Energy Flow Calculation Method for Electric-Gas Integrated Energy System Based on Preprocessed Bi-Conjugate Gradient Stabilized Restarted Holomorphic Embedding

CAO Yuxin; ZHI Menglei; LI Xue

doi:10.12204/j.issn.1000-7229.2026.08.010

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Electric Power Construction ›› 2026, Vol. 47 ›› Issue (8) : 140-160. DOI: 10.12204/j.issn.1000-7229.2026.08.010

Energy Flow Calculation Method for Electric-Gas Integrated Energy System Based on Preprocessed Bi-Conjugate Gradient Stabilized Restarted Holomorphic Embedding

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Abstract

[Objective] Energy flow calculation is fundamental to the planning and operation of integrated energy systems （IES）. The existing holomorphic embedding method requires significant computational effort and long solution times for large-scale IES due to the need for solving coefficient equations of power series and the Padé approximation process. [Methods] This paper proposes an energy flow calculation method for electrical-gas IES based on preconditioned bi-conjugate gradient stabilized （BICGSTAB） restarted holomorphic embedding. The method combines preconditioned BICGSTAB and total multiplication of polynomial （TMP） for grid-side holomorphic embedding power flow calculation， while employing the restarted holomorphic embedding method for gas network energy flow calculation. When solving the power flow at the grid side， the coefficient matrix for solving the power series coefficient equations is first preconditioned； then， the constant value dynamic update mechanism in the Holomorphic Embedding Load Flow Method based on constant values is used to solve the voltage approximation value by using Padé approximation and TMP. When solving the side energy flow of gas network， the restarted holomorphic embedding method is used to calculate the energy flow of gas network， so as to improve the calculation speed and convergence of energy flow. Finally， the proposed method is validated using the E39-G20 and E2383-G60 test systems. [Results] The results demonstrate that the proposed method achieves a maximum relative error of only 0.051 3% compared to the Newton-Raphson method， with a maximum computational speedup ratio of 358.0883. [Conclusions] The proposed method enables accurate and efficient energy flow analysis in the electricity-gas IES， maintains high IES energy flow solution accuracy under various operating conditions， and offers strong robustness， which can provide a new idea for the efficient calculation of IES energy flow.

Key words

energy flow calculation / holomorphic embedding method / total multiplication of polynomial / precondition

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CAO Yuxin , ZHI Menglei , LI Xue. Energy Flow Calculation Method for Electric-Gas Integrated Energy System Based on Preprocessed Bi-Conjugate Gradient Stabilized Restarted Holomorphic Embedding[J]. Electric Power Construction. 2026, 47(8): 140-160 https://doi.org/10.12204/j.issn.1000-7229.2026.08.010

附录 A

（A1）

$\left\{\begin{array}{l}\left\{\begin{array}{l}\sum _{k=1}^{N}{Y}_{ik,\mathrm{t}\mathrm{r}}^{\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}\left[1\right]={S}_{i}^{\mathrm{*}}{W}_{i}^{\mathrm{*}}\left[0\right]-{Y}_{i,\mathrm{s}\mathrm{h}}{V}_{i}\left[0\right]-\sum _{k=1}^{N}{Y}_{ik,\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}\left[0\right]-{I}_{i}^{\left(0\right)}\\ \sum _{k=1}^{N}{Y}_{lk,\mathrm{t}\mathrm{r}}^{\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}\left[1\right]+\mathrm{j}{W}_{i}^{\mathrm{*}}\left[0\right]{Q}_{l}\left[1\right]={P}_{l}{W}_{i}^{\mathrm{*}}\left[0\right]-{Y}_{l,\mathrm{s}\mathrm{h}}{V}_{l}\left[0\right]-\sum _{k=1}^{N}{Y}_{lk,\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}\left[0\right]-{I}_{l}^{\left(0\right)}\\ {V}_{l}\left[0\right]{V}_{l}^{\mathrm{*}}\left[1\right]+{V}_{l}\left[1\right]{V}_{l}^{\mathrm{*}}\left[0\right]={V}_{lm}^{2}-{\left|{V}_{l}^{\left(0\right)}\right|}^{2}\\ {V}_{r}\left[1\right]={V}_{r}^{\mathrm{s}\mathrm{p}}-1\end{array}\right.,n=1\\ \left\{\begin{array}{l}\sum _{k=1}^{N}{Y}_{ik,\mathrm{t}\mathrm{r}}^{\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}\left[n\right]={S}_{i}^{\mathrm{*}}{W}_{i}^{\mathrm{*}}[n-1]-{Y}_{i,\mathrm{s}\mathrm{h}}{V}_{i}[n-1]-\sum _{k=1}^{N}{Y}_{ik,\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1]\\ \sum _{k=1}^{N}{Y}_{lk,\mathrm{t}\mathrm{r}}^{\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}\left[n\right]+\mathrm{j}{W}_{i}^{\mathrm{*}}\left[0\right]{Q}_{l}\left[n\right]={A}_{l}[n-1]-\sum _{k=1}^{N}{Y}_{lk,\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1]\\ {V}_{l,\mathrm{r}\mathrm{e}}\left[0\right]{V}_{l,\mathrm{r}\mathrm{e}}\left[n\right]+{V}_{l,\mathrm{i}\mathrm{m}}\left[0\right]{V}_{l,\mathrm{i}\mathrm{m}}\left[n\right]=\frac{1}{2}\sum _{k=1}^{n-1}{V}_{l}\left[k\right]{V}_{l}^{\mathrm{*}}[n-k]\\ {V}_{r}\left[n\right]=0\end{array}\right.,n>1\\ {A}_{l}[n-1]={P}_{l}{W}_{i}^{\mathrm{*}}[n-1]-{Y}_{l,\mathrm{s}\mathrm{h}}{V}_{l}[n-1]-\mathrm{j}\left(\sum _{k=1}^{n-1}{Q}_{l}\left[k\right]{W}_{i}^{\mathrm{*}}\right[n-k\left]\right)\end{array}\right.$

式中：W_i[0]为PQ节点电压逆幂级数零阶项系数；V_l_，re[0]为PV节点电压实部幂级数零阶项系数；V_l_，im[0]为PV节点电压虚部幂级数零阶项系数。

（A2）

$\mathit{A}=\left(\begin{array}{ccccccc}1& 0& 0& 0& \cdots & 0& 0\\ 0& 1& 0& 0& \cdots & 0& 0\end{array}\right)$

（A3）

$\left\{\begin{array}{l}\mathit{H}=\left(\begin{array}{cccccccc}{\mathit{H}}_{(l,1),1}& {\mathit{H}}_{(l,1),2}& \cdots & {\mathit{H}}_{(l,1),(i,1)}& \cdots & {\mathit{H}}_{(l,1),(l,a)}& \cdots & {\mathit{H}}_{(l,1),(i,b)}\\ ⋮& ⋮& & ⋮& & ⋮& & ⋮\\ {\mathit{H}}_{(i,1),1}& {\mathit{H}}_{(i,1),2}& \cdots & {\mathit{H}}_{(i,1),(i,1)}& \cdots & {\mathit{H}}_{(i,1),(l,a)}& \cdots & {\mathit{H}}_{(i,1),(i,b)}\\ ⋮& ⋮& & ⋮& & ⋮& & ⋮\\ {\mathit{H}}_{(l,a),1}& {\mathit{H}}_{(l,a),2}& \cdots & {\mathit{H}}_{(l,a),(i,1)}& \cdots & {\mathit{H}}_{(l,a),(l,a)}& \cdots & {\mathit{H}}_{(l,a),(i,b)}\\ ⋮& ⋮& & ⋮& & ⋮& & ⋮\\ {\mathit{H}}_{(i,b),1}& {\mathit{H}}_{(i,b),2}& \cdots & {\mathit{H}}_{(i,b),(i,1)}& \cdots & {\mathit{H}}_{(i,b),(l,a)}& \cdots & {\mathit{H}}_{(i,b),(i,b)}\end{array}\right)\\ {\mathit{H}}_{(l,1),1}=\left(\begin{array}{cc}{G}_{(l,1),1}& -{B}_{(l,1),1}\\ {B}_{(l,1),1}& {G}_{(l,1),1}\end{array}\right)\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathit{H}}_{(i,b),(l,a)}=\left(\begin{array}{cc}{G}_{(i,b),(l,a)}& -{B}_{(i,b),(l,a)}\\ {B}_{(i,b),(l,a)}& {G}_{(i,b),(l,a)}\end{array}\right)\end{array}\right.$

式中：G_（_l_，1），1为第1个PV节点与节点1之间的电导值；B_（_l_，1），1为第1个PV节点与节点1之间的电纳值；G_（_i_，_b_），（_l_，_a_）为第b个PQ节点与第a个PV节点之间的电导值；B_（_i_，_b_），（_l_，_a_）为第b个PQ节点与第a个PV节点之间的电纳值；H_（_l_，1），1为由元素G_（_l_，1），1和B_（_l_，1），1组成的矩阵。

（A4）

$\left\{\begin{array}{l}\mathit{W}\left[0\right]=\left(\begin{array}{ccc}{\mathit{W}}_{(l,1)}\left[0\right]& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& {\mathit{W}}_{(l,a)}\left[0\right]\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),{\mathit{W}}_{(l,1)}\left[0\right]=\left(\begin{array}{c}{W}_{(l,1),\mathrm{i}\mathrm{m}}\left[0\right]\\ {W}_{(l,1),\mathrm{r}\mathrm{e}}\left[0\right]\end{array}\right)\\ \mathit{V}\left[0\right]=\left(\begin{array}{cccccccc}0& {V}_{(l,1)}\left[0\right]& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {V}_{(l,1)}\left[0\right]& 0& 0\end{array}\right),{\mathit{V}}_{(l,1)}\left[0\right]=\left(\begin{array}{cc}{V}_{(l,1),\mathrm{r}\mathrm{e}}\left[0\right]& {V}_{(l,1),\mathrm{i}\mathrm{m}}\left[0\right]\end{array}\right)\end{array}\right.$

式中：W_（_l_，1）[0]为第1个PV节点电压逆幂级数零阶项系数；W_（_l_，_a_）[0]为第a个PV节点电压逆幂级数零阶项系数；W_（_l_，1），im[0]为第1个PV节点电压逆虚部幂级数零阶项系数；W_（_l_，1），re[0]为第1个PV节点电压逆实部幂级数零阶项系数。

（A5）

$\left\{\begin{array}{l}\mathit{V}\left[n\right]={\left(\begin{array}{ccccccccc}{V}_{1,\mathrm{r}\mathrm{e}}\left[n\right]& {V}_{1,\mathrm{i}\mathrm{m}}\left[n\right]& {V}_{(l,1)}\left[n\right]& \cdots & {V}_{(i,1)}\left[n\right]& \cdots & {V}_{(l,a)}\left[n\right]& \cdots & {V}_{(i,b)}\left[n\right]\end{array}\right)}^{\mathrm{T}}\\ \mathit{Q}\left[n\right]=\left(\begin{array}{c}{Q}_{(l,1)}\left[n\right]\\ ⋮\\ {Q}_{(l,a)}\left[n\right]\end{array}\right)\\ {V}_{(l,1)}\left[n\right]=\left(\begin{array}{c}{V}_{(l,1),\mathrm{r}\mathrm{e}}\left[n\right]\\ {V}_{(l,1),\mathrm{i}\mathrm{m}}\left[n\right]\end{array}\right)\end{array}\right.$

式中：Q_（_l_，1）[n]为第1个PV节点无功第n阶幂级数系数；Q_（_l_，_a_）[n]为第a个PV节点无功第n阶幂级数系数；V_（_l_，1），re[n]为第1个PV节点电压实部第n阶幂级数系数；V_（_l_，1），im[n]为第1个PV节点电压虚部第n阶幂级数系数。

（A6）

${\mathit{R}}_{\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{k}}[n-1]=\left(\begin{array}{c}{\delta }_{n1}({V}_{1}^{\mathrm{s}\mathrm{p}}-1)\\ 0\end{array}\right)$

（A7）

$\left\{\begin{array}{l}{\mathit{R}}_{\mathrm{P}\mathrm{V},\mathrm{P}\mathrm{Q}}[n-1]=\left(\begin{array}{c}{\mathit{R}}_{(l,1)}[n-1]\\ ⋮\\ {\mathit{R}}_{(i,1)}[n-1]\\ ⋮\\ {\mathit{R}}_{(l,a)}[n-1]\\ ⋮\\ {\mathit{R}}_{(i,b)}[n-1]\end{array}\right)\\ {\mathit{R}}_{(l,1)}[n-1]=\left(\begin{array}{c}\mathrm{R}\mathrm{e}\left({A}_{(l,1)}\right[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(l,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ \mathrm{I}\mathrm{m}\left({A}_{(l,1)}\right[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(l,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\end{array}\right)\\ {\mathit{R}}_{(i,1)}[n-1]=\left(\begin{array}{c}\mathrm{R}\mathrm{e}\left({S}_{(i,1)}^{\mathrm{*}}{W}_{(i,1)}^{\mathrm{*}}\right[n-1]-{Y}_{(i,1),\mathrm{s}\mathrm{h}}{V}_{(i,1)}[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(i,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ \mathrm{I}\mathrm{m}\left({S}_{(i,1)}^{\mathrm{*}}{W}_{(i,1)}^{\mathrm{*}}\right[n-1]-{Y}_{(i,1),\mathrm{s}\mathrm{h}}{V}_{(i,1)}[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(i,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\end{array}\right)\end{array}\right.$

（A8）

${\mathit{R}}_{\mathrm{P}\mathrm{V}}[n-1]=\left(\begin{array}{c}\frac{1}{2}\sum _{k=1}^{n-1}{V}_{(l,1)}\left[k\right]{V}_{(l,1)}^{\mathrm{*}}[n-k]\\ ⋮\\ \frac{1}{2}\sum _{k=1}^{n-1}{V}_{(l,a)}\left[k\right]{V}_{(l,a)}^{\mathrm{*}}[n-k]\end{array}\right)$

（A9）

${\mathit{H}}_{1}=\left(\begin{array}{ccccccc}{\mathit{H}}_{(l,1),2}& \cdots & {\mathit{H}}_{(l,1),(i,1)}& \cdots & {\mathit{H}}_{(l,1),(l,a)}& \cdots & {\mathit{H}}_{(l,1),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {\mathit{H}}_{(i,1),2}& \cdots & {\mathit{H}}_{(i,1),(i,1)}& \cdots & {\mathit{H}}_{(i,1),(l,a)}& \cdots & {\mathit{H}}_{(i,1),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {\mathit{H}}_{(l,a),2}& \cdots & {\mathit{H}}_{(l,a),(i,1)}& \cdots & {\mathit{H}}_{(l,a),(l,a)}& \cdots & {\mathit{H}}_{(l,a),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {\mathit{H}}_{(i,b),2}& \cdots & {\mathit{H}}_{(i,b),(i,1)}& \cdots & {\mathit{H}}_{(i,b),(l,a)}& \cdots & {\mathit{H}}_{(i,b),(i,b)}\end{array}\right)$

（A10）

${\mathit{V}}_{1}\left[n\right]={\left(\begin{array}{ccccccc}{V}_{(l,1)}\left[n\right]& \cdots & {V}_{(i,1)}\left[n\right]& \cdots & {V}_{(l,a)}\left[n\right]& \cdots & {V}_{(i,b)}\left[n\right]\end{array}\right)}^{\mathrm{T}}$

（A11）

$\left\{\begin{array}{c} \boldsymbol{H}_{2}=\left(\begin{array}{ccccccc} \boldsymbol{H}_{21(l, 1), 2} & \cdots & \boldsymbol{H}_{(l, 1),(i, 1)} & \cdots & \boldsymbol{H}_{21(l, 1),(l, a)} & \cdots & \boldsymbol{H}_{(l, 1),(i, b)} \\ \vdots & & \vdots & & \vdots & & \vdots \\ \boldsymbol{H}_{21(i, 1), 2} & \cdots & \boldsymbol{H}_{(i, 1),(i, 1)} & \cdots & \boldsymbol{H}_{21(i, 1),(l, a)} & \cdots & \boldsymbol{H}_{(i, 1),(i, b)} \\ \vdots & & \vdots & & \vdots & & \vdots \\ \boldsymbol{H}_{21(l, a), 2} & \cdots & \boldsymbol{H}_{(l, a),(i, 1)} & \cdots & \boldsymbol{H}_{21(l, a),(l, a)} & \cdots & \boldsymbol{H}_{(l, a),(i, b)} \\ \vdots & & \vdots & & \vdots & & \vdots \\ \boldsymbol{H}_{21(i, b), 2} & \cdots & \boldsymbol{H}_{(i, b),(i, 1)} & \cdots & \boldsymbol{H}_{21(i, b),(l, a)} & \cdots & \boldsymbol{H}_{(i, b),(i, b)} \end{array}\right) \\ \boldsymbol{H}_{21(l, 1), 2}=\binom{-B_{(l, 1), 2}}{G_{(l, 1), 2}} \end{array} \quad ⿱ 1. r r r r r e=1\right),{ }^{\cdots}$

（A12）

${\mathit{H}}_{\mathrm{P}\mathrm{V}}={\left(\begin{array}{ccccccccccc}{G}_{(l,1),2}& {B}_{(l,1),2}& \cdots & {G}_{(i,1),2}& {B}_{(i,1),2}& \cdots & {G}_{(l,a),2}& {B}_{(l,a),2}& \cdots & {G}_{(i,b),2}& {B}_{(i,b),2}\\ ⋮& ⋮& & ⋮& ⋮& & ⋮& ⋮& & ⋮& ⋮\\ {G}_{(l,1),(l,a)}& {B}_{(l,1),(l,a)}& \cdots & {G}_{(i,1),(l,a)}& {B}_{(i,1),(l,a)}& \cdots & {G}_{(l,a),(l,a)}& {B}_{(l,a),(l,a)}& \cdots & {G}_{(i,b),(l,a)}& {B}_{(i,b),(l,a)}\end{array}\right)}^{\mathrm{T}}$

（A13）

${\mathit{V}}_{2}\left[n\right]={\left(\begin{array}{ccccccc}{V}_{(l,1),\mathrm{i}\mathrm{m}}\left[n\right]& \cdots & {V}_{(i,1)}\left[n\right]& \cdots & {V}_{(l,a),\mathrm{i}\mathrm{m}}\left[n\right]& \cdots & {V}_{(i,b)}\left[n\right]\end{array}\right)}^{\mathrm{T}}$

（A14）

$-\mathit{B}=\left(\begin{array}{ccccccc}-{B}_{(l,1),2}& \cdots & -{B}_{(l,1),(l,a)}& \cdots & -{B}_{(l,1),(l,a)}& \cdots & -{B}_{(l,1),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ -{B}_{(i,1),2}& \cdots & -{B}_{(i,1),(i,1)}& \cdots & -{B}_{(i,1),(l,a)}& \cdots & -{B}_{(i,1),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ -{B}_{(l,a),2}& \cdots & -{B}_{(l,a),(i,1)}& \cdots & -{B}_{(l,a),(i,a)}& \cdots & -{B}_{(l,a),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ -{B}_{(i,b),2}& \cdots & -{B}_{(i,b),(i,1)}& \cdots & -{B}_{(i,b),(l,a)}& \cdots & -{B}_{(i,b),(i,b)}\end{array}\right)$

（A15）

$\mathit{G}=\left(\begin{array}{ccccccc}{G}_{(l,1),2}& \cdots & {G}_{(l,1),(l,a)}& \cdots & {G}_{(l,1),(l,a)}& \cdots & {G}_{(l,1),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {G}_{(i,1),2}& \cdots & {G}_{(i,1),\left(i1\right)}& \cdots & {G}_{(i,1),(l,a)}& \cdots & {G}_{(i,1),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {G}_{(l,a),2}& \cdots & {G}_{(l,a),(i,1)}& \cdots & {G}_{(l,a),(l,a)}& \cdots & {G}_{(l,a),(i,b)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {G}_{(i,b),2}& \cdots & {G}_{(i,b),(i,1)}& \cdots & {G}_{(i,b),(l,a)}& \cdots & {G}_{(i,b),(i,b)}\end{array}\right)$

（A16）

${\mathit{G}}_{\mathrm{P}\mathrm{Q}}={\left(\begin{array}{ccccccc}{G}_{(l,1),(i,1)}& \cdots & {G}_{(i,1),(i,1)}& \cdots & {G}_{(l,a),(i,1)}& \cdots & {G}_{(i,b),(i,1)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {G}_{(l,1),(i,b)}& \cdots & {G}_{(i,1),(i,b)}& \cdots & {G}_{(l,a),(i,b)}& \cdots & {G}_{(i,b),(i,b)}\end{array}\right)}^{\mathrm{T}}$

（A17）

${\mathit{B}}_{\mathrm{P}\mathrm{Q}}={\left(\begin{array}{ccccccc}{B}_{(l,1),(i,1)}& \cdots & {B}_{(i,1),(i,1)}& \cdots & {B}_{(l,a),(i,1)}& \cdots & {B}_{(i,b),(i,1)}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {B}_{(l,1),(i,b)}& \cdots & {B}_{(i,1),(i,b)}& \cdots & {B}_{(l,a),(i,b)}& \cdots & {B}_{(i,b),(i,b)}\end{array}\right)}^{\mathrm{T}}$

（A18）

${\mathit{G}}_{\mathrm{P}\mathrm{V}}={\left(\begin{array}{ccccccc}{G}_{(l,1),2}& \cdots & {G}_{(i,1),2}& \cdots & {G}_{(l,a),2}& \cdots & {G}_{(i,b),2}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {G}_{(l,1),(l,a)}& \cdots & {G}_{(i,1),(l,a)}& \cdots & {G}_{(l,a),(l,a)}& \cdots & {G}_{(i,b),(l,a)}\end{array}\right)}^{\mathrm{T}}$

（A19）

${\mathit{B}}_{\mathrm{P}\mathrm{V}}={\left(\begin{array}{ccccccc}{B}_{(l,1),2}& \cdots & {B}_{(i,1),2}& \cdots & {B}_{(l,a),2}& \cdots & {B}_{(i,b),2}\\ ⋮& & ⋮& & ⋮& & ⋮\\ {B}_{(l,1),(l,a)}& \cdots & {B}_{(i,1),(l,a)}& \cdots & {B}_{(l,a),(l,a)}& \cdots & {B}_{(i,b),(l,a)}\end{array}\right)}^{\mathrm{T}}$

（A20）

${\mathit{V}}_{\mathrm{i}\mathrm{m}}\left[n\right]={\left(\begin{array}{ccccccc}{V}_{(l,1),\mathrm{i}\mathrm{m}}\left[n\right]& \cdots & {V}_{(i,1),\mathrm{i}\mathrm{m}}\left[n\right]& \cdots & {V}_{(l,a),\mathrm{i}\mathrm{m}}\left[n\right]& \cdots & {V}_{(i,b),\mathrm{i}\mathrm{m}}\left[n\right]\end{array}\right)}^{\mathrm{T}}$

（A21）

${\mathit{V}}_{\mathrm{P}\mathrm{Q},\mathrm{r}\mathrm{e}}\left[n\right]={\left(\begin{array}{ccc}{V}_{(i,1),\mathrm{r}\mathrm{e}}\left[n\right]& \cdots & {V}_{(i,b),\mathrm{r}\mathrm{e}}\left[n\right]\end{array}\right)}^{\mathrm{T}}$

（A22）

${\mathit{W}}_{\mathrm{i}\mathrm{m}}\left[0\right]={\left(\begin{array}{ccccccc}{W}_{(l,1),\mathrm{i}\mathrm{m}}\left[0\right]& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {W}_{(l,a),\mathrm{i}\mathrm{m}}\left[0\right]& 0& 0\end{array}\right)}^{\mathrm{T}}$

（A23）

${\mathit{W}}_{\mathrm{r}\mathrm{e}}\left[0\right]={\left(\begin{array}{ccccccc}{W}_{(l,1),\mathrm{r}\mathrm{e}}\left[0\right]& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {W}_{(l,a),\mathrm{r}\mathrm{e}}\left[0\right]& 0& 0\end{array}\right)}^{\mathrm{T}}$

（A24）

${\mathit{R}}_{\left(\mathrm{P}\mathrm{V},\mathrm{P}\mathrm{Q}\right),\mathrm{r}\mathrm{e}}[n-1]=\left(\begin{array}{c}\mathrm{R}\mathrm{e}\left({A}_{(l,1)}\right[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(l,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ ⋮\\ \mathrm{R}\mathrm{e}\left({S}_{(i,1)}^{\mathrm{*}}{W}_{(i,1)}^{\mathrm{*}}\right[n-1]-{Y}_{(i,1),\mathrm{s}\mathrm{h}}{V}_{(i,1)}[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(i,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ ⋮\\ \mathrm{R}\mathrm{e}\left({A}_{(l,a)}\right[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(l,a),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ ⋮\\ \mathrm{R}\mathrm{e}\left({S}_{(i,b)}^{\mathrm{*}}{W}_{(i,b)}^{\mathrm{*}}\right[n-1]-{Y}_{(i,b),\mathrm{s}\mathrm{h}}{V}_{(i,b)}[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(i,b),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\end{array}\right)$

（A25）

${\mathit{R}}_{\left(\mathrm{P}\mathrm{V},\mathrm{P}\mathrm{Q}\right),\mathrm{i}\mathrm{m}}[n-1]=\left(\begin{array}{c}\mathrm{I}\mathrm{m}\left({A}_{(l,1)}\right[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(l,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ ⋮\\ \mathrm{I}\mathrm{m}\left({S}_{(i,1)}^{\mathrm{*}}{W}_{(i,1)}^{\mathrm{*}}\right[n-1]-{Y}_{(i,1),\mathrm{s}\mathrm{h}}{V}_{(i,1)}[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(i,1),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ ⋮\\ \mathrm{I}\mathrm{m}\left({A}_{(l,a)}\right[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(l,a),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\\ ⋮\\ \mathrm{I}\mathrm{m}\left({S}_{(i,b)}^{\mathrm{*}}{W}_{(i,b)}^{\mathrm{*}}\right[n-1]-{Y}_{(i,b),\mathrm{s}\mathrm{h}}{V}_{(i,b)}[n-1]-\sum _{k=1}^{N}{Y}_{\left(\right(i,b),k),\mathrm{t}\mathrm{r}}^{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{m}}{V}_{k}[n-1\left]\right)\end{array}\right)$

（A26）

$V_{\mathrm{PV}, \mathrm{re}}[n]=\left(\begin{array}{c} \delta_{n 1} \frac{V_{(l, 1) m}^{2}-\left|V_{(l, 1)}^{(0)}\right|^{2}}{2 V_{(l, 1), \mathrm{re}}[0]}-\frac{\left(\sum_{k=1}^{n-1} V_{(l, 1)}[k] V_{(l, 1)}^{*}[n-k]\right)}{2 V_{(l, 1), \mathrm{re}}[0]}-\frac{V_{(l, 1), \mathrm{im}}[0] V_{(l, 1), \mathrm{im}}[n]}{V_{(l, 1), \mathrm{re}}[0]} \\ \vdots \\ \delta_{n 1} \frac{V_{(l, a) m}^{2}-\left|V_{(l, a)}^{(0)}\right|^{2}}{2 V_{(l, a), \mathrm{re}}[0]}-\frac{\left(\sum_{k=1}^{n-1} V_{(l, a)}[k] V_{(l, a)}^{*}[n-k]\right)}{2 V_{(l, a), \mathrm{re}}[0]}-\frac{V_{(l, a), \mathrm{im}}[0] V_{(l, a), \mathrm{im}}[n]}{V_{(l, a), \mathrm{re}}[0]} \end{array}\right)$

（A27）

$-B_{1}=\left(\begin{array}{ccccccc} -B_{(l, 1), 2}-G_{(l, 1), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & -B_{(l, 1),(l, a)} & \cdots & -B_{(l, 1),(l, a)}-G_{(i, 1),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & -B_{(l, 1),(i, b)} \\ \vdots & & \vdots & & \vdots & \vdots \\ -B_{(i, 1), 2}-G_{(i, 1), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & -B_{(i, 1),(i, 1)} & \cdots & -B_{(i, 1),(l, a)}-G_{(i, 1),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & -B_{(i, 1),(i, b)} \\ \vdots & & \vdots & & \vdots & \vdots \\ -B_{(l, a), 2}-G_{(l, a), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & -B_{(l, a),(i, 1)} & \cdots & -B_{(l, a),(l, a)}-G_{(l, a),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & -B_{(l, a),(i, b)} \\ \vdots & & \vdots & & \vdots & \vdots \\ -B_{(i, b), 2}-G_{(i, b), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & -B_{(i, b),(i, 1)} & \cdots & -B_{(i, b),(l, a)}-G_{(i, b),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & -B_{(i, b),(i, b)} \end{array}\right)$

（A28）

$\boldsymbol{G}_{1}=\left(\begin{array}{ccccrcc} G_{(l, 1), 2}-B_{(l, 1), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & G_{(l, 1),(i, 1)} & \cdots & G_{(l, 1),(l, a)}-B_{(l, 1),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & G_{(l, 1),(i, b)} \\ \vdots & \vdots & \vdots & & \vdots \\ G_{(i, 1), 2}-B_{(i, 1), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & G_{(i, 1),(i, 1)} & \cdots & G_{(i, 1),(l, a)}-B_{(i, 1),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & G_{(i, 1),(i, b)} \\ \vdots & \vdots & & \vdots & \vdots \\ G_{(l, a), 2}-B_{(l, a), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & G_{(l, a),(i, 1)} & \cdots & G_{(l, 1),(l, 1)}-B_{(l, 1),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & G_{(l, a),(i, b)} \\ \vdots & & \vdots & \vdots & \vdots \\ G_{(i, b), 2}-B_{(i, b), 2} \frac{V_{(l, 1), \mathrm{im}}[0]}{V_{(l, 1), \mathrm{re}}[0]} & \cdots & G_{(i, b),(i, 1)} & \cdots & G_{(i, b),(l, a)}-B_{(i, b),(l, a)} \frac{V_{(l, a), \mathrm{im}}[0]}{V_{(l, a), \mathrm{re}}[0]} & \cdots & G_{(i, b),(i, b)} \end{array}\right)$

（A29）

$\boldsymbol{V}_{\mathrm{PV} 1, \mathrm{re}}[n]=\left(\begin{array}{cc} \delta_{n 1} \frac{V_{(l, 1)}^{2}-\left|V_{(l, 1)}^{(0)}\right|^{2}}{2}-\frac{1}{2}\left(\sum_{k=1}^{n-1} V_{(l, 1)}[k] V_{(l, 1)}^{*}[n-k]\right) \\ \vdots \\ \delta_{n 1} \frac{V_{(l, a)}^{2}-\left|V_{(l, a)}^{(0)}\right|^{2}}{2}-\frac{1}{2}\left(\sum_{k=1}^{n-1} V_{(l, a)}[k] V_{(l, a)}^{*}[n-k]\right) \end{array}\right)$

References

List( Publishing order | Descend order by publishing year | Descend order by cited within ) Chart analysis

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Abstract

建设完善的社会综合能源系统是低碳经济背景下解决我国社会供能系统安全性低、自愈能力差、设备利用率低下等一系列问题的一种可能解决方案。文章首先给出综合能源系统的基本概念与主要特征，结合世界各国综合能源系统建设的经验与我国能源资源现状，分别从国家、区域和终端这3个层面探讨了我国未来综合能源系统发展中面临的一些问题，并给出了相应的建议。

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Abstract

可用输电能力(available transmission capability,ATC)反映了电网不同区域间的电能交换能力,为电网的稳定性评估提供了参考。随着电-气综合能源系统(electricity-gas integrated energy system,EGIES)的发展,天然气网络与电网进一步耦合,ATC的计算会更加复杂,从而影响ATC的计算效率。针对上述问题,文章提出了一种面向数字孪生(digital twin)理念的电-气综合能源系统的ATC计算方法。首先将数据驱动和模型驱动进行融合,构建数据-机理融合模型以满足数字孪生理念的指标要求。数据-机理融合模型可以充分挖掘隐藏在海量状态数据下的信息,进而简化传统物理模型迭代计算的过程,缩短计算时间;其次,构建数字虚拟模型以实时处理综合能源系统内不断更新的状态量数据,实现最大输电能力的在线计算并提取系统运行状态的特征;然后,利用提取出来的特征进行综合能源系统的ATC计算;最后,通过IEEE30-NGS10电-气综合能源系统验证了文章所提方法的有效性与高效性。

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Abstract

 摘要：综合能源系统对提高能源利用率、实现能源可持续发展具有重要意义。多能流潮流计算是综合能源系统规划、运行、控制的重要基础工作。为了使相关领域工作者更好地开展综合能源系统方面的研究工作，该文以综合能源系统的多能流潮流计算为对象开展综述，从计算模型和求解方法2个角度，对当前的研究工作和进展进行梳理。首先，对多能流潮流计算问题进行介绍；然后，对电-气-热综合能源系统的多能流潮流计算模型进行分类和总结，在此基础上，对模型的求解方法进行概括和总结；最后，提出当前综合能源系统多能流潮流计算方面存在的问题，并给出一些建议。 

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 ABSTRACT：   Integrated energy system is of great importance to improve energy efficiency and achieve sustainable development of energy.Multi-energy flow calculation is an important basic work for the planning, operation and control of integrated energy system.In order to make the related field workers carry out the research work of integrated energy system better, this paper reviews multi-energy flow calculation of integrated energy system.From two aspects of model and method, current research has been sorted out.First of all, the problem of multi-energy flow calculation is introduced.Then, the classification and generalization of multi-energy flow calculation models in electricity-gas-heat combined system are carried out.On this basis, the methods for solving the models are summarized.At last, some problems existing in multi-energy flow calculation of integrated energy system are put forward and some suggestions are given.<div> </div>

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Abstract

多能流潮流分析是研究综合能源系统（IES）状态估计、安全分析与优化控制的重要基础工作。为方便以潮流分析方法研究电、气、热（冷）等能源间的相互转化与耦合联供，综合分析了IES潮流分析的2个主要研究方向，其一为在某确定工况下根据已知参数计算系统的运行状态，包括稳态和动态潮流；其二为在前者基础上考虑系统不确定性参数对系统运行影响的不确定性潮流，并主要以概率潮流解法进行讨论，指出目前多能流混合潮流的研究进展与解耦方法与局限性，总结了最优潮流的求解方式和应用现状。结果表明，目前多能流稳态计算已较为成熟；动态潮流模型多以差分法求解，但准确度和求解效率仍有待提高；不确定性潮流对数据要求较高，无法兼顾计算精度和效率；最优潮流研究多集中于单一目标优化，较少涉及多目标并考虑不确定因素约束的复杂模型。最后，提出目前混合能源潮流计算在模型约束和算法难度等方面的局限性与未来发展前景。

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Multi-energy flow analysis is a crucial and fundamental link for the state estimation，security analysis and optimal regulation of an integrated energy system（IES）. To study the mutual conversion and coupling of electric power，gas，thermal power and cold energy with power flow analysis，two main research directions of integrated energy system power flow analysis are compared. The first one is to calculate the system's operating state，including steady-state and dynamic power flow，based on known parameters under a certain operating condition.On this basis， the second one considers the impact of uncertain parameters on system operation，and study the uncertain power flow through probabilistic power flow analysis.The research progress made in multi-energy hybrid flow and limitations of current decoupling methods are pointed out，and the optimal power flow solution and its applications are summarized. The results show that the multi-energy steady-state flow calculation is relatively mature，but the dynamic power flow model mostly solved by the differential method lacks accuracy and solution efficiency.Uncertainty power flow requires highly on data and cannot balance computing accuracy and efficiency.The optimal power flow model mainly focuses on single-objective optimization，while seldom considers multiple objectives or uncertain factors.Finally，the limitations of hybrid energy power flow calculation in terms of its model constraints and algorithm complexity are proposed，and its prospects are made.

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Based on large scale sparse and symmetrical matrix of FEM equations,this paper takes the fully sparse strategy and minimum full-in entries algorithm such that makes lowest storing requirement to computer.For the sake of CPU operational time saving to accesses data in matrix decomposition,a symbol LU decomposing method is applied.The combination of the symbol LU decomposition and fully sparse storage structure can greatly improve the algorithmic efficiency for FEM solution of large scale sparse linear equation group.Numerical examples show that the method is available,effective and predominant for time and storage.Therefore it is applicable to solve systems of linear equations from FEM.

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Abstract

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