Typical scenario set generation algorithm for an integrated energy system based on the Wasserstein distance metric

Accepted Manuscript Typical scenario set generation algorithm for an integrated energy system based on the Wasserstein distance metric

Xueqian Fu, Qinglai Guo, Hongbin Sun, Zhaoguang Pan, Wen Xiong, Li Wang PII:

S0360-5442(17)31096-4

DOI:

10.1016/j.energy.2017.06.113

Reference:

EGY 11121

To appear in:

Energy

Received Date:

17 February 2017

Revised Date:

25 May 2017

Accepted Date:

18 June 2017

Please cite this article as: Xueqian Fu, Qinglai Guo, Hongbin Sun, Zhaoguang Pan, Wen Xiong, Li Wang, Typical scenario set generation algorithm for an integrated energy system based on the Wasserstein distance metric, Energy (2017), doi: 10.1016/j.energy.2017.06.113

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ACCEPTED MANUSCRIPT

Typical scenario set generation algorithm for an integrated energy system based on the Wasserstein distance metric Xueqian Fua, Qinglai Guoa, Hongbin Suna,*, Zhaoguang Pana, Wen Xiongb, Li Wangb a

Department of Electrical Engineering, State Key Laboratory of Electricity Systems, Tsinghua University, China Power Supply Co. Ltd., China

bGuangzhou

ARTICLE INFO

ABSTRACT

Keywords:

The stochastic fluctuation characteristics of intermittent renewable energy sources and energy loads, as well as their multi-energy interactions and dependencies, have negligible effects on the

correlation discretization Wasserstein Distance Euclidean Distance

operation and analyses of integrated energy systems. Determining how to model the probability characteristics of such systems with high calculation accuracy using limited scenarios is a major difficulty of uncertainty description. This study proposes the use of an optimum quantile method based on the Wasserstein distance metric to generate a typical scenario set in an integrated energy system considering energy correlations based on weather conditions. The use of discrete variables, as opposed to continuous variables based on sampling techniques such as Monte Carlo simulations, sets this study apart from other studies. The uncertainties of a typical network containing power, heat, and gas are analysed, and the results show that the proposed method can produce a typical scenario set with good precision.

Nomenclature Acronym list MCS LHS HSS IES CHP RES PV PDF CDF

Monte Carlo sampling Latin hypercube sampling Hammersley sequence sampling integrated energy system combined heat and power renewable energy sources photovoltaic probability density function cumulative distribution function

Symbols Xi, Xj K d Wp P1, P2 u r

two n-dimensional data objects clustering number distance measure Wasserstein distance members of a collection of all probability measures set of all laws of marginals P1 and P2 case r =1 gives the usual Wasserstein

Г G Gmax t μ σ fs ft Fs Ft H P Q Y ϕCHP PCHP

Gamma function real-time solar irradiance the maximum solar irradiance real-time temperature mathematical expectation of temperature standard deviation of temperature margin PDF of the solar sample margin PDF of the temperature sample margin CDF of the solar sample margin CDF of the temperature sample two-dimensional subcopula active power of the nodes reactive power of the nodes nodal admittance matrix node voltage phasor heat output of CHP electrical power output of CHP

cm

heat–power ratio

ղe Fin

electrical efficiency of CHP gas input rate

U

* Corresponding author. Present address: Rm. 3-120, West Main Building, Tsinghua University, 100084, Beijing, China. Tel.: +86 137 0107 3689; Fax: +86 10 6278 3086 – 800. E-mail address: [email protected](X. Fu), guoqinglai@ tsinghua.edu.cn (Q. Guo), [email protected](H. Sun).

ACCEPTED MANUSCRIPT pc Q q zq pqd z0 zQ+1 p0d pQ+1d inf det fkernel λ S N m α, β

PDF of a random variable number of optimum quantiles the order of quantile optimum quantile discrete probability of zq lower bound of a variable upper bound of a variable discrete probability of z0 discrete probability of zQ+1 infimum calculation determinant calculation mutivariate kernel density estimate a smoothing parameter covariance matrix sample size variable dimension shape parameters

fr K pj sij pm pn pi po fon fcom fcp fmi kcp Kmi Kon Tgas a

gas flow at steady state a constant which depends on the length pressure of pipeline node j direction of gas flow pressure of pipeline node m pressure of pipeline node n pressure of pipeline node i pressure of pipeline node o export pipeline flow gas flow through the compressor gas consumption of the compressor inlet pipeline flow compression ratio constant coefficient of the inlet pipeline constant coefficient of the outlet pipeline temperature of gas flow polytropic exponent

1. Introduction 1.1 IES characteristics A custom-designed configuration for different renewable technologies, named the integrated energy system (IES), is being deployed rapidly and widely. Combined production of electricity, heat, and cooling power in trigeneration represents a key option for the development of a high-efficiency and cost-effective IES [1]. In addition to trigeneration, hydrogen [2] and biomass [3] linked to heat and electricity are also favourable ways to extend the IES concept and improve energy efficiency. There are three significant differences between modern IES and traditional power systems: uncertainty, interactions of multi-energy systems, and energy correlation based on weather conditions. This paper is concerned with these three characteristics of IES operation and planning, as they can present serious IES challenges. As a result, the problems encountered and the models and methods utilized in IES are different from those of traditional power systems and should be investigated. In terms of uncertainty, the percentage of intermittent renewable energy sources (RES) has been increasing every year, bringing more and more challenges to existing energy systems. Disregarding uncertainty as an innate characteristic of the real world can make optimal design of IES simple but improper [4]. Design and optimization under the uncertainty of RES and combined heat and power (CHP) have become hot topics in IES. Houwing et al. [5] presented a comprehensive framework for analysing the uncertainties, which have a profound impact on the design of residential RES and micro-CHP systems. Qiao et al. [6] built a mathematical model to obtain the maximum permissible capacity of intermittent RES considering the security constraints of an IES. In order to optimize the energy management of a CHP system, Niknam et al. [7] developed a self-adaptive charge search algorithm in an IES with several RES considering the uncertainty of RES and loads. Azizipanah-Abarghooee et al. [8] proposed the use of chance constrained programming to handle CHP’s economic load dispatch considering the uncertainty of RES and load demands. Optimal use of available resources can be an important strategy for coping with increased variability of an IES. Integrated optimization of intermittent RES can reduce the uncertainty in an IES [9]. Based on the existing CHP-dominated district heating schemes, Mikkola and Lund [10] designed optimal RES strategies to increase the flexibility of an IES. Nuytten et al. [11] found that the flexibility at the demand side could be improved to address the uncertainty of RES and CHP, and they evaluated the different types of thermal energy storages in a district. Krad et al. [12] improved the confidence of stochastic optimization using flexibility reserves based on condensed scenarios in an IES with intermittent RES.

ACCEPTED MANUSCRIPT In terms of interactions of multi-energy systems, the operating characteristics of gas and heat can have a considerable impact on the operating characteristics of power in an IES composed of gas, power, and heat systems. The combined heat and power (CHP) device, which can transform natural gas or some other fuel into a combination of heat energy and electrical power, is the most important interaction device of multi-energy systems. The rise of CHP technologies is changing the energy environment, and the power grid is seen not as a primary power supplier, but instead as a back-up of the IES energy supply chain [13]. The multienergy flow can be calculated based on the unified physical equation model [14] and the Newton–Raphson theory [15]. In order to solve multi-energy optimization problems, Geidl and Andersson [16] presented the new concept of energy hubs, offering a general and simple way of modelling power conversion between different energy carriers. The optimization intervals can be considered as they have great impacts on the scheduling performance of domestic energy consumption [17]. In terms of energy correlation, the correlations among the building heat loads, photovoltaic (PV) and wind generations subject to weather conditions [18-19]. The variations in weather conditions have multiple effects on power flows through interactions with multi-energy systems, and the issue becomes more serious when the capacities of CHP, PV, and wind generation connected to the IES become higher. The weather condition correlations have considerable impact on the probability distributions of multi-energy systems, and the accuracy of the IES simulation depends on whether energy correlations are simulated correctly. In order to analyse the economics, security and stability of IES, it is necessary to model the stochastic fluctuation characteristics of gas, power, and heat considering the three particular characteristics. The main problem this study addresses is the mathematical description of stochastic fluctuation characteristics of gas, power, and heat in an IES subject to a multitude of RES and load variability, increasing interactions of multi-energy systems, and prominent correlation relationships. 1.2 Description of uncertainty The previous studies describe the uncertainty of intermittent RES and energy loads by mainly using sampling technologies based on continuous variables, rather than on discrete variables; moreover, only PV and wind generations are used as study objects in scenario generation. The stochastic fluctuation characteristics of gas, power, and heat can be described based on the numerical characters of the probability distributions and stochastic variables. From the probability feature perspective, the mathematical descriptions of stochastic fluctuation characteristics can be divided into two forms: continuous variable formulation and discrete random variable formulation. The most classic sampling method based on continuous variables is the Monte Carlo sampling (MCS). With respect to the stochastic programming problem, Zhou et al. [20] presented a two-stage stochastic programming model based on the genetic algorithm and MCS method that could find the optimum solution of distributed energy systems. Fu et al. [21] presented a chance constrained programming model based on the MCS method, which can be used to solve the RES allocation problem [22]. The MCS simulation method is an effective digital simulation tool, and it can be improved upon by using the Latin hypercube sampling (LHS) strategy from a sampling standpoint. LHS has proven to be a more successful method than the simple random sampling method [23-24] for probabilistic power calculation. Hammersley sequence sampling (HSS) [25] is another important sampling technology, which has similarities to LHS but is more efficient. The convergence of the stochastic annealing algorithm in optimisation under uncertainty can be improved using the HSS approach [26]. When there are dependent variables in power systems or IES, it is indispensable to control the correlations of energies. The copula function method [27] and Nataf transformation [28] can be used to generate the dependent samples. The former controls correlations using the joint distribution of the cumulative probability distribution functions (CDF), and the latter controls correlations using the Pearson coefficient of the equivalent normal distributions. For discrete random variables, an ocean of samples obeying continuous distributions in the MCS and LHS method is hardly necessary. As the analysis efficiency of a deterministic model based on discrete variables is high, it is preferred to use discrete

ACCEPTED MANUSCRIPT variables for the optimal design and operation of an IES with fluctuating RES and load. The typical scenario set generation technology, which balances efficiency and precision, is an effective tool for the generation of discrete random variables from an ocean of continuous variables. So far, there have been two major research studies on this topic, and they have focused on two methods—wind scenario set generation [29] and PV scenario set generation [30]. It has been proved that typical scenario set generation technologies can be used for electrical power dispatch, and can improve the consumptive capacity of intermittent RES [31]. The previous studies did not investigate typical scenario set generation in an IES based on discrete distributions considering the multi-energy correlations and interactions. The typical scenario technology of an IES is different from those of PV and wind generation and power systems, as there are interactions of multi-energy systems and correlations of different energies in an IES. The interactions of multi-energy systems and energy correlations based on weather conditions can make the IES uncertainty analysis more complex than those of the PV and wind generations and power systems. The typical scenario set generation method is proposed to describe the complex stochastic characteristics in an IES. In order to describe clearly the concept of a “scenario”, the classical concept of a “typical day” in power system can be introduced for illustration purposes. The common feature of the two concepts is that they can capture the key points of the problems of energy systems and provide a feasible and simplified analysis method, the accuracy of which can well meet the needs of actual projects. The two concepts’ uncertainty complexities are different, however, which leads to various problems of operation and design. To be more exact, a “typical day” is subjected to the power load uncertainty, while a “scenario” in an IES is subjected to RES and load uncertainty, multi-energy interactions, and energy correlation. The essence of the typical scenario set generation method is that effective scenarios are modelled simplistically to solve the origin and complexity problem of uncertainty at optimal precision. The typical scenario set generation method depends on two key technologies: correlation control and scenario reduction. It should be noted that the weather correlation is controlled in the form of continuous variables, and the scenario reduction can be seen as a discretisation process. A mixed algorithm combining MCS and the copula function can show good performance in controlling the correlations of continuous variables [18]. The degrees of correlation can be different in different weather scenarios, and diversified weather scenarios should be generated using continuous variables. In order to generate diversified weather scenarios, a mixed algorithm combining LHS and the copula function is proposed to cover the weather correlation distribution. Strong correlation, weak correlation, and a certain degree of correlation can be fairly treated in the proposed mixed algorithm. In order to realise scenario reduction, two discretisation technologies are presented: k-means clustering and the optimum quantile method based on the Wasserstein distance metric. The novel contributions of this paper can be summarised as follows. (1) A mixed algorithm combining LHS and the copula function is proposed to improve the correlation control effects, and diversified weather scenarios with different levels of correlation have been given full consideration. (2) For the first time, the typical scenario set generation method of IES is presented, and two different approximate formulas of random variable bounds are proposed to improve the optimum quantile method, which has a better precision than k-means clustering. (3) The study found that different typical scenario sets are needed for different system managers, such as an external power grid, external gas company, and IES manager, and the probability distributions are different for different kinds of energy systems, subject to the nonlinear nature of IES. The paper is organized as follows. First, the description of the uncertainty analysis approach, i.e. the scenario generation approach is described. Second, the discretization algorithms—k-means clustering based on the Euclidean distance metric and the proposed optimum quantile method based on the Wasserstein distance metric—are introduced to generate typical scenario sets for the IES. Third, the energy system models are presented to solve the probabilistic energy flow calculation problems. Finally, two cases are simulated to verify the effectiveness of the proposed scenario generation approach.

ACCEPTED MANUSCRIPT 2. Scenario Generation Algorithms Scenario generation technology provides an effective way to solve the problems related to the planning and operation of IES suffering from load or power complexity and randomness. From an engineering perspective, scenario generation technologies can convert complex and uncertain problems, which are caused by randomicity, into a series of deterministic problems. The difficulties related to constructing and solving stochastic programming models can be reduced by using the scenario generation technologies, which can avoid the establishment of huge complicated stochastic models. From a mathematical perspective, scenario generation technologies will convert the continuous probability distributions of random vectors into a series of discrete distributions of typical scenarios. The generation of a scientific set of typical scenarios is the basic problem in the study of IES uncertainty. A series of scenarios is presented in Ref. [31] to demonstrate the application of the principle, as shown in Fig. 1. The curve in Fig. 1 represents the probability density functions (PDFs) of the original random vectors, and the bar charts represent the discrete distributions of typical scenarios. The vertical axis shows the values of the PDFs or the discrete probability, and the horizontal axis shows the values of the random vectors. The discrete distributions can be obtained from the original PDFs by using scenario generation algorithms. The clustering algorithm is widely used to generate typical wind-power or PV scenarios in power systems. In order to test the performance of the optimum quantile method, the k-means clustering based on the Euclidean distance metric is presented as a reference in this paper. The optimum quantile method based on the Wasserstein distance metric is another effective tool for the generation of scenarios. The discrete quantiles can be calculated based on the Wasserstein distance metric during the generation of scenarios in an IES. The generated scenarios can inherit the necessary probability distribution information from the original continuous probability distribution information. The flow chart of scenario set generation, based on the discretization of continuous variables, is shown in Fig. 2. 2.1 k-means clustering The existing k-means clustering algorithm used to generate typical wind-power scenario set in power systems is presented in Ref. [29]. The Euclidean distance metric can be used for k-means clustering. The Euclidean distance is the most commonly used distance in clustering analysis, and is denoted as d(Xi , X j ) 

2

2

X i1  X j1  X i 2  X j 2  ...  X in  X jn

2

(1)

The k-means clustering algorithm, whose objective function is the minimum of the sum of squares of the Euclidean distances, and the detailed procedures of the k-means clustering algorithm based on Euclidean distance metric are shown below. k-means clustering algorithm Determine the number of scenarios according to the practical problem; assume K scenarios are needed. Choose point K as the initial centre of mass. REPEAT a) Each sample point will be assigned to the position closest to the mass centre, and K clusters will be formed. b) Recalculate the centre of mass of each cluster. UNTIL Cluster does not change or the number of iterations has reached the maximum limit. Calculate the proportion of sample points in each cluster with respect to all sample points in the entire computing space. Notably, the proportions of sample points in the clusters are the probabilities of the discrete scenarios in an IES. 2.2 Optimum quantile method The precision of the simulated scenarios depends on the distance between the simulated probability distribution and the actual probability distribution. The Wasserstein distance metric can be used to generate CHP or compressor typical scenarios in an IES. The Wasserstein distance between two probability distributions can be calculated as [32]

ACCEPTED MANUSCRIPT W p ( P1 , P2 )  (inf  d r ( X , Y )du ( X , Y ))1/ r

(2)

Therefore, the problem of generating the district distribution can be converted into a problem of minimizing the Wasserstein distance from the actual probability distribution. An example of the discretization of continuous variables is presented in Ref. [30] to show the Wasserstein probabilistic distance metric (as shown in Fig. 3). In Fig. 3, six discrete quantiles are set up to obtain the approximate discrete probability distribution from the original PDFs. The smaller the shaded areas, the more accurate the discretization probability distribution will be. Assuming the continuous PDF of the unidimensional random variable x is pc(x), one can obtain the optimum quantile zq (q = 1, 2,…, Q) with the corresponding discrete probability pdq by solving the following equation:



Zq



pc ( x)1/(1 r ) dx 

2q  1  pc ( x)1/(1 r ) dx 2Q 

(3)

Then, the corresponding probability pdq can be obtained based on the calculated zq. Z q  Z q 1

p dq  Z q  2Z q1 pc ( x)dx, q  1,..., Q

(4)

2

The sum of pdq obtained using the optimal quantile theory in Ref. [31] is not equal to 1. As upper and lower limits of energy variables always exist in an IES, this paper proposes two different approximate formulas for variable bounds more adapted to the discretization of IES variables: Z 0  Z1 2 

p 0d  

(5)

pc ( x)dx



p Qd 1  ZQ  ZQ1 pc ( x)dx

(6)

2

2.3 Probability Density Estimation It is important to calculate an accurate continuous variable distribution function for the scenario generation algorithm based on the Wasserstein distance metric. A Gaussian kernel function is introduced in Ref. [33] for an m-dimensional variable set at a coordinate location x. f kernel ( x) 

1 N

N

 (2 ) i 1

1 m/2

 m det( S )1/ 2

exp(

( x  xi )T S 1 ( x  xi ) )x 2 2

(7)

3. Energy System Models There are two significant trends in modern energy systems: the proportion of the intermittent RES is increasing sharply, and the advantage of the multi-energy complement has aroused more and more concerns. In this situation, the correlation between heat loads and renewable power energy, PV generation for example, can have a considerable effect on IES operation and planning projects. Therefore, dependency simulation control can be a key issue in IES modelling. Owing to the coupling mechanism of CHP in an IES, the power, gas, and heating networks can influence each other. Energy flows, including power flows, gas flows, and heat flows, must balance all kinds of energy and satisfy the constraints of the IES. The Newton–Raphson theory based on the Jacobian matrix can be used to calculate the different energy loads under one matrix. The IES model under study consists of a correlation model, coupled multiple energy model, and probabilistic energy-flow calculation model based on the Newton–Raphson theory, as shown in Fig. 4.

ACCEPTED MANUSCRIPT 3.1 Correlation Model Weather conditions are key factors in the assessment of dependent power and heat load in an IES. The copula-based approach, presented by Nelsen [34], can be used to model the joint probability distribution of dependent energy variables to improve the efficiency, resilience, and reliability of an IES. In order to maintain the continuity of IES study, the building heating load and PV models are created in accordance with those in Ref. [18]. The weather variables involved consist of solar irradiance and outdoor temperature. The surface solar irradiance distribution can be denoted by a Beta distribution [21]: fs (G ) 

(   ) G  1 G  1 ( ) (1  ) ( )(  ) Gmax Gmax

(8)

The distribution of outdoor temperature can be denoted by a Gaussian distribution [18]: f t ( Pd ,i ) 

1

 2

exp[

(t   ) 2 ] 2 2

(9)

Considering the complexity of IES sampling, the Latin hypercube sampling (LHS) method is introduced to improve the efficiency of the copula-based approach. LHS is a stratified sampling method, the purpose of which is to ensure that the entire sample space can be covered across the board, avoiding repeated sampling of a certain sample, and the chosen samples can reflect the distribution laws of the random variables under study. Compared to the traditional MCS random sampling method, the LHS method can reduce the sampling size significantly and has good stability, for the same precision. Assuming the sampling sample size to be N, the cumulative distribution function (CDF) of the copula function, denoted as Z = [z1, z2,…, zN], can be calculated as

zi  H ( Fs  Gi  , Ft  ti ),

i  1, 2, , N

(10)

The correlation model based on LHS is shown in Fig. 5. The LHS algorithm can cover the different levels of the cumulative probability distributions of the sample space; the detailed procedures are shown below. Step 1: Determine the sampling sample size N. Step 2: Calculate the sampling intervals of the copula CDF; the intervals can be [0, 1/N], [1/N, 2/N],…, [(N-1)/N, 1]. Step 3: Select a sample point in each interval randomly; the selected copula CDF sample can be denoted as zi = (i−ε)/N. ε is a random number in [0, 1]. Step 4: Calculate the solar and temperature CDF values based on the copula function; the solar and temperature samples can be obtained using the inverse marginal distributions. Step 5: Calculate the PV power and building heating loads using the solar and temperature samples. 3.2 Multiple Energy Model The energy system models, such as the hydraulic model and thermal model, are in accordance with Refs. [14] and [19]. The model descriptions are not repeated in this paper. The power network, CHP, and gas network models in the IES under study are described as follows. First, the power-flow calculation model for power systems is described as follows:

 P  Re{U (YU )*}  * Q  Im{U (YU ) }

(11)

Second, the gas turbine CHP unit model can be described as follows: PCHP 

CHP cm

(12)

ACCEPTED MANUSCRIPT The gas input rate of the CHP unit can be obtained using Eq. (13). Fin 

PCHP

e

(13)

Third, the pipe flow calculation method for a gas network with a compressor is described. The steady-state flow in the gas pipeline can be described using the following formula [35]:

sign( f r ) f r2  K 2 ( pi2  p 2j )

(14)

The pipeline with compressor driven by a gas turbine can be modelled as in Fig. 6 [15]. The mathematical expressions of the gas pipeline with compressor can be described as follows:

f on  K on ( po2  pn2 ) f cp 

kcp f comTgas qgas

(kcp( a 1)/ a  1)

(15) (16)

f mi  f com  f cp

(17)

f mi  K mi ( pm2  pi2 )

(18)

4. Simulation analysis The major difficulty of uncertainty description is determining how to model the probability characteristics with high approximation accuracy using reduced scenarios. To be specific, reduced scenarios can assure feasibility and efficiency of analysis, but they may also reduce approximation accuracy. As in the “typical day” method used in power systems, the balance of accuracy and efficiency is the core problem. There are two different aspects of the accuracy issue in a simulation of scenarios in IES: the accuracy of correlation control and the accuracy of scenarios reduction. The former reacts to the effects of the weather conditions on an IES using continuous distributions, and the level of accuracy can be measured using the mutual information algorithm. The aim of the latter calculation is to reduce huge scenarios by balancing efficiency and precision. Its accuracy can be measured by comparing the numerical characteristics of distributions at the same typical scenario number. The IES simulation is carried out using MATLAB, and two cases are designed in this paper. Case 1 is used to demonstrate the importance of LHS in improving dependency modelling. Case 2 is designed to check the validation of scenario reduction using the optimum quantile method, and an actual IES containing power, heat, and gas networks is used as the simulation study object. 4.1 Case 1 4.1.1 Correlation control In this case, the actual weather data is the same as that in Ref. [18], including the real daytime outdoor temperature and solar irradiance. The marginal distribution of temperature obeys a Gaussian distribution, and the solar irradiance obeys a Beta distribution. The mathematical expectation and standard deviation of temperature are 0.6151 and 4.4845 C, respectively. The shape parameters of the solar irradiance distribution are 1.1112 and 2.4690, respectively. The existing copula function algorithm in Ref. [18], i.e. the mixed algorithm combining MCS and the copula function, is employed to efficiently model the correlation between the outdoor temperature and solar irradiance. HSS is presented as another algorithm to generate the CDFs of the temperature and solar irradiance. It is important to note that the marginal distribution laws of weather, i.e. temperature or solar irradiance, in the HSS simulation are the same as those in the simulation of the mixed algorithm combining MCS and the copula function. One can compare the joint distributions of temperature and solar irradiance calculated using different sampling algorithms in order to

ACCEPTED MANUSCRIPT illustrate the differences between the algorithms. Distributions of the weather conditions in two dimensions have been calculated using actual data, the mixed algorithm, and the HSS algorithm, as shown in in Figs. 7, 8, and 9, respectively. One can see that the distributions of the mixed algorithm are more similar to the actual distributions than those of the HSS algorithm. The distributions of the HSS algorithm are the most uniform, and this uniformity in the two-dimensional space can be identified as a correlation, which is very different from the actual correlation. To describe it differently, although the distribution uniformity of the HSS algorithm would help to solve mathematical optimisation problems, it is hardly necessary for correlation control of temperature and solar irradiance. In order to check the simulation precision of the correlations, the mutual information [19] between the temperature samples and solar irradiation samples is calculated using the actual weather data, HSS data, and mixed algorithm data, with a sample size of 293. As shown in Fig. 10, the mutual information of the mixed algorithm combining MCS and the copula function is closer to the actual value, than that of the HSS algorithm. The more accurate the calculated mutual information is, the more accurate the simulated correlation will be. One could imagine that the mixed algorithm combining MCS and the copula function could control the correlation at a high level. More specifically, the copula function could accurately simulate the joint distribution law of the temperature and solar irradiance CDFs. 4.1.2 Weather scenario The variation in temperature and solar irradiance always comes with the variation in their joint distribution and correlation relationship. One could hold the view that the correlation between temperature and solar irradiance could be related to the sampling time. There is no sunshine at night, so no correlation exists. The correlation between temperature and solar irradiance at dawn would be different from that at noon. That is to say, the correlation between temperature and solar irradiance changes continually. One might think that different weather correlations could indicate a different weather scenario, and different degrees of correlation could stand for different types of weather scenario. In the previous studies, it has been found that the copula function could accurately capture each kind of weather correlation, and MCS could generate a multitude of weather correlations at random [18]. When the sample size is not sufficiently large, the correlation at a certain level may be lost and not generated. Hence, a mixed algorithm combining LHS and the copula function is proposed to ensure that each level of correlations can be generated as a representative type of weather scenario. According to the viewpoints of probability and mathematical statistics, the previous and newly proposed mixed algorithms are identical in terms of the marginal distribution laws and joint distribution laws in continuous distribution conditions, and the difference between them is that they generate different distribution probabilities using different methods of sampling. The previous mixed algorithm simulates the copula CDF at random, while the proposed mixed algorithm simulates the copula CDF based on LHS theory. Five hundred group samples are simulated to show the validity of the proposed mixed algorithm in generating the representative types of weather scenario. It is assumed that at least 50 different types of weather scenario must be generated for the analysis of an IES plan or schedule. As shown in Fig. 11, two types of weather scenario, i.e. CDF in [0.9, 1], using the previous mixed algorithm are lost. One important thing to note about the scenario types is that the number of types of weather scenario must be increased when the precision requirements become more stringent. The time horizon, such as hour, day, and year, and the uncertainty itself will also suffer the strict requirements of weather scenarios. With the same sample size, the proposed mixed algorithm can cover the weather scenario distributions better than the previous mixed algorithm. Essentially, Ref. [18] can ensure an ensemble of weather correlations without any guarantees of a representative level of weather correlations, while the mixed algorithm combining LHS and the copula function can ensure that each representative level of correlations can be generated.

ACCEPTED MANUSCRIPT

4.2 Case 2 In this case, the sample data of CHP, PV, and heat loads are calculated based on the simulated weather samples, as shown in Figs. 12 and 13, which are generated using the mixed algorithm combining LHS and the copula function in case 1. A typical network containing power, heat, and gas systems [15] is presented as the study object, and is shown in Fig. 14. The primary parameters of the power system, heating system, and gas system used in the simulation are listed in Tables 1, 2, and 3. To ensure the safe and stable operation of the IES, three main energy-management bodies are involved. The three managers are the power grid manager, gas manager, and IES manager. The gas station is under the control of the gas manager, whose task is to dispatch gas to ensure IES gas supply. The main concern of the gas station manager is the amount of gas that must be supplied to the IES under different scenarios. As there is no gas load for GB1, GB2, and GB3, the amount of gas supplied to the IES is equal to the amount of gas flowing through GB1 to GB2. The active and reactive power flows, and the voltage of the connection point between the IES and the power grid are the main concerns of the power grid manager. In order to reduce the influence of the IES on the power grid, the power flows and the voltage of the connection point are fixed. Thus, the stochastic behaviour of the IES has no impact on the power grid, and the scenarios do not need much consideration. The 12-busbar electrical power network, 12branches heating network, and 7-pipes gas network with a compressor are under the control of the IES manager. As the PV power and building heating loads are stochastic and uncontrolled factors, the IES manager must dispatch the CHP to ensure multiple energy balance in different scenarios. Two CHP units connect the three different energy systems. CHP 1 operates under the gridorientated operation mode, and CHP 2 operates under the heat-orientated operation mode. CHP 1 undertakes the task of fixed electrical power generation; therefore, there is no uncertainty about the scenario. In conclusion, the gas station and CHP 2 are the scheduling objects with random characteristics, and their scenarios should be generated for the safe and stable operation of the IES. The operational parameters of the gas system and CHP used in the simulation are listed in Table 4. According to the simulated temperatures and solar samples in Case 1 based on the proposed dependency model, one can calculate the PV1 and PV2 voltages using the Newton–Raphson theory, as shown in Figs. 15 and 16. The PDFs of PV1 and PV2 voltages can be estimated using the kernel density estimation technology, as shown in Fig. 17. One can calculate the samples and PDF of the compression ratio of the natural gas compressor based on the proposed pipeline model, as shown in Figs. 18 and 19. The PV voltages and compression ratio are important operation parameters of the IES, and the PDFs can help the IES manager evaluate the operation state of the IES. One can calculate the CHP 2 power and gas supply by using the Newton–Raphson theory, as shown in Figs. 20 and 21. The gas station and IES managers can dispatch the gas flow and CHP 2 output according to the typical scenario set of IES. The results of the proposed scenario method have been compared and checked against Ref. [29]. The major difference between this study and Ref. [29] is that, in this study, the typical scenario set is generated by the optimum quantile method, rather than the kmeans clustering method used in Ref. [29]. Ten typical scenario sets for CHP 2 and the gas station are generated, as shown in Figs. 22, 23, 24, and 25. The operation boundaries of the CHP and gas station can be given full consideration based on the two proposed approximate formulas. After applying Eqs. (5) and (6), one can find that the sum of the probabilities of the different scenarios can equal 1 for the Wasserstein distance metric method. It is important to note that the PDF distributions of the continuous variables are drawn to analyse the accuracy of discretization by comparing the distribution shapes, and not the values, as the comparisons between the PDF curves and the discretization

ACCEPTED MANUSCRIPT probabilities do not make sense. For example, the symmetries of the probability distribution, as well as the bound probability, can be treated as visualizing performance metrics to judge the performance of the discretization methods. By comparing the shapes of PDF distributions of continuous variables, one can conclude that the Wasserstein distance metric can produce a more accurate scenario set. It is important to note that the scenario probabilities of CHP 2 are different from those of the gas station. The complexity and nonlinearity of IES are the major reasons, and different managers face different scenario problems. In order to thoroughly evaluate the accuracy of discretization methods, including the Wasserstein distance metric and Euclidean distance metric, the actual scenarios have been generated using the actual sample data. The numerical features of both actual scenarios and simulated scenarios have been calculated and analysed. The mean values, standard deviations, maximum values, and minimum values of the CHP and gas station are shown in Table 5. The mean values, standard deviations, maximum values, and minimum values obtained using the proposed optimum quantile method based on Wasserstein distance metric are more accurate than those obtained using the k-means clustering based on the Euclidean distance metric. More importantly, extreme scenarios can be generated with the optimum quantile method based on the Wasserstein distance metric; however, k-means clustering based on the Euclidean distance metric cannot be used to obtain the energy limits. The analysis of upper and lower bounds and their probabilities are indispensable for the safe operation of the IES. The IES manager and gas station manager can solve the planning and operation problems by analysing the probability distributions, mean values, standard deviations, maximum values, and minimum values. 5. Conclusions This study focused on addressing the uncertainty involved in the design and operation of CHP systems, including RES. To be specific, this study proposed a description method of stochastic fluctuation characteristics, i.e. typical scenario set generation technology, in IES with good accuracy and a relatively small number of scenarios. The typical scenario set generation technologies for intermittent RES, such as wind power and PV power, have been studied intensively in the field of power systems, and this research extended the study to IES, for ensuring safe and stable operation. In order to cope with the uncertainty problems brought on by the stochastic nature of PV power and building heating loads, technologies to simulate the scenario probabilities were proposed. Specifically, the mixed algorithm combining LHS and the copula function was proposed to model the dependency between PV power and heat load, with good accuracy. The Euclidean distance metric and the Wasserstein distance metric were introduced to discretize the random variables with continuous probability distributions by converting the stochastic problem into a deterministic problem. Compared with the k-means clustering algorithm, the optimum quantile method could ensure discretization with better precision, and the probabilities of the boundaries of the random variables could be calculated using the proposed two different approximate formulas. The new distance metric, i.e. the Wasserstein distance metric, is crucial for extreme scenario generation and accuracy improvement, and the precise and practical scenario sets for CHP and the gas station could be obtained using the optimum quantile method based on the Wasserstein distance metric. Acknowledgements This work was supported in part by National Natural Science Foundation of China (NSFC) (51537006), National Key R&D Program of China (2016YFB0901300), and Project funded by China Postdoctoral Science Foundation (2016M590096). References [1] Chicco, G. and Mancarella, P., Matrix modelling of small-scale trigeneration systems and application to operational optimization. Energy, 2009.34(3): p.261-273. [2] Nastasi, B. and Basso, G. L., Hydrogen to link heat and electricity in the transition towards future Smart Energy Systems. Energy, 2016. 110: p. 5–22. [3] Kalina, J. Complex thermal energy conversion systems for efficient use of locally available biomass. Energy, 2016. 110: p.

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P., et al., Probabilistic load flow evaluation with hybrid Latin hypercube sampling and cholesky decomposition. IEEE Transactions on Power Systems, 2009. 24(2): p. 661-667. [24] Shu, Z., and Jirutitijaroen, P., Latin hypercube sampling techniques for power systems reliability analysis with renewable energy sources. IEEE Transactions on Power Systems, 2011. 26(4): p. 2066-2073. [25] Hammersley, J. M., Monte Carlo methods for solving multivariable problems. Annals of the New York Academy of Sciences, 1960. 86(3): p. 844-874. [26] Papadopoulos, A. I., Giannakoudis, G., Seferlis, P. and Voutetakis, S., Efficient design under uncertainty of renewable power generation systems using partitioning and regression in the course of optimization. Industrial & Engineering Chemistry Research, 2012. 51(39): p. 12862-12876. [27] Stephen, B., Galloway, S. J., McMillan, D., et al., A copula model of wind turbine performance. IEEE Transactions on Power Systems, 2011. 26(2): p. 965-966. 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Table 1. Primary parameters of power system Transmission line

Impedance (p.u.)

Bypass capacitor

13-1

0.02+j0.016

0

1-2

0.008205+j0.019207

0

1-5

0.008205+j0.019207

0

1-3

0.008205+j0.019207

0

5-6

0.008205+j0.019207

0

5-11

0.008205+j0.019207

0

5-9

0.008205+j0.019207

0

2-12

0.008205+j0.019207

0

3-4

0.008205+j0.019207

0

6-7

0.008205+j0.019207

0

6-8

0.008205+j0.019207

0

9-10

0.008205+j0.019207

0

Table 2. Primary parameters of heating system Heat pipe

Length (m)

Diameter (mm)

13-1

500

200

1-2

400

200

2-3

600

200

4-3

400

200

12-4

600

200

1-5

200

200

1-6

150

200

2-7

180

200

2-8

150

200

3-9

100

200

3-10

110

200

4-11

90

200

Table 3. Primary parameters of gas system Gas pipe

Length (m)

Diameter (mm)

1-3

500

150

2-6

2500

150

3-4

500

150

3-5

400

150

5-7

600

150

7-6

200

150

3-2

2500

150

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Table 4. Primary parameters of gas system and CHP Item

Parameter

Value

GB1

0 m3/h

GB2

0 m3/h

GB3

0 m3/h

GB4

1100 m3/h

GB5

1647 m3/h

GB6

2000 m3/h

GB7

2000 m3/h

Power generation

0.77 MW + j0.9 MVar

Voltage

1.05 p.u.

Reactive power limits

−1~2 MVar

Voltage

1.05 p.u.

Fixed gas load

CHP 1 CHP 2

Table 5. Statistics of CHP and gas station

Actual CHP 2

Optimum quantile k-means clustering Actual

Gas station

Optimum quantile k-means clustering

Mean

Standard deviation

Maximum

Minimum

4.542

1.210

8.152

0.708

4.543

1.263

8.152

0.708

4.540

1.173

6.797

1.945

7.570

0.211

8.206

6.907

7.570

0.220

8.206

6.907

7.569

0.204

7.965

7.119

Fig. 1. Sketch map of scenarios

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Continuous variable samples Probability density Estimation Probability distribution function

Euclidean Distance

k-means clustering

Wasserstein Distance

Samples and probabilities of discrete variables Fig. 2. Flow chart of scenario set generation

Fig. 3. Wasserstein’s probabilistic distance metric

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Heat load

PV power

Weather

Dependence Model Copuled Model

Gas network

Heating network

Electricity network

Newton Raphson Method

Heat flow

Gas flow

Power flow

CHP Discretization

Gas scenarios

Power scenarios

Heat scenarios

Scenario generation Fig. 4. IES stochastic model under study

Latin Hypercube Sampling Copula CDF

Copula Function Solar CDF

Temperature CDF

Inverse of Beta Distribution Solar Sample

Inverse of Gaussian Distribution Temperature Sample

PV Generation Model

Building Heat Load Model

Fig. 5. Correlation model based on LHS

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Fig. 6. Pipeline with compressor driven by gas turbine

1 0.9 0.8

CDF of Solar

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CDF of Temperature Fig. 7. Distributions of weather CDFs using actual data

0.9

1

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1 0.9 0.8

CDF of Solar

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF of Temperature Fig. 8. Distributions of weather CDFs using mixed algorithm

1 0.9 0.8

CDF of Solar

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CDF of Temperature Fig. 9. Distributions of weather CDFs using HSS algorithm

0.9

1

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0.09 0.08

mutual information

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

Actual

Mixed algorithm

HSS

Fig. 10. Mutual information between temperature and solar irradiance

40 Previous Proposed 35

30

Frequency

25

20

15

10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

CDF of copula function Fig. 11. Frequency histogram of copula function

0.7

0.8

0.9

1

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20

Temperature (oC)

15

10

5

0

-5

-10

-15

0

50

100

150

200

250

300

350

400

450

500

Sampling points Fig. 12. Temperature samples of the mixed algorithm combining LHS and the copula function

1 0.9 0.8

Solar (kW/m2)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

150

200

250

300

350

400

450

Sampling points Fig. 13. Solar samples of the mixed algorithm combining LHS and the copula function

500

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Fig. 14. Schematic diagram of the typical IES consisting of power, heat, and gas networks

1.05

1.04

Voltage (p.u.)

1.03

1.02

1.01

1

0.99

0.98

0

50

100

150

200

250

300

350

Sampling points Fig. 15. Samples of the PV1 voltages

400

450

500

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1.04

1.03

Voltage (p.u.)

1.02

1.01

1

0.99

0.98

0.97

0

50

100

150

200

250

300

350

400

450

500

Sampling points Fig. 16. Samples of the PV2 voltages

0.4 PV 1 PV 2

0.35 0.3

PDF

0.25 0.2 0.15 0.1 0.05 0 0.96

0.97

0.98

0.99

1

1.01

1.02

Voltage (p.u.) Fig. 17. PDFs of the PV1 and PV2 voltages

1.03

1.04

1.05

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1.6

1.5

1.45

1.4

1.35

1.3

1.25

0

50

100

150

200

250

300

350

400

450

500

Sampling points Fig. 18. Samples of the compression ratio of the natural gas compressor

12

10

8

PDF

Compression ratio

1.55

6

4

2

0

1.3

1.35

1.4

1.45

Compression ratio Fig. 19. PDF of the compression ratio of the natural gas compressor

1.5

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9 8

CHP2 Power （ MW（

7 6 5 4 3 2 1 0

0

50

100

150

200

250

300

350

400

450

500

400

450

500

Sampling points Fig. 20. Samples of the CHP 2 power

8.4

Gas station supply (m3/h)

8.2 8 7.8 7.6 7.4 7.2 7 6.8

0

50

100

150

200

250

300

Sampling points Fig. 21. Samples of the gas supply

350

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0.35 PDF Probability

0.3 0.25

0.2 0.16605 0.16747 0.14685

0.15

0.15553 0.1338

0.1212 0.1

0 0

0.053009

0.044754

0.05 0.0056744 1

0.0056556 2

3

4 5 CHP2 Power （ MW（

6

7

8

9

Fig. 22. Typical scenario set for CHP 2 using optimum quantile method

0.35 PDF Probability

0.3

0.25

0.2

0.19

0.19

0.16 0.142

0.15

0.1 0.072 0.056 0.062 0.05

0 0

0.052

0.034

1

2

3

4 5 CHP2 Power （ MW（

6

Fig. 23. Typical scenario set for CHP 2 using k-means clustering

0.042

7

8

9

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0.2 0.18

PDF Probability

0.16656 0.16748

0.16

0.15102

0.15071

0.14

0.13381 0.12078

0.12 0.1 0.08

0.057143

0.06 0.041233

0.04 0.02 0.0053646 0

0.005902 7000

7200

7400

7600

7800

8000

8200

Gas station supply (m3/h)

Fig. 24. Typical scenario set for the gas station using optimum quantile method

0.2

0.19

0.19

PDF Probability

0.18 0.16

0.16

0.142

0.14 0.12 0.1 0.08

0.072 0.056

0.06 0.04

0.062 0.052 0.042

0.034

0.02 0

7000

7200

7400

7600 Gas station supply (m3/h)

7800

Fig. 25. Typical scenario set for the gas station using k-means clustering

8000

8200

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Highlights 

The paper presents the scenario technology of an integrated energy system.



The copula method is improved by combining it with a sampling technology.



Two formulae of bounds are presented to improve the optimum quantile method.



The study found that different scenario sets are needed for different system managers.



The model is studied using a typical network containing power, heat, and gas.