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Probabilistic duck curve in high PV penetration power system: Concept, modeling, and empirical analysis in China
Qingchun Hou, Ning Zhang, Ershun Du, Miao Miao, Fei Peng, Chongqing Kang
TL;DR
High PV penetration makes deterministic duck curves inadequate because they omit probability and uncertainty in net load and ramps. The paper proposes probabilistic duck and ramp curves using dependence-aware distribution modeling, then evaluates flexible-resource planning in Qinghai. The analysis finds considerable uncertainty in valley net load and ramp demand, while coal-fired-unit retrofits substantially improve flexibility.
Problem
Deterministic duck curves show only selected net-load scenarios and omit probability, net-load uncertainty, ramp variability, and renewable-curtailment information needed for high-PV planning.
Method
The paper models probabilistic duck and ramp curves using kernel density estimation, copulas, and dependent discrete convolution, then applies them to flexible-resource planning.
Results
The Qinghai analysis finds considerable uncertainty in valley net load and ramp demand, and coal-fired-unit retrofits enhance system flexibility.
Takeaways & Limitations
Probabilistic curves indicate how flexible capacity should cover likely net-load and ramp conditions rather than rare worst-case scenarios.
Takeaways & Limitations
The methodology is especially focused on high-PV-penetration power systems and flexible-resource planning, though the authors note it could be extended further.
Abstract
from arXiv · showhide
The high penetration of photovoltaic (PV) is reshaping the electricity net-load curve and has a significant impact on power system operation and planning. The concept of duck curve is widely used to describe the timing imbalance between peak demand and PV generation. The traditional duck curve is deterministic and only shows a single extreme or typical scenario during a day. Thus, it cannot capture both the probability of that scenario and the uncertainty of PV generation and loads. These weaknesses limit the application of the duck curve on power system planning under high PV penetration. To address this issue, the novel concepts of probabilistic duck curve (PDC) and probabilistic ramp curve (PRC) are proposed to accurately model the uncertainty and variability of electricity net load and ramp under high PV penetration. An efficient method is presented for modeling PDC and PRC using kernel density estimation, copula function, and dependent discrete convolution. Several indices are designed to quantify the characteristics of the PDC and PRC. For the application, we demonstrate how the PDC and PRC will benefit flexible resource planning. Finally, an empirical study on the Qinghai provincial power system of China validates the effectiveness of the presented method. The results of PDC and PRC intuitively illustrate that the ramp demand and the valley of net load face considerable uncertainty under high PV penetration. The results of flexible resource planning indicate that retrofitting coal-fired units has remarkable performance on enhancing the power system flexibility in Qinghai. In average, reducing the minimal output of coal-fired units by 1 MW will increase PV accommodation by over 4 MWh each day.
Nomenclature
The paper’s nomenclature defines probability-distribution, dependence, and uncertainty terms used to model probabilistic duck and ramp curves.
- PV net load, total load, and total PV generation denote the principal time-period variables.
- Kernel density estimation is represented by f̄(·) and bandwidth h.
- Copula functions represent dependencies among PV generation, loads, total PV generation and load, and consecutive net-load periods.
- Dependent discrete convolution terms describe aggregated PV generation, loads, PDC, and PRC distributions.
- N and M are the numbers of PV farms and nodes; MOU denotes minimal output at period t.
- Lα% denotes the α% quantile, while CLα% denotes the α% confidence level of net-load or ramp distributions.
1. Introduction
High PV penetration reshapes net load and increases flexibility challenges, while deterministic duck curves omit probability and ramp uncertainty. The paper introduces probabilistic curves and applies them to flexible-resource planning in Qinghai.
- Motivation: Qinghai planned over 10 GW of PV in 2020, approximately 77% of peak load demand, creating major flexibility challenges.
- Existing duck curve: The conventional duck curve falls toward noon and rises sharply at sunset, stressing conventional generators’ peaking and ramping regulation.
- Research gap: Deterministic duck curves omit net-load ranges, scenario probabilities, ramp variability, and accurate renewable-curtailment quantification.
- Proposed approach: The probabilistic duck curve models period-by-period net-load distributions, while the probabilistic ramp curve represents probabilistic ramp-capacity requirements.
- Proposed approach: Copulas and dependent discrete convolution model complex dependencies among PV generation and loads, and several indices quantify curve characteristics.
- Application: The framework supports planning coal-unit retrofits, energy storage, and PV curtailment, with empirical analysis on Qinghai’s high-PV system.
2. Methodology
The paper defines probabilistic duck and ramp curves to represent distributions, uncertainty, and dependence in net load under high PV penetration. It models these curves with kernel density estimation, copulas, and dependent discrete convolution, then derives indices for flexible-resource planning.
- Concepts: The probabilistic duck curve (PDC) is a set of net-load distributions for each daily period, incorporating load and PV-generation uncertainty over the long term.Unlike the deterministic duck curve, it represents probabilistic net-load behavior rather than a single typical or extreme scenario.
- Concepts: The probabilistic ramp curve (PRC) gives distributions of net-load changes between adjacent periods, capturing uncertainty in ramp demand.Its ramp calculation accounts for dependence between net load at consecutive periods.
- Modeling method: PDC and PRC modeling proceeds through marginal-distribution estimation, dependence modeling, dependent discrete convolution, and derivation of the probabilistic curves.Kernel density estimation models PV and load marginals; copulas model spatiotemporal dependencies; DDC calculates total distributions.
- Modeling method: Kernel density estimation is selected because it accommodates arbitrary PV-generation and load distributions without assuming a predefined parametric family.The bandwidth is adjusted to avoid overfitting or underfitting, and samples farther from the evaluation point receive smaller weights.
- Modeling method: Copula functions connect modeled marginal distributions into joint distributions while representing dependencies among PV farms, loads, total PV, total load, and adjacent-period net load.The method assumes Gaussian copulas for specified dependence structures.
- Planning indices: Four indices quantify expected behavior, uncertainty, peak regulation demand, and PV curtailment for flexible-resource planning.They are the expected value curve, α% confidence-level curve, peak-to-valley difference, and probabilistic area; peak and valley times are derived from the PDC expected-value curve.
3. Empirical study of the Qinghai power system
The Qinghai case study validates probabilistic modeling of PV, load, net-load, and ramp distributions under high PV penetration, showing substantial uncertainty and informing flexible-resource planning.
- Case-study setting: 10 GW of PV capacity, approximately 77% of peak load, frames Qinghai’s high-penetration planning case.PV generation is expected to meet over 20% of total load demand; hydropower and coal-fired capacities are 15 GW and 5 GW, respectively.
- Dependency modeling: Kernel density estimation best fits the skewed, long-tailed distribution of a single PV farm’s generation.At 12:00 PM, generation is mainly concentrated between 0.6 p.u. and 0.9 p.u.
- Dependency modeling: Modeling dependencies among PV farms captures fatter tails and lower peak values, producing an aggregated distribution closer to realistic statistical data.The result reflects correlated PV farms reaching maximal or minimal generation simultaneously, especially at noon.
- Dependency modeling: Weak total-load/PV-generation correlation has limited effect on noon net-load distribution because industrial load exceeds 90% of Qinghai’s total load.By contrast, adjacent-period net-load dependence is remarkable, and ignoring it introduces considerable ramp-distribution error.
- Probabilistic duck and ramp curves: Approximately 4000 MW of possible ramp-up demand occurs at 6:00 PM, while ramp-down demand peaks near 3000 MW at 8:00 AM.Ramp-up uncertainty width is approximately 4000 MW/h, compared with approximately 2000 MW/h for ramp-down demand.
- Flexible-resource planning: Reducing coal-unit minimum output from 5000 MW to 3000 MW decreases daily PV curtailment from 15,000 MWh to 5000 MWh.When system minimum output exceeds 2700 MW, each 1 MW reduction increases PV accommodation by over 4 MWh daily.
- Flexible-resource planning: Retrofitting available coal capacity to reduce system minimum output to 3000 MW increases daily PV accommodation by 10,000 MWh; storage can reduce it further to 2700 MW.The analyzed storage deployment is 1500 MWh, equivalent to 300 MW × 5 h; demand response may address the 1500–2700 MW interval.
4. Discussion
The PDC and PRC guide flexible-resource planning by showing how net-load valley conditions determine the appropriate mix of flexibility measures in Qinghai.
- Flexible resource planning: The PDC and PRC indicate required flexible generation capacity and how frequently flexibility is needed to accommodate intermittent PV generation.Planning should match peak-regulation and ramp capacity to the net load represented by the PDC and PRC.
- Peak-regulation sections: Valley loads above 5000 MW require no critical peak-regulation intervention because the system’s flexible generation schedule can follow demand.
- Peak-regulation sections: Valley loads from 3000 MW to 5000 MW suggest retrofitting coal-fired units, while loads from 2700 MW to 3000 MW suggest deploying energy storage.The discussion links coal retrofits to relatively low investment and operating costs, whereas storage addresses more severe conditions while maintaining economic efficiency.
- Peak-regulation sections: Valley loads below 2700 MW occur with very low probability, making demand response and PV curtailment the suggested choices.These options have nearly zero investment cost but high operating cost.
5. Conclusions and future works
The paper extends duck-curve analysis into probabilistic net-load and ramp modeling, then applies it to flexible-resource planning in high-PV systems. Its conclusions identify substantial uncertainty, potential benefits from coal-unit retrofits, and extensions involving wind and other uncertainty-driven systems.
- Conclusions: The PDC and PRC extend the duck curve to describe uncertainty in net load and net-load ramp using copula and DDC techniques.The methodology models dependence among PV generation and loads and includes indices for representative scenarios, uncertainty, curtailment, and peak regulation.
- Conclusions: Kernel density estimation outperforms parameter estimation for PV marginal distributions, while copula and DDC techniques effectively model dependence among PV generation and loads.
- Conclusions: High PV penetration moves Qinghai’s valley period from early morning to noon, enlarges PTV nearly three times in 2020, and creates major minimal-load uncertainty.
- Conclusions: High PV penetration increases average ramp-capacity demand and its uncertainty, requiring larger high-ramp-capacity generators whose utilization rate decreases as uncertainty rises.
- Conclusions: The PDC and PRC are valuable for flexible-resource planning, including assessing coal-unit retrofits for improving flexibility and renewable accommodation.
- Future works: The concepts, modeling methods, and planning method are presented as general for high-PV systems, while the paper specifically focuses on flexible planning in such systems.Future extensions include systems combining high wind and PV penetration, broader operating applications, and other systems with uncertainty and dependence structures.