e-Highway 2050: The importance of spatial correlations in Monte Carlo adequacy simulations

Challenge
What is the bias induced when neglecting spatial correlations in Monte Carlo adequacy simulations in the framework of planning methodologies?

Background and assumptions
Monte Carlo methods are commonly used for the reliability assessment of power systems which are subject to stochastic phenomena. They are particularly of interest in contexts with a large integration of Renewable Energy Sources (RES) or in countries with volatile consumptions. They consist in analyzing the response of the power system to many possible situations with various degrees of occurrence.
Monte Carlo methods allow to broadly explore the space of possible situations that the system could face. Starting from a set of uncertain events randomly sampled, simulations of the system behavior are carried out. These simulations are repeated for a sufficient number of samples so that aggregated results lead to a good approximation of the probabilistic response of the power system.
Monte Carlo methods require the generation of Time Series (TS), for instance on consumption or on RES generation. TS need to be sampled in accordance with their probability of occurrence. To do so, TS generation aims to reproduce the intrinsic characteristics of the considered phenomena learnt in past realizations, such as their seasonality, temporal and spatial correlations.
The focus of the present method is on the reproduction of the spatial correlations. Spatial correlations are the inter-dependencies between time series of a same phenomenon (e.g. loads and generation) in different locations of the system, and typically result from the weather and climatic conditions which tend to affect simultaneously neighbouring areas. Neglecting spatial correlations induces a bias in the sampling of the TS.
The generation of spatially correlated TS is a topic which has already been addressed in the literature, notably for WPG (Wind Power Generation). However studies on their impact on the operation of the power system remain rare. The present article addresses the analysis of the influence of spatial correlations in Monte Carlo system adequacy  studies.

1.      Methodology
The objective is to sample inter-dependent time series for n locations of the studied system. The proposed methodology can be adapted to different continuous stochastic phenomena, notably the electrical consumption, photovoltaic / wind power generation and hydrological inflows.
The proposed generating process is described in Figure 1. The main principle is to first generate the random variables so as to build the stochastic component of the phenomenon (step 1 to 3) and then to combine it with two deterministic patterns: the trend and the seasonality (step 4).
First,  independent Gaussian white noises are sampled with a pseudo random-number generator and are then used to generate correlated Gaussian white noises. Then, for each zone of the system, a new stochastic component is computed. The stochastic component is constructed using either a seasonal ARMA (autoregressive–moving-average) or a diffusion-type model.

Figure 1: Proposed methodology correlated time series of a stochastic phenomena

Finally, the stochastic component is combined with a trend and a seasonality, into a multiplicative model, resulting in the final time series. The trend describes the multi-year evolution of the TS and is deduced from the state of the power system, or from the hypotheses on its future development. The seasonality includes the intra-year variations of the TS and represents the expected average behaviour of the studied phenomenon along the year. It is mainly explained by the yearly, weekly and daily cycles of the climatic conditions and consumption patterns.
A critical step is the estimation of the correlation matrix. Classical methods, which include the capture of cross-correlations observed in the historical data, face tractability issues for large problems. The proposed approach adopts an empirical search through an iterative update of such a matrix: for each iteration, long time series of stochastic components are generated until the correlations observed in the historical data are obtained. On the tested datasets, convergence is reached after a few iterations.

2.      Test Case
In order to assess the sensitivity of some adequacy results to spatial correlations, the TS generation methodology has been tested within Monte Carlo adequacy simulations. An adequacy simulation consists in determining how the generating units operate to meet the electrical demand. In the Monte Carlo approach, such simulations are performed under several patterns of uncertainties, i.e. using several randomly drawn TS of the stochastic phenomena as inputs.

The French power system has been modelled using the hypotheses of the energy scenario “Nouveau Mix” proposed in the 2012’s adequacy report of RTE [5]. This scenario assumes a favourable RES integration at the horizon 2030, with respectively 14% and 6% of the load provided by wind and solar energy.
The generation fleet has been divided up between the seven zones (regional dispatching) presented in Figure 2 assuming a proportional development of the demand and RES capacities.


Figure 2: Seven zones of the modelled French power system

For the adequacy simulations, the stochastic phenomena considered in this test case are the load, the wind power generation, the PV generation, the inflows and the outages of the thermal units (considered uncorrelated here). TS models of WPG, PV generation and demand have been calibrated for the seven zones of France. The correlation matrices have been calculated for these three phenomena, based on historical data. The observed spatial correlations are presented in Figure 3.


Figure 3: Spatial correlation between seven zones in France, calculated from historical data, after removing their trend and their seasonality; Correlations between zones below 30% are not shown

Two cases are distinguished for TS generation: with (using the correlation matrix) and without spatial correlation (the TS of the seven zones of France are sampled independently).  Once the TS are generated, they are used in a sequential adequacy model which aims at simulating the behaviour of the hydro and thermal units by finding the most economic generation schedule which balances the load. 100 Monte Carlo years have been simulated for both cases. 

3.      Results
Reliability of the system
For each Monte Carlo year, a few reliability indicators have been retrieved: the energy not served (ENS), the loss of load duration (LOLD) the energy in excess (EIE) and the total operational cost. The sensitivity of the reliability indicators to spatial correlations is reported in Table 1 (assuming the value of ENS at 10 k€/MWh).


Table 1: Sensitivity of reliability indicators to spatial correlations

Note that in this test-case, France is supposed to be an isolated system and international exchanges are not modelled. It explains the high value of ENS and LOLD.

The case study shows that not taking into consideration spatial correlations clearly biases the results of the adequacy simulations. ENS and LOLD are low when the spatial correlations are neglected, while they become substantial when correlations are accounted for: positive spatial correlations tend to gather extreme events (such as consumption peaks) of the different zones at the same moment in time. Equivalently, simultaneous peaks in RES generation increase the risk of having RES curtailments. EIE is indeed much higher when the correlations are taken into account.
In other words, the variability of the residual load is undervalued when spatial correlations are neglected and the probability of occurrence of the situations which are critical for the system is biased. 

Inter-zonal flows
Impact assessment of the correlations on the inter-zonal exchanges in the power system is based on an analysis of the time series of injections (generation minus consumption) of each zone.
The average injection of each zone is slightly affected by the modelling of the spatial correlations. The spatial correlations tend to enhance the intermittencies of the load and RES, which results in an increased use of peaking units. That is to say, spatial correlations affect both the energy mix and the average inter-zonal exchanges in the system.
The zone with the highest sensitivity is the North-West zone, i.e. the zone with the highest installed nuclear capacity. When spatial correlations are neglected, the generation of this zone increases by 7% (64 MWh/h in average). On the contrary, the generation of the zones with few nuclear power plants and a larger fleet of more expensive thermal units tends to decrease.
Moreover, the variability of the injections is also impacted by spatial correlations. For example Figure 4depicts the distribution of the injection of the West zone over the 100 Monte Carlo years. For this zone, the distribution is less spread out when spatial correlations are modelled. Four other zones of the system follow the same trend (North, North West, North East and South East). A possible explanation is the variations of the residual load which tend to be synchronised in different zones of the system when correlations are modelled. As a consequence, inter-zone flows are lower during the occurrence of extreme events as these events are more likely to be global (i.e. occur simultaneously in several zones).
The modelling of spatial correlations therefore impacts the distribution of injections. These effects depend on the energy sources of each zone so that average injections can be increased or reduced and the distribution around this average can be spread out or tightened.
In the context of transmission expansion and long-term planning studies, reinforcement options can therefore be gauged incorrectly if spatial correlations are neglected.


Figure 4: Distribution of the injections of zone West, with and without spatial correlations

4      Conclusions
This work highlights the importance of taking into account spatial correlations in Monte Carlo studies and their impact on the results of adequacy simulations. Neglecting spatial correlations can lead to an over-estimation of the reliability of the system, and can clearly result in a bias when estimating the inter-zonal exchanges.

Significant spatial correlations exist at the pan-European scale (e.g. strong dependencies of the North Sea winds affecting the wind power production of France, Great Britain, Belgium, the Netherlands and Denmark). It is strongly recommended that spatial correlations are taken into account in future studies of the pan-European power system.

References
[1]    A. Papavasiliou and S. S. Oren, "Stochastic modeling of multi-area wind power production “, in Proceedings of PMAPS 2012, Istanbul, Turkey, June 10-14, 2012.
[2]    D.C. Hill, D. McMillan, K.R.W. Bell and D. Inflied, "Application of Auto-regressive Models to UK Wind Speed Data for Power System Impact Studies," IEEE Transactions on Sustainable Energy, vol.3, no.1, pp.134-141, Jan. 2012.
[3]    J.M. Morales, R. Mínguez, A.J. Conejo, "A methodology to generate statistically dependent wind speed scenarios," Applied Energy, vol. 87, no. 3, Pages 843-855, March 2010.
[4]    RTE, Generation Adequacy Report 2012
[5]    B. Seguinot, A. Zani “The role of spatial correlations in Monte Carlo studies on power system”, Powertech, 2015