An operating state estimation model for integrated energy systems based on distributed solution

Abstract

In view of the disadvantages of the traditional energy supply systems, such as separate planning, separate design, independent operating mode, and the increasingly prominent nonlinear coupling between various sub-systems, the production, transmission, storage and consumption of multiple energy sources are coordinated and optimized by the integrated energy system, which improves energy and infrastructure utilization, promotes renewable energy consumption, and ensures reliability of energy supply. In this paper, the mathematical model of the electricity-gas interconnected integrated energy system and its state estimation method are studied. First, considering the nonlinearity between measurement equations and state variables, a performance simulation model is proposed. Then, the state consistency equations and constraints of the coupling nodes for multiple energy sub-systems are established, and constraints are relaxed into the objective function to decouple the integrated energy system. Finally, a distributed state estimation framework is formed by combining the synchronous alternating direction multiplier method to achieve an efficient estimation of the state of the integrated energy system. A simulation model of an electricity-gas interconnected integrated energy system verifies the efficiency and accuracy of the state estimation method proposed in this paper. The results show that the average relative errors of voltage amplitude and node pressure estimated by the proposed distributed state estimation method are only 0.0132% and 0.0864%, much lower than the estimation error by using the Lagrangian relaxation method. Besides, compared with the centralized estimation method, the proposed distributed method saves 5.42 s of computation time. The proposed method is more accurate and efficient in energy allocation and utilization.

Keywords

integrated energy system / state estimation / electricity-gas coupling energy system / nonlinear coupling / distributed solution

Introduction

Integrated energy systems focus on building the interconnection framework of smart grid, thermal pipeline network, natural gas pipeline network, transportation network and other types of networks [¹], so as to achieve the goals such as the coordinated transformation of multiple energy forms and coordinated operating of centralized and distributed energy sources [²]. Compared with the traditional energy supply systems which are planned, designed, and operated independently, the integrated energy system can organically integrate the energy supplies such as electricity, gas, heating or cooling with supporting systems such as transportation and information [³]. Through scientific dispatch of the various energy forms, like traditional energy, renewable energy, cold, heat, electricity, gas, etc., in the integrated energy system, energy can be efficiently utilized, and energy supply can be ensured and enhanced [⁴]. Hence, the integrated energy system has attracted wide attention as soon as it was proposed. However, researches on integrated energy systems are mainly focused on basic theoretical studies or analyses of demonstration project construction [⁵], lacking researches on operating state estimation of the integrated energy systems, to support the optimization of energy conversion and energy dispatch for the increasingly severe energy situation.

At present, with the deepening of the energy reform, the coupling between various energy sub-systems has become increasingly prominent, especially for the power system and the natural gas system. On the one hand, natural gas networks consist of different components, most of which are driven by electricity, such as motor-driven compressor units, gas storage facilities, key valves, pressure regulators, etc. In addition, due to the low cost of installation and maintenance, motor-driven compressor units are increasingly utilized. In view of environmental friendliness, some areas even enforce the use of motor-driven compressors to satisfy the emission requirements [⁶], which greatly increases the interaction between the natural gas network and the power grid. On the other hand, as a safe and reliable fossil fuel, natural gas is increasingly used as primary energy for gas-fired generating stations in the power grid. Driven by a large number of cheap shale gas and low carbon policies, the installed capacity of gas-fired generators has increased substantially [⁷]. Therefore, there is a strong bi-directional coupling between the power grid and the gas network. By the end of 2017, the installed capacity had reached 76.29 million kW, maintaining an 8.8% growth rate in China. In 2017, the national gas-fired generators generated 152.8 billion kWh of electricity, accounting for 2.4% of the whole electricity generation [⁸]. In addition, under the condition that a large number of intermittent renewable energy has been connected to the grids, the large number of gas-fired power plants can be used as peak shaving or standby power to ensure the stable operating of the grids. Due to the close interaction between the power grid and natural gas network, independent analysis of a single system has been unable to provide sufficient information to ensure safe energy supply and efficient energy utilization [⁹]. In this context, it is particularly necessary to make a comprehensive analysis and unified planning of the power system and natural gas system. Shao et al. developed a two-stage optimization planning algorithm for an integrated power and natural gas transmission system by describing the influence between the state and extreme events of the power system through a variable uncertainty set, expecting to improve the restoration capacity of the power grid through integrated planning [¹⁰]. Hence, the present paper attempts to study the operating state estimation of the electricity-gas interconnected integrated energy system. In recent years, for the electricity-gas integrated energy system, Ling et al. established a general electricity-gas multi-energy system model based on the consideration of the characteristics of flexible load on the user side and conducted optimal scheduling analysis, which reduced emissions and costs, and improved the efficiency of electricity-gas conversion [¹¹]. Li et al. reported a security-constrained dispatch model for integrated natural gas and electricity systems, which contains security constraints and coupling constraints. And numerical case studies demonstrate the effectiveness of the proposed model [¹²]. Clegg et al. established a multi-stage integrated gas and electrical transmission network model to quantify the flexibility of the gas network for the power system [¹³]. Fang et al. studied the dynamic optimal energy flow in natural gas and power systems and obtained the optimal operating strategy for the integrated gas and power system [¹⁴].

The precondition of operating optimization of integrated energy system is to accurately, quickly, and comprehensively obtain the precise information about the actual operating state of the system. In the state estimation of the power grid, unknow state parameters, like the voltage and phase of several nodes, are estimated via measurement parameters like power, current, voltage, and phase. There are many main calculation methods that can be applied to a state estimate. Cattivelli et al. used a diffusion recursive least-squares algorithm to study the distributed estimation of adaptive networks, which has no topology constraints and reduces complexity [¹⁵]. Li et al. proposed a heterogeneous decomposition algorithm to optimally dispatch the generation resources and evaluate the power system, and the generation resources are optimized by the efficient and robust algorithm [¹⁶]. Monticelli et al. adopted orthogonal transformation methods to overcome the numerical ill-conditioning of the gain matrix and to realize hybrid state estimation based on observability analysis [¹⁷]. Ni et al. used the measurement transformation method to conquer the variety of measurement categories in the power transmission system and distribution system and the shortcomings of traditional fast decoupling state estimation, and the bad data are effectively processed [¹⁸]. The state estimation of natural gas networks has the power function, which is different from the state estimation of power grid networks with the trigonometric function. Hence, the method used in power grid networks cannot be directly applied to natural gas networks. For natural gas networks, Jalving et al. used steady-state information to initialize a priori and constructed an optimized state estimation framework to track the gas flow and pressure distribution in the dynamic natural gas network, which solved the instability of state estimation [¹⁹]. Behrooz et al. proposed an algorithm to deal with the discontinuity problem in the dynamic model of a high-pressure long-distance gas transmission network. The extended Kalman filter was used to estimate the state of the reference point, and the recursive observability of large-scale network was analyzed [²⁰]. Although some good results have been achieved in the separate state estimation of the power grid and natural gas networks, it is difficult to meet the actual needs and realize rational allocation of multiple energy carriers. It is a general practice to develop integrated energy state estimation.

Compared with the power grid and natural gas networks, the structure of the integrated energy system is more complex, the scale of nodes increases sharply, and a huge difference exists in mathematical descriptions between different energy sub-systems. As a result, it will lead to instability and difficulty for iterative convergence if existing state estimation methods of the power networks and natural gas networks are directly applied to the integrated energy system. For integrated energy systems, Ma et al. proposed an optimal dispatching method by analyzing the quantity and grade of energy loss of different subsystems. The feasibility and validity of this method were verified by simulation analysis [²¹]. Ge et al. estimated the state of a power network and gas network coupling system based on the least square method and studied the influence of multi-system joint state estimation on estimation error [²²]. Dong et al. considered the coupling between the heat network and the power network, completed the state estimation of the combined heat and power networks, and compared the performance of the combined state estimation with the single state estimation in a Monte Carlo experiment [²³]. Zhang et al. studied the application of micro combined cooling, heating, and power generation systems with natural gas as input in the household integrated energy system. By establishing a mathematical model of the household energy system, the coordination and complementarity between electricity and natural gas were analyzed [²⁴]. Given the lack of unified identification and location of integrated energy system faults in existing research, Zhong et al. presented a method for unified identification and location of faults using big data analysis based on the Isomap algorithm, local sparsity coefficient, and node association [²⁵]. Considering parameter redundancy among multiple energy systems, joint state estimation of multiple systems was selected in the present paper to improve estimation accuracy and false data detection rate. Due to the increase in the number and types of equations caused by the joint estimation of multiple subsystems, the primary challenge was to overcome these shortcomings, in order to improve computational speed and estimated accuracy.

As the scale of the integrated energy system increases, the dimension of traditional centralized state estimation increases correspondingly [²⁶], resulting in a longer computational time of state estimation. Traditional centralized state estimation methods are applied with difficulty to integrated energy systems due to different types of mathematical models for different subsystems. Aimed at the complex electricity-gas network system, the distributed computing method is proposed which is beneficial to improving computational efficiency, accuracy, and convergence. With the a deep understanding of the state estimation of integrated energy systems, the latest research focuses on distributed state estimation, which reduces the computation time and improves the redundancy of local measurements by reducing the dimension of the whole state estimation [²⁷,²⁸]. Moreover, the influence of local bad data can be limited to the current region to avoid algorithm non-convergence [²⁹], which will improve computational efficiency. Zhang et al. studied the nonlinear constraints and differential terms in the state estimation of the power network and heat network using the asynchronous alternating direction multiplier method (ADMM). However, in the process of state estimation, the information of each subsystem needs to be collected first, and the complete decentralized solution is not achieved [³⁰]. Ahmadian et al. determined the optimal location and scale of distributed energy using the improved interactive optimization method. This method realizes the optimal allocation and utilization of multiple energies based on the principle of the maximum objective function [³¹]. Jiang et al. put forward a direct regional hot water system model and operating optimization strategy for the comprehensive utilization of wind energy, solar energy, natural gas and electric energy, which can save energy and reduce consumption remarkably in operating [³²].

Distributed computing has advantages in solving multi-system coupling problems. At present, distributed computing has been applied in the state estimation of integrated energy systems. However, the state estimation based on the least-squares method needs to reconstruct the Jacobian matrix in each iteration process, which seriously affects the algorithm efficiency. At the same time, the state estimation model based on the weighted least squares (WLS) method is usually a non-convex optimization problem, and many convex optimization decentralized algorithms are difficult to apply directly. Aiming at the practical operation demands of the integrated energy system and the difficulty in solving the strong coupling state estimation of multi-energy systems, a distributed state estimation method based on ADMM is proposed in this paper, which combines observation model with solution algorithm. The proposed method is more accurate and efficient in energy allocation and utilization.

The present paper summarizes the state estimation problems of the integrated energy systems based on the measurement equations of electric power and natural gas system and the coupled node equations. Besides, it introduces the system state estimation model and studies the distributed state estimation method of an integrated energy system based on the proposed model. Moreover, it proves the accuracy and effectiveness of the proposed method in the state estimation of the integrated energy system through the simulation analysis.

Background of state estimation

State estimation

State estimation was first proposed by Fred Schweppe in 1970 [³³–³⁵]. Since then, state estimation has been extensively studied in several fields such as control theory and engineering. State estimation is to improve the data accuracy and system operation ability, automatically eliminate the uncertainties caused by measurement accuracy, mathematical model error, unexpected system change, random interference, etc., and estimate or predict the operation state of the system by using the redundancy of real-time measurement system [³³]. The ratio of independent measurement variables to state variables is usually 1.5–3.0. According to different types of the model used in state estimation, there are steady-state estimation, which is the basis of research [³⁶], and dynamic state estimation [³⁷]. The research of integrated energy system state estimation in the present paper is a “steady state” state estimation. The maximum likelihood estimation criterion is most commonly used in state estimation, that is, the probability that the measured value \mathit{z} is observed to the greatest extent by finding the optimal estimated value {\mathit{x}}^{*}.

In integrated energy systems, the WLS method [³⁸] is often used to obtain the optimal state vector {\mathit{x}}^{*} from the measured vector \mathit{z}. Its objective is to minimize the square sum J\left(x\right) of deviations between the measured value \mathit{z} and the estimated value, as expressed in Eq. (1).

(1)\min \mathrm{\hspace{0.17em}}J\left(\mathit{x}\right)={[\mathit{z}-h(\mathit{x}\left)\right]}^{\mathrm{T}}{W}^{-1}[\mathit{z}-h(\mathit{x}\left)\right],

where h\left(\mathit{x}\right) is the process simulation model, which describes the relationship between the measured variable vector \mathit{z} and the state variable vector \mathit{x}. W is the measurement accuracy variance matrix. Each diagonal element in the diagonal matrix W corresponds to a sensor accuracy, and the value of the element is determined by the measurement accuracy of the sensor. In actual measurement, the measurement accuracy between different sensors is relatively independent. Therefore, the measured variance matrix is generally a diagonal matrix. W^{-1} denotes the weight of each parameter in the objective equation. As the measurement accuracy of the sensor decreases, the weight of the measurement parameter in the state estimation process increases. Besides, as the accuracy of the sensor decreases, the credibility of the measurement result of the sensor decreases. The influence of the deviation of the measurement result needs to be weakened in the objective function. Therefore, it is hoped its weight is reduced in the calculation, so the inverse matrix method is adopted. In the gas-electricity integrated energy system, the accuracy of the sensor on the gas network side is obviously lower than that on the power grid side. Therefore, the accuracy of the sensors on the power grid side needs to be provided to improve the accuracy of state estimation. Setting the values of \mathit{W} matrix reasonably can improve the state estimation of the integrated energy system.

Integrated energy system state estimation

The structure of a typical electricity-gas integrated energy system is shown in Fig. 1. The power grid system and the gas networks system are connected by the gas power plant and the compressor unit. Although the sub-systems are coordinately operated, each of them enjoys certain autonomy and independence.

In the electricity-gas integrated energy system, the state variable {\mathit{x}}_{{\mathrm{ac}}_k} represents the kth alternating current power grid sub-system, and the state variable {\mathit{x}}_{{\mathrm{gas}}_m} represents the mth gas network sub-system. Subscripts ac and gas respectively represent alternating current power grid and natural gas network.

(2){\mathit{x}}_{{\mathrm{ac}}_k}={[V_{{\mathrm{ac}}_k,i},{\theta }_{{\mathrm{ac}}_k,i}]}^{\mathrm{T}},

where V_{{\mathrm{ac}}_k,i} is the voltage of node i while {\theta }_{{\mathrm{ac}}_k,i} is the phase of node i. The state variable {\mathit{x}}_{{\mathrm{gas}}_m} representing the gas network sub-system can be expressed as

(3){\mathit{x}}_{{\mathrm{gas}}_m}={[p_{m,i},q_{m,ij}]}^{\mathrm{T}},

where p_{m,i} denotes the pressure of node i while q_{m,ij} denotes the pipeline flow. The state variables of the integrated energy system are a set of state variables of all sub-systems, \mathit{x}={[V_{\mathrm{ac},i},{\theta }_{\mathrm{ac},i},p_i,q_{ij}]}^{\mathrm{T}}. Therefore, the relationship between the measured values and the state values of the integrated energy system can be expressed as

(4)\left[\begin{array}{c}\hfill z_{{\mathrm{ac}}_{\mathrm{1}}}\hfill \\ \hfill ⋮\hfill \\ \hfill z_{{\mathrm{ac}}_K}\hfill \\ \hfill z_{{\mathrm{gas}}_{\mathrm{1}}}\hfill \\ \hfill ⋮\hfill \\ \hfill z_{{\mathrm{gas}}_M}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill h_{{\mathrm{ac}}_{\mathrm{1}}}(V_{{\mathrm{ac}}_{\mathrm{1}},i},{\theta }_{{\mathrm{ac}}_{\mathrm{1}},i})\hfill \\ \hfill ⋮\hfill \\ \hfill h_{{\mathrm{ac}}_K}(V_{{\mathrm{ac}}_K,i},{\theta }_{{\mathrm{ac}}_K,i})\hfill \\ \hfill h_{{\mathrm{gas}}_{\mathrm{1}}}(p_{\mathrm{1},i},q_{\mathrm{1},ij})\hfill \\ \hfill ⋮\hfill \\ \hfill h_{{\mathrm{gas}}_M}(p_{M,i},q_{M,ij})\hfill \end{array}\right]+\left[\begin{array}{c}\hfill r_{{\mathrm{ac}}_{\mathrm{1}}}\hfill \\ \hfill ⋮\hfill \\ \hfill r_{{\mathrm{ac}}_K}\hfill \\ \hfill r_{{\mathrm{gas}}_{\mathrm{1}}}\hfill \\ \hfill ⋮\hfill \\ \hfill r_{{\mathrm{gas}}_M}\hfill \end{array}\right],

where z_{{\mathrm{ac}}_k}, h_{{\mathrm{ac}}_k}, and r_{{\mathrm{ac}}_k} represent the measured values, state function, and measurement error of the kth power grid sub-system, respectively; z_{{\mathrm{gas}}_m}, h_{{\mathrm{gas}}_m}, and r_{{\mathrm{gas}}_m} represent the measured value, state function, and measurement error of the mth gas sub-system, respectively.

The operating state estimation model of an integrated energy system based on the least square method is constructed as

(5)\min \left[{\displaystyle {\sum }_{k=1}^K}{J}_{{\mathrm{ac}}_k}\left({\mathit{x}}_{{\mathrm{ac}}_k}\right)+{\displaystyle {\sum }_{m=1}^M}{J}_{{\mathrm{gas}}_m}\left({\mathit{x}}_{{\mathrm{gas}}_m}\right)\right],

(6)J_{{\mathrm{ac}}_k}\left({\mathit{x}}_{{\mathrm{ac}}_k}\right)={\mathit{r}}_{{\mathrm{ac}}_k}^{\mathrm{T}}{\mathit{W}}_{{\mathrm{ac}}_k}^{-1}{\mathit{r}}_{{\mathrm{ac}}_k},

(7)r_{{\mathrm{ac}}_k}=\left(z_{{\mathrm{ac}}_k}-h_{{\mathrm{ac}}_k}(V_{{\mathrm{ac}}_{k,i}},{\theta }_{{\mathrm{ac}}_{k,i}})\right)(k=\mathrm{1},\mathrm{2},...,K),

(8)J_{{\mathrm{gas}}_m}\left({\mathit{x}}_{{\mathrm{gas}}_m}\right)=r_{{\mathrm{gas}}_m}^{\mathrm{T}}{W}_{{\mathrm{gas}}_m}^{-1}{r}_{{\mathrm{gas}}_m},

(9)r_{{\mathrm{gas}}_m}=\left(z_{{\mathrm{gas}}_m}-h_{{\mathrm{gas}}_m}(p_{m,i},q_{m,ij})\right)(m=\mathrm{1},\mathrm{2},...,M),

where W_{{\mathrm{ac}}_k}^{-1} and W_{{\mathrm{gas}}_m}^{-1} are the inverse matrices of the measurement covariance matrices W_{{\mathrm{ac}}_k}^{} and W_{{\mathrm{gas}}_m}^{} of the kth power grid sub-system and mth natural gas network sub-system, respectively; r_{{\mathrm{gas}}_m}^{} is I-dimensional vector in which I is the number of nodes of the mth sub-system; r_{{\mathrm{ac}}_k}^{} is J-dimensional vector in which J is the number of the nodes of the kth sub-system; and r_{{\mathrm{ac}}_k}^{} and r_{{\mathrm{gas}}_m}^{} are measurement errors of the kth power grid sub-system and mth gas network sub-system, respectively.

Power grid system measurement equation

The power grid sub-system vector {\mathit{z}}_{\mathrm{ac}} contains four types of parameters:

(10){\mathit{z}}_{\mathrm{ac}}={\left[P_{ij},Q_{ij},P_i,Q_i\right]}^{\mathrm{T}},

where P_{ij} and Q_{ij} are the active and reactive power flow of branch ij, while P_i and Q_i are the active and reactive power of node i.

As can be seen from the power system model, {\mathit{z}}_{\mathrm{ac}} can be described by the state value {\mathit{x}}_{\mathrm{ac}}=[V_i,{\theta }_{ij}].

(11)z_{\mathrm{ac}}=h\left({\mathit{x}}_{\mathrm{ac}}\right)=h(V_i,{\theta }_{ij}),

(12)P_{ij}=V_i^{\mathrm{2}}{g}_{ij}-V_i{V}_j{g}_{ij}\cos {\theta }_{ij}-V_i{V}_j{b}_{ij}\sin {\theta }_{ij},

(13)Q_{ij}=-V_i^{\mathrm{2}}{b}_{ij}-V_i{V}_j{g}_{ij}\sin {\theta }_{ij}+V_i{V}_j{b}_{ij}\cos {\theta }_{ij},

(14)P_i=V_i{\displaystyle {\sum }_{j\in i}}{V}_j(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}),

(15)Q_i=V_i{\displaystyle {\sum }_{j\in i}}{V}_j(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}),

where g_{ij} is the conductance of branch ij, b_{ij} is the susceptance of branch ij, while G_{ij} and B_{ij} are the real and imaginary parts of element ij in the known admittance matrix, respectively. Equations (12)–(15) constitute a set of overdetermined equations. To make the results of state estimation closer to the true value, the overdetermined problem is transformed into an optimization problem. An approximate solution can be obtained by minimizing Eq. (1).

Natural gas system measurement equation

The operating state of the natural gas pipeline network is decided by pipeline friction resistance C_{ij} and node flow rate q_i, which respectively represents the natural gas transmission capacity and the load of the pipelines. The hydraulic characteristics of the natural gas pipeline network are described by the node pressure p_i and the pipeline flow rate q_{ij}. In the present paper, p_i and q_{ij} are defined as state variables, while the pipeline friction resistance C_{ij} and the node flow rate q_i are defined as measured variables, as expressed in

(16)z_{\mathrm{gas}}={\left[C_{ij},q_i\right]}^{\mathrm{T}},

(17)x_{\mathrm{gas}}={\left[p_i,q_{ij}\right]}^{\mathrm{T}}.

The relationship between the measured variables and state variables of a natural gas subsystem can be derived from the natural gas network model and the pipeline model [³⁹]:

(18){\mathit{z}}_{\mathrm{gas}}=h\left({\mathit{x}}_{\mathrm{gas}}\right)=h(p_i,q_{ij}),

(19)q_i=-{\displaystyle {\sum }_{j=\mathrm{1}}^P}{a}_{i,ij}{q}_{ij},

(20)C_{ij}={\displaystyle \frac{\mathrm{1}}{q_{ij}}}\sqrt{|p_i^{\mathrm{2}}-p_j^{\mathrm{2}}|},

where P represents the number of natural gas pipelines related to node i in a subsystem; a_{i,ij} denotes the element of the i\mathrm{th} row and ij\mathrm{th} column in the associated matrix of the nodes and pipelines; a_{i,ij} is 1, –1, or 0, whose positive or negative attribution is determined by the flow direction of the pipelines. In essence, the matrix associated represents the relationship between the pipelines and the nodes.

Coupled node equation

As illustrated in Fig. 2, there is a coupling between the power system and the natural gas system. Besides, energy transfer is achieved through the gas-fired power plant and the compressor unit. The compressor power consumption P_{{\mathrm{gc}}_m}, pressure ratio p_{\mathrm{gc},i}/p_{\mathrm{gc},j}, power plant fuel consumption G_{{\mathrm{pp}}_{,i}}, and power generation P_{{\mathrm{pp}}_m} satisfy

(21)P_{{\mathrm{gc}}_m}=q_{\mathrm{gc},i}\times C_{\mathrm{gc}}\times \left[{\left(p_{\mathrm{gc},i}/p_{\mathrm{gc},j}\right)}^{v^{\prime }}\right],

(22)v^{\prime }={\displaystyle \frac{v-\mathrm{1}}{v}},

(23)P_{{\mathrm{pp}}_m}=G_{\mathrm{pp},i}\times H{u}_{\mathrm{pp},i}\times {\eta }_{\mathrm{pp},i},

where subscripts gc and pp respectively represent gas compressor and power plant. C_{\mathrm{gc}} and q_{\mathrm{gc},i} are the power coefficient and pipeline flow of the compressor, respectively; G_{\mathrm{pp},i} is the load node flow; H{u}_{\mathrm{pp},i} is the natural gas calorific value; {\eta }_{\mathrm{pp},i} is the power generation efficiency of the gas-fired power plant; and n is polytropic coefficient.

In fact, the common parameters at the coupling nodes need to be calculated at the power grid side and the natural gas networks side respectively. However, they are assigned to the gas network side when calculating the measured parameters.

(24){\mathit{z}}_{\mathrm{gas}}={\left[C_{ij},q_i,P_{\mathrm{gc}},P_{\mathrm{pp}}\right]}^{\mathrm{T}}.

Usually, there is a nonlinear relationship between the measurement equation and the state variables in the power system and the natural gas system. The nonlinearity of the natural gas network and the power grid network equations results in the slow solving speed. Linearization is to accelerate the solution and solve the nonlinear problems. The strong coupling of sub-systems usually results in a non-convex search domain. Therefore, the problem of state estimation may be non-convex. The distributed solution adopted in the present paper is to improve the convexity of the objective function.

Methodology

Classical optimization method

The state variables of the power subsystem and natural gas sub-system are defined as {\mathit{x}}_{{\mathrm{ac}}_k} and {\mathit{x}}_{{\mathrm{gas}}_m}. The state variables \mathit{x} of the integrated energy system are the union of {\mathit{x}}_{{\mathrm{ac}}_k} and {\mathit{x}}_{{\mathrm{gas}}_m}.

(25)\mathit{x}={\left[x_{\mathrm{a}{\mathrm{c}}_1}^{\mathrm{T}},\mathrm{...},x_{{\mathrm{ac}}_K}^{\mathrm{T}}\mathrm{;}\mathrm{}{x}_{{\mathrm{gas}}_1}^{\mathrm{T}},\mathrm{...},x_{{\mathrm{gas}}_M}^{\mathrm{T}}\right]}^{\mathrm{T}}.

The state estimation problem of the integrated energy system can be expressed as a optimization problem with equality constraints in Eqs. (26) and (27).

(26)\min \left[J_{{\mathrm{ac}}_k}\left({\mathit{x}}_{{\mathrm{ac}}_k}\right)+J_{{\mathrm{gas}}_m}\left({\mathit{x}}_{{\mathrm{gas}}_m}\right)\right],

(27)\mathrm{s}\mathrm{.t}\mathrm{.}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}A\left[\begin{array}{c}\hfill x_{{\mathrm{ac}}_{\mathrm{1}}}\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_{{\mathrm{ac}}_K}\hfill \end{array}\right]+B\left[\begin{array}{c}\hfill x_{{\mathrm{gas}}_{\mathrm{1}}}\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_{{\mathrm{gas}}_M}\hfill \end{array}\right]=c,

where J_{{\mathrm{gas}}_m}\left({\mathit{x}}_{{\mathrm{gas}}_m}\right) and J_{{\mathrm{ac}}_k}\left({\mathit{x}}_{{\mathrm{ac}}_k}\right) are given in Eqs. (6) and (8). Equation (27) shows the constraints of the coupling node between the natural gas network subsystem and the power grid subsystem; A and B are the coefficients of the equality constraint. The Lagrangian relaxation method is a common method for convex quadratic programming with equality constraints. A linear term is added to the original objective function in order to achieve the same solution to the original problem when the optimal multiplier is obtained. The augmented Lagrange equation can be transformed as

L_{\rho }({\mathit{x}}_{{\mathrm{ac}}_k},{\mathit{x}}_{{\mathrm{gas}}_m},\mathit{\lambda })

(28)=J_{{\mathrm{ac}}_k}\left({\mathit{x}}_{{\mathrm{ac}}_k}\right)+J_{{\mathrm{gas}}_m}\left({\mathit{x}}_{{\mathrm{gas}}_m}\right)+{\mathit{\lambda }}^{\mathrm{T}}(A{\mathit{x}}_{{\mathrm{ac}}_k}+B{\mathit{x}}_{{\mathrm{gas}}_m}-c)+\rho /2{\left|\right|A{\mathit{x}}_{{\mathrm{ac}}_k}+B{\mathit{x}}_{{\mathrm{gas}}_m}-c\left|\right|}^2,

where \mathit{\lambda } is the Lagrangian multiplier and \mathit{\rho } is the penalty coefficient.

Standard ADMM algorithm

To further improve the accuracy of the state variables {\mathit{x}}_{\mathrm{gas}} and the state variables {\mathit{x}}_{{\mathrm{ac}}_k}, the ADMM method is proposed to calculate {\mathit{x}}^{o+1} through a nonlinear iteration from the calculation result {\mathit{x}}^o and of the state estimation model. The iteration process for the power subsystem and the natural gas sub-system is expressed as

(29){\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}=J\left({\mathit{x}}_{{\mathrm{ac}}_k}^o\right)+{\mathit{\lambda }}^o(A{\mathit{x}}_{{\mathrm{ac}}_k}^o+B{\mathit{x}}_{{\mathrm{gas}}_m}^o-c)+\rho /2\left|\right|A{\mathit{x}}_{{\mathrm{ac}}_k}^o+B{\mathit{x}}_{{\mathrm{gas}}_m}^o-c|{|}^2,

(30){\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}=J\left({\mathit{x}}_{{\mathrm{gas}}_m}^o\right)+{\mathit{\lambda }}^{\mathrm{T}}(A{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}+B{\mathit{x}}_{{\mathrm{gas}}_m}^o-c)+\rho /2\left|\right|A{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}+B{\mathit{x}}_{{\mathrm{gas}}_m}^o-c|{|}^2,

(31){\mathit{\lambda }}^{o+1}={\mathit{\lambda }}^o+\rho (A{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}+B{\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}-c),

(32){\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}=\arg \mathrm{\hspace{0.17em}}\min L_{\rho }({\mathit{x}}_{{\mathrm{ac}}_k}^o,{\mathit{x}}_{{\mathrm{gas}}_m}^o\mathrm{,}{\mathit{\lambda }}^o),

(33){\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}=\arg \mathrm{\hspace{0.17em}}\min L_{\rho }({\mathit{x}}_{{\mathrm{ac}}_k}^{o+1},{\mathit{x}}_{{\mathrm{gas}}_m}^o,{\mathit{\lambda }}^o),

(34){\mathit{\lambda }}^{o+1}={\mathit{\lambda }}^o+\rho (A{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}+B{\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}-c),

where o is the number of iterations; \mathit{\lambda } is the Lagrangian multiplier; and \mathit{\rho } is the penalty coefficient. Therefore, the result of the state variable of the state estimation model can be calculated. When {\mathit{x}}_{{\mathrm{ac}}_k}^{o+1} is calculated first, {\mathit{x}}_{{\mathrm{gas}}_m}^{o+1} can be calculated through the functions described above and the value of {\mathit{\lambda }}^{o+1} can finally be obtained.

Parallel ADMM algorithm

Based on the results of the average value v^o=\frac{{\mathit{x}}_{{\mathrm{ac}}_k}^o+{\mathit{x}}_{{\mathrm{gas}}_m}^o}{2} from the last time calculation, the Parallel ADMM algorithm is established. The state variables {\mathit{x}}_{{\mathrm{ac}}_k}^{} and {\mathit{x}}_{{\mathrm{gas}}_m}^{} representing the power subsystem and the natural gas subsystem and the iterative calculation process is changed as

{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}=f\left({\mathit{x}}_{{\mathrm{ac}}_k}^o\right)+{\mathit{\lambda }}^o(A{\mathit{x}}_{{\mathrm{ac}}_k}^o+B{v}^o-c)

(35)+\rho /2\left|\right|A{\mathit{x}}_{{\mathrm{ac}}_k}^o+B{v}^o-c|{|}^2,

(36){\mathit{\lambda }}^{o+1}={\mathit{\lambda }}^o+\mathit{\rho }(A{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}+B{v}^o-c),

{\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}=g\left({\mathit{x}}_{{\mathrm{gas}}_m}^o\right)+{\mathit{\lambda }}^{\mathrm{T}}(A{v}^o+B{\mathit{x}}_{{\mathrm{gas}}_m}^o-c)

(37)+\rho /2\left|\right|A{v}^o+B{\mathit{x}}_{{\mathrm{gas}}_m}^o-c|{|}^2,

(38){\mathit{\lambda }}^{o+1}={\mathit{\lambda }}^o+\mathit{\rho }(A{v}^o+B{\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}-c).

Distributed state estimation model for integrated energy system

In the state estimation of the integrated energy system, the steady-state model is used without considering the dynamic process. For the integrated energy system, the measured parameters of the gas networks side are pressure and pipeline flow while the parameters of the power grid side are generating power and load power. The coupling between the gas network and the power grid can be described as Eqs. (21)–(23).

Based on these equations, the power of the coupling node is defined as {\mathit{U}}_{\mathrm{cp}}^{\mathrm{ac}}=\{P_{\mathrm{gc,}k}^{\mathrm{ac}},P_{\mathrm{pp,}k}^{\mathrm{ac}}\} and {\mathit{U}}_{\mathrm{cp}}^{\mathrm{gas}}=\{P_{\mathrm{gc,}m}^{\mathrm{gas}}, P_{\mathrm{pp,}m}^{\mathrm{gas}}\}, where subscript cp represents coupling point, while superscripts ac and gas represent the alternating current power grid and natural gas network. Because of the coupling relationship between the power grid subsystem and the gas network subsystem, the estimated power of the coupling nodes should be equal, that is, {\mathit{U}}_{\mathrm{cp}}^{\mathrm{ac}}={\mathit{U}}_{\mathrm{cp}}^{\mathrm{gas}}. In addition, {\widehat{P}}_{\mathrm{gc,}k}^{} and {\widehat{P}}_{\mathrm{pp,}k}^{} are considered as new measurement variables on the power grid side while P_{\mathrm{gc,}k}^{} and P_{\mathrm{pp,}k}^{} are used as the gas networks side measurement. Therefore, some new constraints as expressed in Eqs. (39)–(40) will be introduced into the model.

(39){\widehat{P}}_{\mathrm{gc,}k}^{}=P_{\mathrm{gc,}k}^{},

(40){\widehat{P}}_{\mathrm{pp,}k}^{}=P_{\mathrm{pp,}k}^{},

(41){\widehat{P}}_{\mathrm{gc,}k}^{}=V_i{\displaystyle {\sum }_{j\in i}}{V}_j(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}),

(42){\widehat{P}}_{\mathrm{pp,}k}^{}=V_i{\displaystyle {\sum }_{j\in i}}{V}_j(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}).

According to the objective function, the power grid subsystem and the natural gas network subsystem are decoupled. In addition, the equality constraints constitute the coupling phenomenon occurring in the state estimation of each subsystem. In the present paper, the alternating direction multiplier method [⁴⁰] based on the weighted least squares (WLS-ADMM) is used to solve the problem of system coupling. The basic idea is to relax the equality constraints into the objective function so that the original problem is decomposed into multiple sub-problems and solved in parallel. When the original coupling problem is a convex function, the ADMM method achieves convergence theoretically. First, the objective function is constructed as

(43)\min \left[{\displaystyle {\sum }_{k=1}^K}{J}_{{\mathrm{ac}}_k}\right({\mathit{x}}_{{\mathrm{ac}}_k})+{\displaystyle {\sum }_{m=1}^M}{J}_{{\mathrm{gas}}_m}({\mathit{x}}_{{\mathrm{gas}}_m}\left)\right],

(44)\mathrm{s}.\mathrm{t}.\mathrm{\hspace{0.17em}}{\widehat{P}}_{\mathrm{ac}}^{}=P_{\mathrm{gas}}.

The classical ADMM equations (Eqs.(29)–(34)) and parallel ADDM equations (Eqs. (35)–(38)) can be used in an integrated energy system when \mathit{x} and \mathit{z} are replaced with {\widehat{P}}_{\mathrm{ac}}^{} and P_{\mathrm{gas}}. Hence, the optimization problem for the integrated energy system would be divided into two subproblems of the power grid and gas networks. The key to solving this problem is to choose the appropriate Lagrangian multiplier iteration mechanism. The estimation problem of the power grid and gas neteork can be described as

{\mathit{x}}_{{\mathrm{ac}}_k}^{o+1}=\arg \mathrm{\hspace{0.17em}}\min \left[J_{{\mathrm{ac}}_k}\right({\mathit{x}}_{{\mathrm{ac}}_k}^o)+{\mathit{\lambda }}^o({\widehat{P}}_{\mathrm{ac}}^{}-v^{\mathrm{e},o})

(45)+\rho /2\left|\right|{\widehat{P}}_{\mathrm{ac}}^{}-v^{\mathrm{e},o}\left|{|}^2\right],

(46){\mathit{\lambda }}^{\mathrm{e},o+1}={\mathit{\lambda }}^{\mathrm{e},o}+\mathit{\rho }({\widehat{P}}_{\mathrm{ac}}^o-v^{\mathrm{e},o+1}),

(47){\mathit{x}}_{{\mathrm{gas}}_m}^{o+1}=\arg \mathrm{\hspace{0.17em}}\min \left[J_{{\mathrm{gas}}_m}\right({\mathit{x}}_{{\mathrm{gas}}_m}^o)+{\mathit{\lambda }}^o(P_{{\mathrm{gas}}_m}^{}-v^{\mathrm{g},o})+\rho /2||P_{{\mathrm{gas}}_m}^{}-v^{\mathrm{g},o}|{|}^2],

(48){\mathit{\lambda }}^{\mathrm{g},o+1}={\mathit{\lambda }}^{\mathrm{g},o}+\mathit{\rho }(P_{{\mathrm{gas}}_m}^{o+1}-v^{\mathrm{g},o+1}),

where, the superscripts e and g represent electricity and gas respectively. Equations (45)–(48) have unique common variables \mathit{\lambda } and \mathit{\rho }. The power subsystem and the natural gas subsystem can be solved in parallel. Equations (46) and (48) are the iterative formulas for Lagrange multipliers, and its calculation involves {\widehat{P}}_{\mathrm{ac}}^{} and P_{\mathrm{gas}}. Therefore, it is necessary to continuously update Lagrange multipliers according to the state estimation results of each sub-system in the solution process. Each subsystem processes its own state estimation via the boundary parameters calculated by another subsystem at the last step, and the Lagrange multiplier is updated separately to realize the distributed solution. The update equation of the average value of the variable {\widehat{P}}_{\mathrm{ac}}^o and P_{\mathrm{gas}}^o is expressed as

(49)v^{\mathrm{e},o}=v^{\mathrm{g},o}={\displaystyle \frac{{\widehat{P}}_{\mathrm{ac}}^o+P_{\mathrm{gas}}^o}{\mathrm{2}}}.

The convergence condition of the iterative process is expressed as

(50)

where \mathit{\epsilon }=\left[{\epsilon }_{\mathrm{1}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{\epsilon }_{\mathrm{2}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{\epsilon }_{\mathrm{3}}\right] is the convergence threshold.

Figure 2 demonstrate the calculation process for the state estimation of the integrated energy system, which can be divided into 7 steps.

Step 1: The values of state variables x of the power grid and gas networks system, multiplier, and convergence threshold are initialized. The measurement variables \mathit{z} is inputted into calculation program. The value of state variable x is constantly modified in the iteration.

Step 2: The solution for the power gird and gas networks can be calculated in parallel. {\mathit{x}}_{\mathrm{ac}}^{o+1} and {\mathit{x}}_{\mathrm{gas}}^{o+1} would be solved through the result of {\mathit{x}}_{\mathrm{ac}}^o and {\mathit{x}}_{\mathrm{gas}}^o, while the value of o is increased from zero in the first step.

Step 3: The boundary variables {\widehat{P}}_{\mathrm{ac}}^{} and {\widehat{P}}_{\mathrm{gas}} would be updated according to the new calculated result.

Step 4: The Lagrangian multiplier {\mathit{\lambda }}^{\mathrm{e},o+1} and {\mathit{\lambda }}^{\mathrm{g},o+1} for the power gird and gas networks are updated.

Step 5: Eq. (50) is considered to determine whether the calculation algorithm is convergent or not. The optimal solution would be printed out directly in Step 7 for the convergent condition. Otherwise, the calculation process would be continued to Step 6.

Step 6: The average value of variables in the power gird and gas networks would be updated to obtain v^{\mathrm{e},o} and v^{\mathrm{g},o} according to Eq. (49). The value of o is increased to o + 1 and the next step is repeated form Step 2.

Step 7: Calculation process comes to an end and optimal result prints out.

Results and discussion

Simulation model

To verify the effectiveness of the distributed state estimation method for the electricity-gas integrated energy system, a comprehensive simulation model is built, as depicted in Fig. 3.

The integrated energy system consists of an IEEE 39-nodes power grid system and a 15-nodes natural gas network system. There are 12 power generation nodes (including wind power stations and solar power stations) in the power grid system, of which, node 2 is a balanced node, the reference voltage e_{\mathrm{0},t} is 345 kV, and the reference power W_{\mathrm{0},t} is 100 MVA. Ten gasfired power plants are connected to nodes 30–39. The wind power plant is connected to node 37, and the solar power plant is connected to node 38. To study the impact of load characteristics on system performance, 21 load nodes are divided into residential loads and commercial loads. The specific network parameters and connections can be referred to in Ref. [⁴¹].

The natural gas network consists of 15 nodes which are connected to each other through 12 pipelines. Natural gas is injected into the network through nodes 1 and 2, while nodes 3–12 are the distribution nodes. Natural gas is distributed to the gas-fired power plants. Nodes 13–15 distribute natural gas to meet other gas loads. The diameter of the natural gas pipeline is 500 mm, and the gas pressure in the pipeline varies from 5 to 11 MPa. Due to the frictional resistance of the pipeline, the natural gas pressure is reduced after being transported through the pipeline. To meet the requirements of gasfired power plants on the natural gas pressure and flow, several compressor units are required to increase natural gas pressure along the pipelines. There are four compressor stations in the natural gas network, of which, No. 1 and No. 2 are motor-driven ones. Nodes 22 and 26 of the power grid provide power for No. 1 and No. 2 compressor stations . In No. 3 and No. 4 compressor stations, the natural gas consumed by the gas turbine is taken from the intake pipeline. Tables 1 and 2 list the connections of the gasfired power plants and motor-driven compressors. The main parameters of natural gas networks are given in Table 3.

The measured parameters and points of the integrated energy system are tabulated in Table 4. There are 96 measured parameters in the 39-nodes power sub-system. The standard deviation of the actual measurement random error is 1% of the measured value. There are 45 measured parameters in the natural gas sub-system. The pipeline frictional resistances corresponding to 12 pipelines cannot be measured, which are usually estimated based on the pipeline length and roughness. Hence, there is a certain error between the estimated value and the true value. As a result, there is a deviation between the model simulation results and the state estimation results. Therefore, the pipeline resistance coefficient will be estimated based on the model and other measurement parameters. The error standard deviation of the natural gas sub-system is also defined as 1% of the measured value. The convergence iterations and the operating time are the averages of the multiple simulation results. The measurement error {\overline{S}}_{\mathrm{M}} is defined as

(51){\overline{S}}_{\mathrm{M}}={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{t=\mathrm{1}}^N\sqrt{{\displaystyle \frac{\mathrm{1}}{s}}{\displaystyle \sum _{i=\mathrm{1}}^s{\left({\displaystyle \frac{Z_i-h_i\left(x\right)}{{\sigma }_i}}\right)}^{\mathrm{2}}}}},

where t and N are the sampling time and the number of samples, respectively; i and s are the ordinal number and the total number of measurement points, respectively; Z is the analog value of the ith measurement point for t samplings; and h_i\left(x\right) denotes the measuring equation.

Comparative analysis of simulation results

The state estimation process is tested by a PC with 3.3 GHz, i5 processor and 8 GB RAM. The integrated energy system and state estimation methods are modeled using the MATLAB software. To test the computational performance of the proposed distributed state estimation method, it is compared with the Lagrangian relaxation method based on weighted least squares (WLS-LR), power grid state estimation, and natural gas network state estimation. The three methods listed and compared in the result are Method A which is the proposed method with a coupling node model and ADMM; Method B which is the Lagrangian relaxation method based on WLS-LR; and Method C which is the separated sub-system state estimation without coupled nodes and restraints.

In the simulation, 100 Monte Carlo simulation experiments were performed. The analog measurement value was generated by adding random white noise into the actual value. The measurement value in the power grid is obtained by adding the random error between [0, 1%] to the actual value while the measurement value in the gas network is obtained by adding the random error between [0, 5%] to the actual value. The results of the state estimation for the state variables and measurement variables are presented in Tables 5 and 6, respectively.

The results in Table 5 indicate that the maximum relative errors of voltage amplitude and phase are \mathrm{0.0151}\mathrm{\%} and \mathrm{0.0175}\mathrm{\%} in the proposed method (Method A), respectively while the average relative errors are \mathrm{0.0132}\mathrm{\%} and \mathrm{0.000545}\mathrm{\%}, respectively. In the comparative study of the Lagrange relaxation method (Method B), the maximum relative errors of voltage amplitude and phase are \mathrm{0.0176}\mathrm{\%} and \mathrm{0.0296}\mathrm{\%}, respectively while the average relative errors are \mathrm{0.0144}\mathrm{\%} and \mathrm{0.00783}\mathrm{\%}. Compared with the results of direct measurement, the errors of the proposed method in estimating voltage amplitude and node pressure are reduced by 4 times and 3 times, respectively. When the state estimation of the subsystem is transformed into the state estimation of the integrated system, the grade of accuracy improvement of power grid state estimation is higher than that of gas network state estimation. From the single estimation to the distributed estimation, the proposed method is more accurate than the estimated voltage to estimate the phase of the grid and more accurate than the estimated node pressure to estimate the pipeline flow of the gas network. Even for Method C, the results of state estimation are more accurate than those of direct observation.

From the perspective of state estimation accuracy, the proposed method is more accurate in estimating voltage amplitude and voltage phase based on the aforementioned analyses. The proposed method does not require centralized coordination. From a mathematical point of view, the estimation result of the distributed method is equivalent to that of the centralized method. The algorithm convergence can be quickly achieved because of the consistency of boundary variables. The calculation efficiency is significantly improved for the proposed method. Under normal circumstances, the calculation time of state estimation increases exponentially as the size of the system increases. Therefore, when the scale of the integrated energy system increases, the distributed state estimation method will show greater advantages.

Figures 4 and 5 exhibit the relative errors of estimating voltage amplitude and voltage phase by using the three methods, respectively. For the state estimation of voltage amplitude in Fig. 4, the estimation accuracy is greatly improved by using the state estimation of the integrated system, and it is not obviously improved by further using the distributed calculation. For the state estimation of the voltage phase in Fig. 5, from the subsystem estimation to the distributed calculation, the estimation accuracy is gradually improved. The accuracy obtained by using distributed calculation is the highest.

Figure 6 displays the state estimation results of node pressure. The accuracy of subsystem estimation is significantly improved by using distributed computing and integrated system estimation. Compared with the state estimation of the integrated system, the estimation accuracy of distributed calculation is not always the highest. Figure 7 gives the state estimation results of the pipeline flow rate. The state estimation accuracy of using distributed calculation is the highest. The comprehensive results indicate that the proposed method achieves more accurate estimation results in estimating the gas network pipeline flow rate.

Correction of measurement parameters

In addition to the accurate estimation of the state value, state estimation can also reduce the noise contained in the measurement value and identify bad data. The error is implanted into measurement parameters to serve as the bad data caused by sensor faults. Figure 8 displays the relative errors of gas network state estimation after implanting 5% error of the measured value into the flow rate of Node 7. The three methods can suppress the influence of bad data, but the proposed method can achieve a higher estimation accuracy. Figure 9 shows the relative error of the power grid state estimation after implanting 1% error of the measured value into the active power rate of Node 1. The results indicate that the three methods have achieved the aim of suppressing the bad data contained in measured parameters. The bad data contained in measurement parameters will cause the whole system to oscillate, but the adjustment effect of the node with bad data are not worse than that of other measurement nodes. The proposed method has more excellent performance in suppressing the bad data of gas network than Methods B and C.

Comparison of convergence speed

To verify the calculation speed of different state estimation methods, the average calculation time of 100 Monte Carlo simulation experiments is summarized in Table 7. With the increase of calculation scale, the state estimation of sub-systems only needs to solve the estimation problems of the power grid and gas network respectively, while the state estimation of the integrated system needs to solve the whole system coupling estimation problem. It takes about three times longer for Method B to estimate the integrated energy system by distributed calculation, as shown in Table 7. As the complexity of the system increases, it takes more time for centrlized estiamtion to compute than the proposed distribution method. Therefore, it is necessary to choose the distributed state estimation method according to the actual demands. The convergence processes of different methods are shown in Fig. 10. It is worth noticing that there is no iterative optimization process between sub-systems in Method C. The coupling parameters between the subsystems do not need to be balanced. Therefore, the estimation error is larger. The convergence of the proposed method (Method A) is better than that of Method B.

Conclusions

The integrated energy system is a complex system composed of different subsystems. To be aware of the state of the integrated energy system and suppress the influence of measurement noise, a novel state estimation method is proposed. Compared with the traditional independent estimation of each subsystem, the proposed method can obtain a higher estimated accuracy based on the coupling node model. To further improve the estimated accuracy and speed of the integrated system, a distributed state estimation calculation framework is proposed. The simulation results validate the feasibility of the proposed method. In the simulation study of an integrated energy composed of a power grid with 39 nodes and a natural gas network with 15 nodes, the accuracy, robustness, and calculation speed of the distributed state estimation method have been compared. Besides, the proposed method has been used to elimilate the effects of measurement noise and sensor fault. Based on the findings in this paper, the following conclusions can be reached.

The integrated energy system state estimation method based on the segmentation model and the distributed calculation framework is feasible. The proposed distributed state estimation method can quickly estimate the accurate state of the whole system.

After considering all state variables in the estimation process, the proposed distributed state estimation method can improve the estimated accuracy of the power grid by four times and the estimation accuracy of the natural gas networks by three times.

When the bad data contained in the measurement parameters are suppressed, the proposed method improves the estimated results in the natural gas network more accurately than that in the power grid.

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