D planning, adaptable to the two T&D ownership models that exist in current regulatory frameworks:

1. 1.

   Integrated utilities: when T&D is managed by the same entity, NRCCs can be used to communicate least-cost grid investment options from distribution to transmission planning teams.

2. 2.

   Separated utilities: when T&D is managed by two distinct entities, NRCCs can be seen as a marginal capacity product that can be offered by the distribution company. Since marginal costs and netload range guarantees are clearly identified in NRCCs, these products are verifiable and therefore can be regulated.

Thus, NRCCs are a straightforward and transparent mechanism for quantifying the contribution of distribution-side investments, in particular BESS and other forms of NWAs, on transmission level.

Our computational experiments are based on the San Francisco distribution network provided in the SmartDS dataset [32], which includes one full year load timeseries with a 6.5 MW peak value and has a 12.47 kV base voltage, 126 existing lines, and 354 buses. This distribution system has been modified to have an initial 1.68 MW RPV capacity and an existing voltage regulator near the main substation. A 10-year planning horizon is considered in the simulations, for which three load growth scenarios with different growth rates per year are considered: Low (2%), Medium (3%) and High (4%); yielding a total of 81 candidate line reinforcements, and 5 candidate buses to receive the installation of new BESS and/or voltage regulators (see Fig. 3). Parameters and costs associated with candidate line reinforcements, BESSs, and voltage regulators are extracted from [33, 34].

We consider a total of 100 RPV adoption scenarios to characterize realistic DER growth patterns. These scenarios are obtained from an agent-based extension of the NREL’s dGen model [35] (see Section IV-B for more details). To reduce computational burden, 3 of the 100 DER growth scenarios are selected based on the resulting aggregate RPV capacities, yielding the Maximum, Average, and Minimum RPV adoption scenarios indicated in Figs. 4 and 5. These scenarios are then combined with the 3 aforementioned load growth scenarios, therefore a total of 3x3=9 netload scenarios $\bm{\xi}_{k}$ are considered for this planning exercise. Such netload scenarios are further simplified by picking the two representative days that are associated with the highest and lowest peak netload values (see Fig. 6).

For benchmarking purposes, we compare three different distribution planning strategies in our experiments. The first one corresponds to the deterministic planning model presented in Section II-A, which minimizes overall costs for distribution network upgrades while taking into account a single deterministic load/DER growth projection scenario [4, 5]. This deterministic scenario is based on the previously described Medium load growth and Average RPV adoption scenarios. The second planning strategy corresponds to the scenario-based planning model described in Section II-B, in which all 9 load/DER growth scenarios are considered in the planning process while maintaining a cost-minimization approach focused on the distribution-side. The third one is the proposed transmission-aware distribution planning strategy, described in Section III, in which a transmission-aware planning model embeds DER growth uncertainty and is used to build NRCCs that describe possible distribution network upgrade options with different associated budgets and peak netload intervals. Under this strategy, a symmetric weighting factor of $W=0.5$ is considered for both direct and reverse peak netloads.

The conventional and proposed approaches are implemented in Python, with Pyomo and Gurobi building and solving the optimization problems, and our computational experiments have been performed on a PC with an Intel Core i7 2.20-GHz processor and 32 GB of RAM under a 64-bit Windows 11 operating system. RPV projects are simulated using the PySAM package [36]. In addition, we run AC powerflows for result validation using the OpenDSSDirect package [37].

Here we describe the methodology used to generate realistic DER growth scenarios for distribution planning. This methodology is based on simulating future RPV adoption patterns, while considering the uncertainty related to the location and size of the new RPV projects in the distribution grid.

Modern RPV adoption models are typically based on Innovation Diffusion theory, which characterizes how a new technology is adopted over time. Under this approach, RPV adoption is characterized as follows [35, 38]:

As a first step, each bus $n\in\mathcal{N}=\{1,\ldots,N\}$ in the distribution network is treated as an independent agent, for which an adoption status $x_{n,t}$ at time $t$ is defined ($1$ if adopted, $0$ if not). Thus, based on (3) and [38], the probability of adoption at time $t+\Delta t$ can be computed as follows:

which can be used to run Monte-Carlo simulations describing potential adoption scenarios for each agent $n\in\mathcal{N}$. As an example, the results of running 100 independent agent-based simulations is shown in Fig. 7, in which the fraction of adopting agents over time for each simulated scenario is presented, and compared to the one estimated by the aggregate continuous model (3).

Another relevant output of the modified agent-based simulation framework is the size (in kW) of the RPV project assigned to each agent. To capture the economic rationality of the agents, this aspect is modeled by solving an optimization problem in which the RPV capacity that maximizes the Net Present Value (NPV) of each potential project is calculated independently. Thus, when a new RPV project in installed into the distribution grid (i.e., $x_{n,t}=1$), this pre-determined RPV capacity value is used. Furthermore, similar to [35], a maximum market share (MMS) dependent on the project’s NPV and payback period is also considered here. Within the modified agent-based framework, this means first running a Bernoulli trial (with a success probability equal to the MMS) to determine the subset of agents $n\in\mathcal{N}^{\prime}\subset\mathcal{N}$ that will be considered for an eventual adoption in the future resulting from (4).

The Minimum, Average, and Maximum RPV scenarios resulting from running 100 agent-based simulations and picking the 3 most representative cases (in terms of aggregate RPV capacity) are illustrated in Figs. 4 and 5. Observe that not only the total installed RPV capacity varies significantly across the 3 scenarios (from 3.97 to 10.60 MW), but also the spatial distribution of the new RPV projects. Therefore there is a two-dimensional RPV growth uncertainty related to the size and location of new projects. As described in Section IV-A, the above 3 RPV adoption scenarios are the ones considered for distribution planning.

The results obtained from implementing the deterministic planning model are presented in Fig. 8a. Under this approach, a total investment cost of 0.98 MUSD is obtained as a result of line reinforcements. Moreover, minimum and maximum peak netload values for the single considered long-term scenario are equal to -0.42 MW and +8.62 MW, respectively. These observed results stem from the deterministic planning model’s objective to minimize investment costs only. Hence, the cheapest network upgrade option is preferred (line reinforcements in this case) without considering the potential impact of the distribution system’s peak netload on the transmission-side.

The above planning results were tested under all 9 load and DER growth scenarios by running detailed AC powerflow simulations, which revealed significant line capacity violations (see Fig. 9). These violations result from the fact that only a single expected scenario of load growth and RPV adoption is considered under the framework of the deterministic planning model, which underestimates the network upgrades required for most extreme scenarios with higher load and RPV growth.

The results obtained from implementing the scenario-based planning model are presented in Fig. 8b, for which all selected 9 load/DER growth scenarios are considered in the planning process. Observe that unlike indicated by the solution attained via the deterministic planning model (see Fig. 8a), additional distribution line reinforcements are required in this case to ensure a secure system operation under the realization of all selected scenarios. As a result, the total investment cost is $1.24>0.98$ MUSD. Moreover, note that in this case, a wider peak netload range equal to [-3.66, +9.50] MW is obtained, which is significantly larger than the one originally estimated by the deterministic planning model, i.e., [-0.42, +8.62] MW. This larger range of peak netloads is a natural consequence of considering different long-term scenarios.

The performance of the proposed transmission-aware distribution planning strategy described in Section III is tested by solving planning model (2) considering all 9 load/DER growth scenarios for different available budget values $\Gamma$. A direct and reverse ‘expected’ transmission reinforcement need of $\Lambda^{D}=8.62$ MW and $\Lambda^{R}=0.42$ MW is used in (2), based on the peak netloads estimated by the deterministic planning model (see Section IV-C).

As a first analysis, the planning model (2) is solved considering $\Gamma=1.24$ MUSD, which corresponds to the total investment cost obtained via the scenario-based planning model, as discussed in Section IV-D. The planning results for this case are illustrated in Fig. 10a, which as anticipated, are consistent with the ones obtained through the scenario-based planning model (see Fig. 8b), including the resulting direct and reverse peak netloads $\lambda^{D}=9.50$ and $\lambda^{R}=3.66$ MW.

As a second analysis, the available budget was increased to $\Gamma=2$ MUSD, which resulted in the network upgrade plan presented in Fig. 10b. Observe how unlike the case with $\Gamma=1.24$ MUSD (see Fig. 10a), a 440kW BESS is installed in this new case, which translates into reduced direct and reverse peak netloads of $\lambda^{D}=9.06<9.50$ MW and $\lambda^{R}=3.22<3.66$ MW. In addition this solution indicated the deferral of some of the originally required distribution line upgrades due to the BESS smoothing dispatch. We observe similar results by further raising the available budget to $\Gamma=3$ MUSD. As a consequence of this larger budget, the BESS capacity is increased to $1007>440$ kW, and the peak netloads are further reduced to $\lambda^{D}=8.62<9.06$ MW and $\lambda^{R}=2.65<3.22$ MW (see Fig. 10c), therefore very likely decreasing the levels of needed investments in the transmission system.

A summary of the resulting direct and reverse peak netloads $\lambda^{D}$ and $\lambda^{R}$ for different available budgets $\Gamma$ is presented in Fig. 11, which corresponds to the NRCCs originally described in Section III-B. It is worth noting how peak netloads $\lambda^{D}$ and $\lambda^{R}$ decrease as a larger budget $\Gamma$ is made available for the distribution network, ultimately converging to the ‘expected’ requirements $\Lambda^{D}$ and $\Lambda^{R}$ originally estimated by the deterministic planning model. Such reduction results from the installation of new BESSs in the distribution network, which reduces the system’s minimum and maximum peak netloads across all the selected 9 load/DER growth scenarios. Note that Fig. 11 also includes dispersion bars to depict the peak netloads obtained by implementing and assessing the performance of the attained planning decisions (which were determined while considering 9 scenarios as previously mentioned) under the realization of 300 scenarios of DER penetration and load growth. These 300 scenarios are a combination of the original 100 RPV adoption scenarios (see Fig. 4) with the 3 load growth scenarios under consideration. As can be seen, the dispersion bars indicate that the peak netload for all scenarios are within the bounds of the NRCC, which were built using only the Maximum, Average, and Minimum RPV adoption scenarios.

From the observation of Fig 11, it is possible to derive the three main characteristics of the proposed NRCCs:

1. 1.

   The notion of distribution-side planning flexibility, i.e. the exploration of feasible investment solutions that go beyond what is strictly necessary to ensure distribution grid security to decrease the uncertainty “seen” from the transmission side;

2. 2.

   The expression of this planning flexibility in the form of netload range guarantees at the distribution feeder head or substation transformer;

3. 3.

   The ability to link these netload range guarantees to concrete tractable distribution investments and corresponding incremental costs, which facilitates the regulatory processes around T&D.

After disaggregating the planning solutions by investment type, it is possible to observe the benefit of strategically locating new BESS investments to avoid reconductoring some of the distribution’s network corridors (see Fig. 10). This benefit can directly be quantified, as illustrated in Fig. 12, in which it can be observed that efficiently-located BESS investments result in a reduction of line reinforcement needs at a distribution level. In particular, for $\Gamma=3$ MUSD, line savings equal to 0.25 MUSD are obtained as a result of strategically locating a 1007kW BESS in the middle of the feeder. This illustrates how, in our proposed framework, the value of strategically locating BESS (and other possible NWAs) is captured from a T&D point of view.

Thus, from the case study above, it results clear that NRCCs provide a series of practical benefits that facilitate their implementation in the context of a coordinated T&D planning. First, they represent the distribution-side’s planning flexibility in a simple and compact manner, which facilitates its integration within existing transmission planning workflows. Second, from a distribution grid perspective, NRCCs are able to distinguish assets strictly required to ensure system security from those that can be installed to additionally benefit the transmission side.