Integrated energy operation considering the dependence of multiple wind turbines

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Introduction
In recent years, global warming and pollution have become increasingly important issues. For the sustainability of human society, it is important to promote the reduction of carbon dioxide for moving to a low-carbon economy [1]. With this in mind, the technology of integrated energy systems (IES), consisting of wind power, gas-fired generation and combined heat and power (CHP), has attracted more attention in recent years, due to its high efficiency in generating electricity and low carbon intensity [2]. Complementarity between different energy sources is the key feature of the IES, but it does not effectively address the problem of carbon dioxide emissions. Research into an effective low-carbon IES operating framework is therefore urgently needed in the era of low-carbon development around the world.
Additionally, in the energy system with large centralized wind turbine integration, the performance of different wind turbines is generally considered to be independent [3]. Indeed, the spatial dependency and coupling of wind power cannot be ignored. Failure to do so will have the effect of operation due to inaccurate power flow calculations. In [4], Linear regression and the Cholesky decomposition are each developed to describe the correlation between the two. However, the above techniques are based on the assumption that the relationship is linear with constant matrices. The nonlinear revelation approach based on the Nataf transformation, a kind of copula technique, is proposed to address this problem [5]. It is an approach that is revelatory. It is possible to arrive at a recognized practice in a short space of time schedule by assuming that the errors will follow the Gaussian distribution [6]. Moreover, a sufficiently large non-Gaussian variable follows an almost Gaussian distribution, which is based on the central limit theorem [7]. In fact, obtaining the joint distribution is the simplest way to fully describe the correlations of random variables. To represent the joint distribution of wind speeds associated with different wind turbines and capture dependencies between different wind turbines, the multivariate Weibull function is derived, but the assumed distribution may not correspond to reality [8]. A powerful tool for fitting the joint distribution of wind power is the Gaussian Mixture Model. However, it is difficult to implement a multivariate parameter estimation when the dimension of the random inputs is very large. Therefore, to obtain the power correlation between multiple wind turbines, an advanced parametric estimation method, multivariate Gaussian copula, is proposed.
Low carbon emissions under the dependence on multiple wind turbines, however, are not adequately considered in existing work on integrated IES, which is reflected in a lack of economic efficiency in the planning decisions. Therefore, the low-carbon oriented integrated energy operation considering the dependence of multiple wind turbines is proposed to study the couplings of outputs of multiple wind turbines, which targets determining the dependent structure among multiple wind turbines in IES, so as to decouple the dependence of regional wind turbines.

Integrated energy system description and mathematical modelling
The main objective aims to achieve the coordinated operation for a low-carbon oriented integrated electricitygas-heat energy system, which is established on the basis of multiple energy sectors including natural gas-fired generators, coal-fired generators, and CHP units, electric boilers (EB) and wind turbines.

Objective
In order to realize the economic operation of the lowcarbon oriented IES, the objective of the proposed IES framework for coordination operation is to minimize the total operation cost. The objective function is presented as follows: represent the carbon emission volume of coal-fired and gas-fired generation at time t, respectively,

Electricity network constraints
The power flow where , v t  and w,t  denote respectively the voltage angle of nodes v and w in line l at time t, and l Im represents the impedance of line l.
The operation constraints of generation including output constraint, ramping up and ramping down constraints, are formulated as: where D t P , W t P and CHP t P are the load demand, power output of wind turbine and CHP generator at time t, respectively.

Natural gas network constraints
In this paper, the model of the main components of the proposed natural gas network involves gas well, gas pipeline, compressor and gas load.
The gas productions of the gas well Moreover, the relationship between outlet pressure Pj,t through the compressor and inlet pressure Pn,t can be expressed as Pj,t=ρt×Pn,t, where ρt is the compression ratio of compressor.
The gas balance equation in the natural gas network is expressed as: are respectively the natural gas load demand, the consumption of gas-fired generators and CHP generators at time t.

Heating network constraints
The heat sources incorporate CHP generator and electric boiler. The relationship between heat production where Tms,in, Tms,out, Tmr,in and Tmr,out are respectively the nodal temperature before and after the water mixes in the supply and return pipes, msin, msout, mrin and mrout denote the mass flow rates before and after the water mixes in the supply and return pipes, respectively. The water outflowing temperature equals to the mixed temperature, i.e., Tms,out=Tms, Tmr,out=Tmr, Tms and Tmr are the mixing nodal temperature of the water supply pipes and the return pipes, respectively. The relationship between the heat load and temperature can be expressed as: H is the heat load at time t, p C is the heat capacity of water, and t m is the amount of water injected at time t.
The heat load balance of heating network is

The dependence of multiple wind turbines
Wind turbines always gather in the region with rich wind resources. Hence, the dependence of regional wind power for multiple wind turbines should be considered under the circumstance of large-scale wind energy integration. In this paper, we propose a multivariate Gaussian copula function for the sake of obtaining the correlation among multiple wind turbines and sampling procedure of wind power. With regard to a vector w = [w1,…,wd] of wind power outputs from d wind turbines, the joint distribution function M can be obtained by the connection of marginal distribution functions with the unique copula function C.
where ui = F(wi), F(wi) is the cumulative distribution function (CDF) of wi for ith wind turbine. Then, the probability density function (PDF) m of M can be obtained by The multivariate Gaussian copula function of d different wind turbines is given by where A is the covariance matrix of wind power outputs from different wind turbines. The Sobol sequence is introduced to study the sampling of wind power for different wind turbines.

Case study
We make a comparison for the IES with and without dependence of multiple wind turbines of wind power from different wind turbines, which is shown in Fig. 1. The case without the dependence of multiple wind turbines is involved in "Case 2" and the case considering the dependence of multiple wind turbines is set in "Case 2".  Fig. 1 and Table 1, it can be observed in Case 2 that the power curves of wind turbines are more compact in the situation of considering the uncertain couplings, and the power curves of wind turbines in Case 1 are loosely distributed nevertheless. Furthermore, the power output of wind turbines in Case 2 improves 17.3887 MWh and the operation cost reduces 0.2352×10 6 $ comparing to that of Case 1. Therefore, Case 2 considering the uncertain couplings promotes the correlation of multiple wind turbines, has the advantages of compact distribution, and reduces the uncertain risk of integrated wind power on the basis of ensuring operation costs. Furthermore, the Spearman correlation coefficient ρs and the Kendal correlation coefficient ρk are employed to describe the linear correlation and consistent correlation between the marginal distributions. By the calculation, ρs and ρk in Case 2 are respectively 0.8696 and 0.9637, larger than 0.7841 and 0.8945 of Case 1, and the probability of the same variation trend in Case 2 is higher than that of Case 1. Therefore, the IES operation considering the uncertain couplings of wind turbines can better reflect the mutual relationship between wind turbines, thereby reducing the error caused by the correlation of wind turbines output.

Conclusion
In this paper, a novel model for integrated energy system considering carbon emission and dependence of multiple wind turbines is proposed consisting of electricity, gas and heating systems with the dependence of multiple wind turbines. This paper aims to meet and balance the operation economy, dependence requirements, wind power accommodation, carbon emission reduction effectively and efficiently by controlling the level of carbon intensity. Case studies demonstrate that the coordinated operation of the integrated electricity-gasheat energy system considering the carbon emission can reduce the system operation costs and carbon emission to some extent. Detailedly, the IES operation cost with the dependence of multiple wind turbines reduces 0.2352×10 6 $, and the operation cost reduces 5.83% compared to those without the dependence of multiple wind turbines. Furthermore, it is necessary to consider the correlation of uncertain wind power, and multivariate Gaussian Copula is effective in describing the dependence and complicated correlation among random variables of wind power among different wind turbines.