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A Multi-State Power Model for Adequacy Assessment of Distributed Generation via Universal Generating Function
Yan-Fu Li, Enrico Zio
TL;DR
Reliability assessment for renewable distributed-generation networks is challenged by stochastic generation and the computational cost of simulation-based approaches. The paper develops an analytical multi-state UGF model that combines renewable-source and mechanical states, then applies it to a modified IEEE 34-node feeder. The authors present analytical modeling as efficient and accurate for small-scale power grids, while noting scope limitations involving over-intensity conditions.
Problem
Existing reliability approaches often rely on computationally costly Monte Carlo simulation and commonly use binary-state representations for renewable-generation components.
Method
The paper uses universal generating functions with a multiplication-type composition operator to combine renewable-source, mechanical, generation-unit, and load states analytically.
Results
The model is demonstrated on a distributed-generation system adapted from the IEEE 34-node distribution test feeder.
Takeaways & Limitations
For small-scale power grids, analytical modeling can provide efficient and accurate reference results.
Takeaways & Limitations
The study does not consider over-intensity of irradiation and over-intensity conditions affecting the generators.
Abstract
from arXiv · showhide
The current and future developments of electric power systems are pushing the boundaries of reliability assessment to consider distribution networks with renewable generators. Given the stochastic features of these elements, most modeling approaches rely on Monte Carlo simulation. The computational costs associated to the simulation approach force to treating mostly small-sized systems, i.e. with a limited number of lumped components of a given renewable technology (e.g. wind or solar, etc.) whose behavior is described by a binary state, working or failed. In this paper, we propose an analytical multi-state modeling approach for the reliability assessment of distributed generation (DG). The approach allows looking to a number of diverse energy generation technologies distributed on the system. Multiple states are used to describe the randomness in the generation units, due to the stochastic nature of the generation sources and of the mechanical degradation/failure behavior of the generation systems. The universal generating function (UGF) technique is used for the individual component multi-state modeling. A multiplication-type composition operator is introduced to combine the UGFs for the mechanical degradation and renewable generation source states into the UGF of the renewable generator power output. The overall multi-state DG system UGF is then constructed and classical reliability indices (e.g. loss of load expectation (LOLE), expected energy not supplied (EENS)) are computed from the DG system generation and load UGFs. An application of the model is shown on a DG system adapted from the IEEE 34 nodes distribution test feeder.
2 Politecnico di Milano, Milan, Italy, Dipartimento di Energia
The paper defines notation for multi-state power-output and probability models for renewable generators, electric vehicles, transformers, system generation, and load. These quantities are represented through u-functions for reliability analysis.
- Multi-state notation: The notation covers solar, wind, electric-vehicle, transformer, system-generation, and load power states and their associated probabilities.The listed variables include component outputs, operation or mechanical states, state probabilities, and u-functions.
- Renewable generators: Solar and wind models represent renewable power output across discretized irradiance or wind-speed states.The solar notation includes discretized solar irradiance, while the wind notation includes discretized wind speed and turbine output.
- Mechanical states: Mechanical-state variables describe renewable-generator condition values, probabilities, and their contribution to generator power-output u-functions.The notation distinguishes mechanical condition from the resulting solar-generator performance states.
- System quantities: The system model includes state probabilities and u-functions for aggregate generation and uncertain load demand.System-generation and load quantities are represented separately before reliability indices are computed.
1 Introduction
The introduction motivates reliability assessment for distributed generation because renewable output is stochastic and conventional simulation can be costly. It presents an analytical multi-state UGF approach that combines source, component, generation, and load models to compute reliability indices.
- Motivation: Renewable generators increase the need for distribution-network reliability assessment because solar irradiation and wind speed are random.Their outputs depend substantially on external natural resources.
- Motivation: Monte Carlo simulation is widely used for reliability indices but may have unstable accuracy and lengthy computation time.These costs can motivate simplifying components into binary working or failed states.
- Prior modeling: Multi-state models provide greater flexibility than binary-state models for representing generation randomness and component degradation or repair.The paper motivates multi-state descriptions as more realistic approximations of power-system state evolution.
- Proposed approach: The paper applies UGF techniques to model generation-source outputs and component degradation, failure, and repair behavior.UGFs are presented as analytical tools for describing multi-state components and constructing complex systems.
- Proposed approach: A multiplication operator combines source and component u-functions into renewable-generator power-output models.The distributed-generation model combines solar generators, wind turbines, electric vehicles, and transformers, while uncertain load demand is also represented by a u-function.
- Evaluation: The resulting overall model computes reliability indices from distributed-generation and load u-functions and is demonstrated in a case study.The paper reports a modified IEEE 34-node test feeder application and formulates reliability indices in terms of UGFs.
System
The section develops multi-state UGF models for distributed-generation components, representing renewable-source variability and mechanical condition jointly. Solar and wind models discretize environmental inputs, combine them with mechanical states, and derive component power-output distributions.
- The DG component models use UGF representations to describe multi-state probability distributions and derive power-output distributions.
- Solar Generator: Solar generation models combine discretized irradiation states with mechanical degradation, failure, and repair states.Irradiation and mechanical behavior are treated as independent sources of randomness.
- Solar Generator: The solar-generator power output multiplies individual-module output determined by irradiance by the number of functioning modules.For example, 50% module availability and 50% rated module output yield 25% of rated generator power.
- Solar Generator: The solar model uses a multiplication-type composition operator to combine the U-functions of irradiance and mechanical condition.The operator corresponds to multiplication of two random variables and was not formally specified previously.
- Wind Turbine: Wind-turbine models similarly discretize wind speed and combine it with mechanical condition to represent power generation.Wind speed is divided into equal-sized states, with probabilities derived from its distribution; the wind turbine includes generator, gearbox, and rotor subassemblies connected in series.
3 Multi-State Model for the Distribution Network and Reliability Assessment
The model represents distributed-generation components and loads with multi-state universal generating functions, combines generation states analytically, and computes adequacy indices from generation and load distributions.
- Network assumptions: The distribution network is modeled as a local radial system whose component loads are represented by an aggregate load model.The network includes one transformer and distribution nodes sharing a common feeder.
- Component assumptions: Solar and wind source states are assumed perfectly correlated within each technology, while their internal degradation and repair mechanisms are mutually independent.The independence assumption is motivated by limited data on intercomponent dependence.
- UGF construction: A multiplication-type UGF composition combines mechanical states and renewable-source states into solar, wind, and overall generation UGFs.The construction successively combines the solar generators, wind generators, and all generator types, then reduces redundant states.
- UGF construction: The resulting generation UGF and aggregated load UGF provide the state distributions required for multi-state adequacy assessment.Load states are formed by sorting load values into states, while generation states are reduced to distinct energy levels and redundant-state counts.
- Reliability assessment: The distributive operator computes LOLE and EENS from the system-generation and load-demand UGFs.LOLE represents expected periods when demand exceeds available generation, whereas EENS represents expected unsupplied energy.
4 Case Study
The case study applies the multi-state framework to a modified IEEE 34-node feeder containing wind, solar, and aggregated EV generation. Component reliability, renewable-source, and load models are parameterized and combined into system reliability calculations.
- System configuration: The case study uses a modified IEEE 34-node distribution test feeder with a radial network downscaled to 4.16 kV.The distributed-generation system is represented with a reliability block diagram.
- System configuration: Wind, solar, and EV generation contribute 60%, 30%, and 10% shares, corresponding to rated outputs of 750 kW, 375 kW, and 125 kW.The system includes five solar generators, five wind turbines, and an EV aggregation of 25 vehicles.
- Component models: All generation sources are modeled in parallel for supplying load, while the transformer uses a two-state working/failed Markov model.The transformer has failure and repair rates of 0.0004/yr and 0.013/yr, with steady working and failure probabilities of 0.97 and 0.03.
- Renewable models: Solar and wind models each use five renewable-source states and two mechanical states, with distributions fitted from daily irradiation and wind-speed data.Solar irradiation uses a Beta distribution, while wind speed uses a Rayleigh distribution; both are divided into five states.
- System assessment: The component UGFs are combined into a composite generation UGF and then used with the load UGF to compute the DG system reliability indices.The combined generation model covers the five-state wind-speed and solar-irradiation divisions.
- Load model: The load model uses 8736 hourly IEEE-RTS values, groups demand into ten equal intervals, and assigns state probabilities from interval frequencies.The load range spans 1863.5 to 5500 kW, balancing modeling accuracy and evaluation efficiency.
5 Discussion and Conclusion
The paper presents a UGF-based multi-state analytical model for distributed-generation reliability assessment and illustrates it on a modified IEEE 34-node feeder. The discussion identifies modeling assumptions, neglected network effects, and computational challenges that limit scope and motivate efficiency improvements.
- Contributions: The study proposes a UGF-based multi-state analytical model for reliability assessment of distributed-generation systems.
- Contributions: Multi-state sub-models represent solar generators, wind generators, transformers, electrical vehicles, and loads within a unified distribution-network model.
- Application: The composition operator combines component states into a system model that computes reliability indices, with an illustration using a modified IEEE 34-node test feeder.
- Limitations: The model assumes no dependence between renewable energy variables and mechanical states, although extreme irradiation or wind speed may damage generator components and cause failures.
- Limitations: Transmission lines are neglected, even though their failures can cause islanding of downstream network areas.
- Limitations and future efficiency: Computational effort may become critical as the numbers of components and states increase, motivating state clustering and fast convolution algorithms such as FFT.