### Partners

### Theme

### Promotor / Supervisor

Prof. dr. ir. D. Van Hertem

### Samenvatting van het onderzoek / Summary of Research

The increased penetration of renewable resources, such as photovoltaics (PV) and new loads, such as electric vehicles (EV) and heat pumps (HP), have increased uncertainty levels in the low voltage distribution system (LVDS). The traditional approach to planning such LVDS is the conservative ``fit and forget'' approach investing for the worst-case scenario, which means installing a lot of additional infrastructure. However, considering LVDS planning as a stochastic problem is deemed more efficient, reducing grid infrastructure investment cost. Determining the PV hosting capacity (HC) is one such planning problem, yielding the capacity of the system to incorporate new PV resources. Stochastic HC calculations are currently done using computationally expensive, iterative Monte Carlo (MC) based methods, which require solving the power flow equations thousands of times for each iteration of the PV installation scenario. To calculate the PV HC of a large service area consisting of thousands of LV feeders, a computationally tractable and accurate probabilistic power flow (PPF) tool is required. Various approaches exist to make MC-based iterative HC calculation method computationally more efficient, e.g., linearizing the power flow equations, replacing some of the uncertain variables with deterministic values, etc. Alternatively, MC-based methods can be replaced by faster analytic methods, or the iterative HC calculation method can be replaced by stochastic optimization. This thesis aims to build analytical alternatives for MC-based methods and stochastic optimization tools to replace iterative LVDS HC calculation methods.

First, comparing the existing PV HC methods is challenging as they are demonstrated on different test feeders using different assumptions. Therefore, this thesis proposes a benchmark wherein the effects of assumptions made for calculating the HC are evaluated. The comparison of the PV HC obtained from existing methods shows a huge spread, mainly affected by the assumptions on the grid and stochastic limits and the size and number of the PV installations.

Standardizing these limits is a major step towards defining a realistic HC of LVDS feeders. As a benchmark, a stochastic limit of 5% and a grid voltage limit of 0.95-1.05 pu are recommended.

Second, MC simulations are the bottleneck in computing the stochastic PV HC of a large service area. This thesis proposes non-intrusive general Polynomial Chaos (gPC) expansion based PPF as an alternative. The gPC-based PPF outweighs MC and quasi-Monte Carlo (qMC) methods clearly in terms of computational effort with comparable accuracy, as the complex power flow equations are replaced by their polynomial surrogate. The proposed PPF, when used to calculate the congestion probability of a European LVDS, is ten times faster than the MC-based method with the same accuracy.

Third, stochastic HC calculations require an understanding of the uncertainties in LVDS. The operational uncertainties due to load and generation variations and PV scenario uncertainties due to size, type and phase of the PV installations are usually sampled together in MC-based probabilistic HC approaches. This dissertation presents a decoupled approach to calculate the stochastic PV HC, where the probability of violation of the operational limits is computed for possible PV planning scenarios. Then, the PV scenario that results in the highest total PV power installed without violating the stochastic operational limits of the LV feeder is proposed to be the stochastic HC of that feeder. Decoupling the PV (planning) scenarios from the operational uncertainties instead of sampling them together enables to study of the impact of planning policies and operational rules on the overall PV HC.

Fourth, the iterative approach of computing PV HC, where the PV size installed is increased in every iteration, is a computationally demanding process. The decoupled method is also a brute force approach where the probability of congestion is calculated for each possible scenario. In contrast, a chance-constrained stochastic optimal power flow (SOPF) based method reduces the decoupled HC calculation into a single-shot problem. This thesis proposes an intrusive gPC-based SOPF for calculating stochastic PV HC. This method reduces the computation time from days to seconds while giving the upper bound of the HC that can be identified using conventional methods.

Finally, when investigating the needs for a larger region, it is generally too hard to compute the stochastic HC for all individual LV feeders, as a small service area already can have hundreds of feeders. An approximation of the PV HC of the entire service area can be obtained by scaling up the HC of a small number of representative feeders. This thesis presents a clustering scheme that captures the most relevant features of LVDS feeders for PV HC to obtain such representative feeders. A case study shows that the PV HC of a small service area can be approximated by using only 3% of the carefully chosen representative feeders with an error of 20%.