JP3654451B2 - Pipe network analysis method, network minimum cost flow analysis method, and pipe network management system - Google Patents

Description

【００１３】
ここに、
Ｎ ：全節点の集合
Ｎ_IN ：供給点の集合
Ｎ_OUT ：需要点の集合（分岐点は需要量０の需要点と考える）
ｘ_j ：管路ｊの流量（ｍ³/ｈ）
ｗ_i ：供給点ｉの供給量（ｍ³/ｈ）
ｙ_i ：需要点ｉの需要量（ｍ³/ｈ）
Ａ₊（ｉ）：節点ｉを始点とする管路の集合
Ａ_-（ｉ）：節点ｉを終点とする管路の集合
節点が完全な供給点のときは、数１の左辺の第１項は“０”になり、節点が完全な需要点のときは、数１の左辺の第２項は“０”になり、節点が供給、需要に関係のない分岐点のときは、数１の右辺は“０”になる。
更に、各管路において、次の圧力平衡条件が成り立つ。
【００１４】
【数２】

【００１５】
ここに、
Ｂ ：全管路の集合
ｐ_i ：節点ｉの圧力（kgf/cm²）
ｓ（ｊ） ：管路ｊの始点
ｅ（ｊ） ：管路ｊの終点
Ｒ_j ：管路ｊの抵抗係数
sgn（ｘ）：ｘの符号
である。
ガスの場合ｒ＝２であり、Ｒ_jは、Ｗｅｙｍｏｕｔｈ氏の実験式によれば、以下のようになる。
【００１６】
【数３】

【００１７】
【数４】

【００１８】
ここに、
ｓ：ガスの比重
Ｌ_j：管路ｊの管長（ｍ）
Ｄ_j：管路ｊの管径(cm）
である。
節点数をｎ、管路数をｍとすると、式の数は（数１）、（数２）で、計ｍ＋ｎ個となる。一方、未知数の数は、管路流量ｍ個、供給点の供給量または圧力と需要点の圧力でｎ個で、計ｍ＋ｎ個である。よって、連立方程式を解けば、管網上の流量、圧力分布を求めることができる。
【００１９】
管網の規模が増大して方程式の数が増すと、上記方程式の解を求めるのは非常に困難となる。本発明では、管網の中で木構造となっている部分のデータを記憶しておくことによって管網を効果的に縮約し、連立させる方程式の数を大幅に削減して計算効率を上げる。
一般に、管網上でループを含まない部分、すなわち木構造部分については、圧力を考慮することなく需要量から一意に流量を決定することができる。従って、木構造部分の根となっている節点の圧力さえ求まれば、（数２）を用いて末端へ向かって順次各節点の圧力を求めることができ、連立方程式を解く必要はない。このことに着目して、管網の木構造部分を、根となっている節点で代表させることにより管網を縮約する。
木構造部分のデータは、木構造記憶部１０５に記憶させておいて元の管網から削除し、根となっている節点には、木構造部分を構成する全ての節点の需要量の総和量を割り付ける。縮約管網について（数１）および（数２）で表される方程式を解くことによって、根となっている節点の圧力が求まり、この圧力と木構造記憶部１０５のデータから削除された木構造部分の流量と圧力も計算することができる。この方法では、縮約されたことによる誤差は全く生じない。
【００２０】
図３に縮約の例を示す。実際の管網は、節点３０１、３０２、及び管路３０８から構成されるループ構造部３１４と、節点３０３〜３０７、管路３０９〜３１３から構成される木構造部３１５とに分割できる。木構造部３１５の総需要量を、木構造部３１５の根となっている節点３０２の需要量に加え、木構造部を削除することにより、元の管網は、ループ構造部のみからなる管網に縮約できる。
【００２１】
管網から木構造部分を見つけだして縮約するには、例えば図４のフローチャートで示す処理を用いればよい。まず、供給量、又は需要量が既知の節点を、縮約可能節点（供給圧のみが既知の節点は対象外）とする（４０１）。
次に、唯一の管路ｊの節点であり、かつ管路ｊのもう一方の節点ｉ₂が縮約可能節点であるような縮約実行節点ｉ₁を探す（４０２）。
もし上記縮約実行節点ｉ₁が存在しなければ、管網縮約処理を終了する（４０３）。
存在すれば、縮約実行節点ｉ₁と管路ｊを管網から削除し、節点ｉ₂の需要量ｑ₂に、縮約実行節点ｉ₁の需要量ｑ₁を加えた量ｑ₁＋ｑ₂を、新たに節点ｉ₂の需要量とする（４０４）。但し、供給量は負の需要量と考える。
続いて、削除した管路の番号および縮約実行節点の番号とその時点での需要量（即ち、修正後の需要量）を順に木構造記憶部１０５に記憶する（４０５）。再び、ステップ４０２に戻り、上記処理を繰り返す。
【００２２】
図３の（ａ）に示す管網モデルは（ｂ）に示す管網モデルに縮約される。（ａ）の管網のどの節点も縮約可能節点であるとする。
節点３０２〜３０７の需要量をｙ₂〜ｙ₇とすると、（ｂ）の縮約管網では、節点３０２の需要量がｙ₂（管路３０９に流れる流量とは別の節点３０２自体の需要量）からｙ₂＋ｙ₃＋ｙ₄＋ｙ₅＋ｙ₆＋ｙ₇になる。縮約後の需要量が負の場合、その節点は供給点と考える。木構造記憶部１０５に記憶されるデータは、削除した節点と管路の番号、および削除された節点のその時点での需要量の組である。これを削除した順に記憶していく。図３に示した縮約の結果、（節点３０３、管路３１０：ｙ₃）、（節点３０６、管路３１２：ｙ₆）、（節点３０７、管路３１３：ｙ₇）、（節点３０５、管路３１１：ｙ₅＋ｙ₇）、（節点３０４、管路３０９：ｙ₃＋ｙ₄＋ｙ₅＋ｙ₆＋ｙ₇）となる。ｙ₄は節点３０４自体の需要量である。
ここで記憶される需要量は、削除された管路における流量も同時に意味している。記憶されたときとは逆に、管路３０９、管路３１１、管路３１３、管路３１２、管路３１０の順に、各管路における圧力平衡式を解くことにより、削除された節点における圧力を求めることができる。
本発明による木構造部分のデータを記憶することにより管網を効果的に縮約することについて上記に述べた。図８、図９に縮約前と、縮約後の実際のガス管網データの例を示す。この例では縮約によりノード数が２８３から１０４に減少し、管路数が２９６から１１６に減少し、約１／３の規模になる。
【００２３】
次に、縮約された管網について（数１）、（数２）で記述された連立方程式を解く方法を説明する。
この方法の中でも、木構造部分を記憶しておく方法を応用することによってさらに計算効率を上げることができる。この方程式は、従来の最小費用流問題（ガス管網解析問題においては、費用とは圧力の２乗の損失と考えることができる。すなわち、流れにくさが費用に対応する。）に帰着させることによって解くことができる。
最初にその定式化について説明する。まず、「仮想的な供給点」（ソース）と「仮想的な需要点」（シンク）、及びソースと供給点、需要点とシンクを接続する管路を管網に新たに設定する。このとき、各管路ｊの単位流量当りの費用関数ｕ_j(ｘ_j)を、以下のように設定する。
【００２４】
【数５】

【００２５】
ここに、
Ｂ_IN1：ソースと圧力指定供給点とを接続する管路の集合
Ｂ_IN2：ソースと供給量指定供給点とを接続する管路の集合
Ｂ_OUT：需要点とシンクとを接続する管路の集合
である。
また、ソースと供給量指定供給点とを接続する管路は、指定供給量が容量であるものとし、同じく需要点とシンクとを接続する管路には、需要量が容量であるとする。その他の管路の容量は無制限とし、各管路ｊの容量ａ_jを、以下のように設定する。
【００２６】
【数６】

【００２７】
このとき、管網解析問題は、以下のような最小費用流問題に定式化される。
制約条件：
【００２８】
【数７】

【００２９】
【数８】

【００３０】
目的関数：
【００３１】
【数９】

【００３２】
ここで、Ａ₊(ｉ）、Ａ_-(ｉ）は、ソース、シンクに流出入する仮想的な管路も含む集合とする。従来法ではすべての供給点を圧力指定としていたが、供給量指定の供給点についても、（数５）、（数６）に示すように、費用関数は十分小さく（−∝）、容量は供給量に等しくすることで定式化可能である。
【００３３】
上記最小費用流問題は（数５）に示されるように、非線形に可変な費用関数を有する。これを、階段関数近似することにより、Ｐｒｉｍａｌ−Ｄｕａｌ法を適用できるようにする。この近似方法について説明する。図２において２０１は、管路ｊ∈Ｂの費用関数ｕ_j（ｘ_j）を表しており、該費用関数は増加関数である。流量軸を複数区間［ａ_j（ｋ），ａ_j（ｋ＋１）］に分割し、階段関数２０２によって費用関数２０２を近似する。
区間幅を決定する場合、流量軸を等分割すると、流量が増大したとき費用の誤差が増大する。しかし、最大流量近傍での誤差が許容範囲に入るように区間幅を決めると、０近傍で不必要に細かい分割となり、計算量が増える。費用軸を常に等分割すると逆に、流量が減少したときは流量の誤差が増大し、０近傍での誤差が許容範囲に入るように区間幅を決めると、最大流量近傍で不必要に細かい分割をすることになる。
そこで、精度を保証しかつ計算の高速化を図るため、両者を融合した区間幅決定をする。流量軸と費用軸に関し、精度を保証するのに十分な区間幅δ₁、δ₂を設定し、分割の起点をｘ_j＝０にとる。分割点ｘ_j＝ａ_j（ｋ）が求まっているとき、正方向の次の分割点ｘ_j＝ａ_j（ｋ＋１）は、次式で求める。
【００３４】
【数１０】

【００３５】
但し、Δｘ_jは次式の解である。
【００３６】
【数１１】

【００３７】
負方向の分割についても同様である。このとき、区間［ａ_j（ｋ），ａ_j（ｋ＋１）］におけるｕ_j（ｘ_j）を、次式で近似する。
【００３８】
【数１２】

【００３９】
この方法によれば、図２の階段関数２０２が示すように、流量が小さい値のとき流量軸が幅δ₁で分割され、流量が大きな値になると費用軸がδ₂で分割される。そこで、２種類の分割の境界点となる境界流量値α_j≧０を予め求めておき、流量軸に対して、区間［−α_j，α_j］のときは、流量軸を幅δ₁で分割し、それ以外の区間のときは、費用軸を幅δ₂で分割するようにして、次の分割点を決定する。α_jは、次式を同時に満たしかつδ₁の整数倍であるという条件で一意に定まる。
【００４０】
【数１３】

【００４１】
【数１４】

【００４２】
ガスの場合ｒ＝２であるから、α_jは解析的に次式で得られる。
【００４３】
【数１５】

【００４４】
よって、各管路ｊに対して、境界流量値α_jを記憶しておくことにより、区間［−α_j，α_j］のときは、（数１１）によるΔｘ_jの計算を省略することができる。以降で述べるように、最小費用流問題の求解過程において、管路の流量が更新される度に、次の分割点を決定する計算が必要となるため、上記方法が効果的となる。
【００４５】
図５のフローチャートに基づいて、最小費用流問題の解法であるＰｒｉｍａｌ−Ｄｕａｌ法による処理について説明する。この処理の特徴は、全ての管路流量が０の状態から計算を開始し、最小費用経路問題と最大流問題を交互に繰返し解いて流量を積み重ねていき、最小費用の飽和した流れを求める点にある。
通常のＰｒｉｍａｌ−Ｄｕａｌ法では、費用関数が一定であることを前提としているが、以下に示す方法は、前述したような階段関数状に可変な費用関数の場合に適用できるように拡張されている。
各管路ｊの流量、容量、費用関数をそれぞれｘ_j、ａ_j、Ｕ_j、所定の総需要量をｖ、計算過程での総流出量をｑとする。初めにｑ＝０、即ち各管路ｊの流量をｘ_j＝０とし、各管路ｊの正方向の費用関数ｄ_j+、負方向の費用関数ｄ_j-を以下のように設定する（５０１）。
【００４６】
【数１６】

【００４７】
【数１７】

【００４８】
ただし、ｋ＝０である。
上記費用関数に基づいて、ソースからシンクに至る最小費用経路を求める（５０２）。最小費用経路の探索においては、ソースから各節点に至るまでの最小費用も共に算出される。そのため、実際にはソースからシンクに至る最小費用経路木が生成される。
ｑ＝Ｑの時の各管路ｊの流量をｘ_j(OLD)とするとき、最小費用経路に沿って増加可能な最大量Δｑを、次式によって算出し、
【００４９】
【数１８】

【００５０】
【数１９】

【００５１】
各管路ｊの新たな流量ｘ_j(NEW)を、
【００５２】
【数２０】

【００５３】
と変更し、ｑ＝Ｑ＋Δｑとする（５０３）。ただし、
Ｐ₊：最小費用経路に順方向で含まれる管路の集合
Ｐ_-：最小費用経路に逆方向で含まれる管路の集合
である。
このとき、流れが飽和しているかどうか、即ちｑ＝ｖかどうかを判定する（５０４）。もし流れが飽和しているならば計算を終了する。
飽和していないならば、最小費用経路上で、変更後の流量が区間分割点に一致した管路はｋを更新して新たな区間分割点ａ_j（ｋ＋１）および費用関数Ｕ_j（ｋ＋１）を求め、再び（数１６）、（数１７）により正負両方向の費用関数を設定する（５０５）。但し、流量が区間の下限に一致したときはｋをｋ−１に、上限に一致したときはｋをｋ＋１に更新する。
【００５４】
従来法ではここでステップ５０２に戻り、更新された費用関数に基づく最小費用経路木Ｔ_NEWを新たに生成していた。しかし、反復の度に新たな最小費用経路木Ｔ_NEWを生成するのは計算負荷が大きい。そこで本発明では、前回の最小費用経路木Ｔ_OLDの一部を木構造データとして記憶しておき、これを利用することによって次回の最小費用経路木Ｔ_NEWの生成の手間を省く。
ステップ５０５で費用を変更した管路以降の部分木Ｔ_SUB2をＴ_OLDから除いて残る部分木Ｔ_SUB1の木構造データを記憶する（５０６）。そして、Ｔ_SUB1のいずれかの適当な節点から、シンクに至る最小経路木Ｔ_SUB3を探索し、記憶しておいたＴ_SUB1に結合することによってＴ_NEWを生成する（５０７）。ここでステップ５０３に戻りステップ５０４の条件が満たされるまで反復する。
【００５５】
図６は、本発明による方法の過程で、最小費用経路木Ｔ_OLDから新たな最小費用経路木Ｔ_NEWが生成される具体例を示したものである。
図６（ａ）のネットワークの太線部分は、前回生成されたソース６０１からシンク６０８に至る最小費用経路木Ｔ_OLDを表している。最小費用経路６１０、６１３、６１６、６２０に最大流を流したとき、管路６１６において初めて、流量がある区間の端点に到達したとする。
このとき、管路６１６以前の管路は費用が修正されないため、管路６１６以降の部分木Ｔ_SUB2（管路６１６、６２０）を木Ｔ_OLDから除いた木構造部分Ｔ_SUB1は、新たに生成される最小費用経路木Ｔ_NEWの部分木になる。
残りの部分のネットワーク（節点６０６、６０７、６０８、管路６１６、６１７、６１９、６２０、６２１）において、Ｔ_SUB1のいずれかの節点から、シンク６０８に至る最小経路木Ｔ_SUB3（管路６１７、６２０）を新たに生成する。
例えば、シンクからソースに向けて、Ｔ_SUB1上の節点に到達するまで最小費用経路木Ｔ_SUB3の生成を行う。Ｔ_SUB1とＴ_SUB3結合することにより、図６（ｂ）のネットワークの太線部分によって表されているＴ_NEWを生成できる。ソースからシンクに至る最小費用経路木が求まり、最小費用経路探索の高速化を図ることができる。
【００５６】
以上述べてきた管網解析方法により、従来に比べて高速に解析結果を得ることができる。
本発明による方法は、木構造部分を効果的に利用することが特徴であり、図１に示したシステムでは、管網解析部１０２において解析演算部１０４の他に木構造記憶部１０５を設けたことが特徴である。
通常の計算機上で、解析演算部１０４はソフトウエアモジュールで、木構造記憶部１０５はメモリの一部分であってもシステムは構築可能であるが、専用の装置とすればさらに効果的である。
高速に解析結果を得ることができるため、運用計画を目的とした管網管理システムでは、計画の周期を短縮できるようになり、より正確な管理を行うことができる。また計算手法の効率化であるため、解析演算を行う装置の低コスト化にも寄与する。
【００５７】
〈実施例２〉
本発明の第２の実施例として水道事業における管網管理システムを取り上げ、以下図面に基づいて説明する。
図７は、配水管網における流量圧力分布の監視や制御を目的とした管網管理システムの全体構成を示す図である。本システムによる監視制御のながれを図７に基づいて説明する。
まず、需要分布予測部７０６によって得られた需要分布および管網制御部７１０から出力されるポンプやバルブの操作指令値に基づき、管網データ作成部７０７において、管網解析計算の入力とする管網データを作成する。
管網データは、節点の需給データ、すなわち管網の各需要点における需要量や各供給点における供給量や供給圧力、および管路の諸元データ、すなわち接続状態、抵抗係数、管長、管径等である。
操作指令値であるポンプの運転状態やバルブの開度は、各管路の抵抗係数に反映される。上記管網データを入力として、管網解析部１０２において管網解析を行い、解析結果を出力する。
管網制御部７１０では、解析結果に基づいて操作指令値を決定し、ポンプ７１１やバルブ７１２へ送信する。解析結果出力部７０９では、図面データ管理部７０８から出力される図面データと解析結果に基づいた流量圧力分布図を作成して表示や印刷を行う。解析結果出力部７０９による出力は、管網監視に用いられる。
【００５８】
管網解析部１０２において行われる管網解析問題は、各節点における流量収支条件、
【００５９】
【数２１】

【００６０】
ここに、
Ｎ ：全節点の集合
Ｎ_IN ：供給点の集合
Ｎ_OUT ：需要点の集合（分岐点は需要量０の需要点と考える）
ｘ_j ：管路ｊの流量（ｍ³/ｈ）
ｗ_i ：供給点ｉの供給量（ｍ³/ｈ）
ｙ_i ：需要点ｉの需要量（ｍ³/ｈ）
Ａ₊（ｉ）：節点ｉを始点とする管路の集合
Ａ_-（ｉ）：節点ｉを終点とする管路の集合
と、各管路における圧力平衡条件、
【００６１】
【数２２】

【００６２】
ここに、
Ｂ ：全管路の集合
ｐ_i ：節点ｉの圧力（kgf/cm²）
ｓ（ｊ） ：管路ｊの始点
ｅ（ｊ） ：管路ｊの終点
Ｒ_j ：管路ｊの抵抗係数
sgn（ｘ）：ｘの符号
からなる連立方程式である。
実施例１のガスとの相違点は、圧力平衡条件の左辺の節点圧力の指数が１で、右辺の管路流量の指数がｒ＝１．８５である点である。また、Ｒ_jは、
Ｈａｚｅｎ−Ｗｉｌｌｉａｍｓ氏の実験式によれば、以下のようになる。
【００６３】
【数２３】

【００６４】
ここに、
Ｃ_j：管路ｊの流速係数
Ｌ_j：管路ｊの管長（ｍ）
Ｄ_j：管路ｊの管径(cm）
である。
この問題は、実施例１と全く同様にして解くことができる。ただし、費用関数を階段関数近似する場合の区間幅決定法において、境界流量値α_jは、配水系の場合ｒ＝１．８５であるため、（数１５）のように解析的には得られない。しかし、Ｎｅｗｔｏｎ法等の繰り返し計算により予め求めておくことは可能であるため、同様の方法がとれる。従って、管網解析部１０２においてリアルタイムな管網解析を実行し、きめ細かい制御や監視が可能となる。
【００６５】
尚、本発明の求解方法は、本実施例では管網解析問題の求解方法として記述したが上記実施例１、２から判るように、本来は最小費用流問題に対する求解方法である。上記管網解析問題は実施例１で示したように最小費用流問題に変換できるため本方法が適用できるのであって、本方法はノード（節点）、及びアーク（管路）からなるネットワークにおいて、各アークの費用関数が増加関数で表わされるような全ての最小費用流問題に対して適用可能である。
「ネットワーク理論」 １９７６年９月２７日、日化技連発行、伊理正夫、古林 隆著（ｐ８１〜）では、最小費用流問題の求解方法をいくつか紹介し、最小費用流問題の一例であるヒッチコック型輸送問題にその方法を適用している。
本発明の方法は勿論輸送問題に対しても適用可能であり、例えば、電力事業において配電管網の電力分布を把握するには、電圧が圧力、電流が流量に対応していると考えれば、上記水やガスの管網解析方法がそのまま適用できる。
【００６６】
【発明の効果】
以上述べたように、本発明によれば、大規模管網に対しても、十分な精度でかつ短時間に管網解析結果を得ることが可能になるため、よりきめ細かい管網管理が可能になる。更に、高速かつ高価な計算機を使う必要がなくなり、低価格な計算機へのダウンサイジングが可能になる。
【図面の簡単な説明】
【図１】ガス管網管理システムの全体構成を示す図である。
【図２】費用関数の階段関数近似を説明する図である。
【図３】管網縮約の例を示す図である。
【図４】本発明を適用した管網縮約処理のフローチャートを示す図である。
【図５】本発明を適用したＰｒｉｍａｌ−Ｄｕａｌ法による処理のフローチャートを示す図である。
【図６】最小費用径路木から新たな最小費用径路木が生成される具体例を示す図である。
【図７】配水管網管理システムの全体構成図を示す図である。
【図８】縮約前の実際のガス管網データを示す図である。
【図９】縮約後の実際のガス管網データを示す図である。[0001]
[Industrial application fields]
The present invention relates to a pipe network analysis method, a network minimum cost flow analysis method, and a network minimum cost flow analysis method for calculating the flow rate of each pipe line and the pressure of each node in a large-scale transportation pipe network such as a water pipe network and a gas pipe network. It relates to a pipe network management system.
[0002]
[Prior art]
In the transportation business such as water supply business and gas business, in order to produce an appropriate amount of fluid (water, gas, etc.) and stably supply the fluid with sufficient pressure to consumers, the fluid pressure in the pipe network is properly adjusted It is required to keep. In addition, it is necessary to optimize the pressure distribution on the pipe network at the design stage in order to create an appropriate facility plan in the new expansion of transportation facilities such as pipes and pressure regulators. Therefore, in order to perform appropriate pipe network management, it is essential to properly grasp the pressure distribution and flow rate distribution on the pipe network.
Understanding the pressure flow distribution on the pipe network based on the pipe network data is called pipe network analysis. The pipe network analysis problem is nothing but a problem in which nonlinear algebraic equations representing the flow rate balance condition at each node and the pressure balance condition at each pipe are solved simultaneously.
[0003]
The above equation is very large and has a large computer load in the actual problem that is usually dealt with. Therefore, attempts have been made to speed up the solution. One example of this is the IEEJ Transactions C101 Volume 11 (Showa 56), page 261, Paper 56C-34, “A Pipeline Network Analysis Method Using Minimum Cost Flow Calculation”.
This method pays attention to the property regarding the loss energy of the steady flow on the pipe network, converts the pipe network analysis problem into the minimum cost flow problem on the network, and the Primal-Dual method which is a fast solution of the minimum cost flow problem. This solves this.
In this method, the pressure equilibrium condition, which is a nonlinear relationship established between the flow rate and the pressure, is regarded as a function of a cost coefficient with respect to the flow rate, and this is approximated by a step function. Based on this function, the minimum cost path on the network (actually, the minimum cost path tree) is obtained, the maximum flow is made to flow through the minimum cost path, and each pipe flow rate is changed. By repeating the above process until the flow is saturated, a saturated flow with minimum cost is obtained.
In the above paper, a function approximation method as shown in FIG. 2 is proposed in order to guarantee accuracy and speed up calculation.
Regarding flow rate and pressure, section width δ from each required accuracy ₁ , Δ ₂ When the flow rate is small, set the flow axis to the width δ ₁ When the flow rate is large, the cost axis is ₂ It is divided by. Based on these section widths, the original function 201 is approximated by a step function like a function 202. This method eliminates unnecessarily fine division and can speed up calculation.
[0004]
Also, in the IPSJ Transactions Vol. 29, No. 11 (1988), p. 1079, the paper “A Solution of the Pipe Network Analysis Problem Based on the Minimum Cost Flow Algorithm”, when the cost coefficient function is approximated by a step function, the cost Always make the axis δ ₂ The restriction of dividing by is added. As a result, the minimum cost path tree for each execution of the Primal-Dual method has the property that it is obtained by performing the primary transformation of the tree once on the previous minimum cost path tree, and the calculation time for obtaining the minimum cost path tree It proposes an algorithm to reduce
On the other hand, in Japanese Patent Laid-Open No. 2-209700, an actual pipe network is reduced to an equivalent pipe network model, a pipe network analysis is performed on the reduced pipe network model, and the pipe network analysis result is based on the result. A pipe network analysis system is described which reduces the calculation time required for pipe network analysis by deploying to the other pipe network. The basic concept of the contraction method is to replace two pipe lines connected to the same start point and end point with one pipe line. By reducing the scale of the pipe network by contraction, the time required for pipe network analysis is shortened.
[0005]
[Problems to be solved by the invention]
In recent years, due to the large scale of the pipe network and detailed services to customers, the scale of the pipe network to be subjected to pipe network analysis has been increasing, and accordingly the time required for pipe network analysis has increased. Yes.
In order to perform more appropriate management of the pipe network, it is essential to speed up the pipe network analysis method, and there is a demand for executing pipe network analysis of a large-scale pipe network with thousands of nodes within practical time. It is getting stronger. Here, the practical time is, for example, about several minutes as a time that a human can wait for the pipe network analysis without feeling stress.
Due to changes in computer system configuration due to downsizing, there is a demand for pipe network analysis on smaller computers such as workstations and personal computers, and real-time computations, and faster analysis methods are desired. . However, in order to analyze a large-scale pipe network, it is necessary to solve thousands of nonlinear simultaneous equations. However, due to the problems of scale and nonlinearity of equations, there has been a problem that conventional pipe network analysis methods cannot be solved within practical time.
[0006]
The pipe network analysis method described in the above mentioned IEEJ Transaction C101 Vol. 11 (Showa 56), page 261, paper 56C-34 "Pipe network analysis method by minimum cost flow calculation" guarantees accuracy and performs calculations. In order to increase the speed of the cost coefficient, in the step function approximation of the cost coefficient, every time the pipe flow reaches the end point of the section during execution of the Primal-Dual method, the next section width is changed to the flow axis width δ. ₁ Or split the cost axis with width δ ₂ The calculation to determine whether to divide by. Since this calculation is repeated for all pipes, a certain amount of calculation time is required to determine the section width.
Further, in the pipe network analysis method described in the above-mentioned Journal of Information Processing Society of Japan, Vol. 29, No. 11 (1988), page 1079, the paper “A Method for Solving the Pipe Network Analysis Problem Based on the Minimum Cost Flow Algorithm”, the Primary-Dual In order to reduce the computation time for finding the minimum cost path tree each time the method is executed, the cost axis is always set to the width δ when the cost coefficient is approximated by a step function. ₂ The restriction of dividing by is added. At this time, in order to guarantee sufficient accuracy, the division width δ ₂ Need to be small. But δ ₂ When the pipe flow rate is increased, the smaller the is, the more unnecessary fine division is increased, and the number of repeated calculations of the Primal-Dual method is increased.
Further, the contraction by the pipe network analysis system described in the above-mentioned Japanese Patent Application Laid-Open No. 2-209700 is very limited and ineffective, and the pipe network analysis within a practical time for a large-scale pipe network. To do this, it is necessary to reduce it to a smaller pipe network.
Thus, with the conventional pipe network analysis method, it has been difficult to obtain the pipe network analysis result with sufficient accuracy within a practical time.
The present invention has been made in view of the above circumstances,
An object of the present invention is to provide a pipe network analysis method capable of obtaining a pipe network analysis result within a practical time with sufficient accuracy even for a large-scale pipe network.
Another object of the present invention is to provide a network minimum cost flow analysis method with one input and one output.
Another object of the present invention is to provide a pipe network management system using the pipe network analysis method.
[0007]
[Means for Solving the Problems]
In order to achieve the above object, the present invention provides:
In the pipe network analysis method, the supply and demand data of each node in the pipe network that transports fluid and the specification data of each pipe line are input to the processing device, and the pressure at each contact point and the flow rate in each pipe line are obtained.
The connection state of the tree structure part of the pipe network and the demand amount of each node of the tree structure part are stored in the storage means, and the node that is the root of each tree structure part is connected to all the nodes constituting the tree structure part. Allocate the total amount of demand, and reduce the pipe network by representing each tree structure part with one representative node. For the new reduced pipe network, the flow balance equation at each node and each pipe By finding the solution of simultaneous equations consisting of pressure balance equations on the road Flow rate The pressure distribution is obtained, and each tree structure part is the root of the tree structure part based on the flow rate of each pipe and the specification data of each pipe uniquely determined based on the information stored in the storage means. The flow pressure distribution is obtained by sequentially obtaining the pressure at each node from the pressure of the node toward the end of the tree structure portion, and the flow pressure distribution of the entire pipe network is output.
Further, a source that is a virtual supply source, each supply point, a sink that is a virtual consumption point, and each demand point are connected to each other by a virtual pipeline to obtain an extended pipeline network.
The flow rate of all pipelines is set to 0, and the coefficient for the flow rate of the pressure equilibrium equation for each pipeline obtained from the specification data is regarded as a cost function. On the basis of the cost function, searching the minimum cost path tree from the source to the sink based on the cost function, and on the minimum cost path from the source to the sink obtained from the minimum cost path tree Step 3 of adding the maximum possible flow rate obtained from the section width of the flow rate section corresponding to the cost function of each pipeline to the current flow rate of each pipeline, and the flow rate of the virtual pipeline from all demand points to the sink When the step 4 for determining whether or not the demand amount is equal to the determination at the step 4 is equal to the step 5 for ending the flow rate calculation, and when the determination at the step 4 is not equal, the calculation at the step 3 is performed. For a pipeline whose amount matches the segment division point of the current flow rate section, step 6 for obtaining a segment cost point and a new cost function for the adjacent flow segment, and the processing of step 6 from the minimum cost path tree. Step 7 for storing the data of the remaining tree structure except from the target pipe line to the sink, and the minimum cost path from the sink to any terminal node of the tree structure part stored in the processing of step 7 Step 8 of updating the minimum cost path tree by searching and connecting to the tree structure part, and repeating the processing of Step 3 to Step 8 until the determination of Step 4 is equal, The flow distribution is calculated, and the pressure at each node is calculated sequentially toward the sink based on the source pressure using the specification data of each pipe, and the flow pressure distribution of the entire pipe network is output. To have.
In addition, the step 6 includes
The first flow rate width based on the flow rate tolerance and the pressure zone width based on the pressure tolerance are determined in advance, and the second flow rate width corresponding to the pressure zone width is set to the first for each pipe line. A boundary flow value that is smaller than the flow rate interval width is obtained in advance and stored, and in a zone where the absolute value is smaller than the boundary flow value, the first flow rate width is adopted, and the absolute value is greater than the boundary flow value. In the section, by obtaining a second flow section width, a section dividing point of the adjacent flow section is obtained, and a function value of a cost function for the flow rate at a midpoint of the adjacent flow section is obtained. The cost function is used.
In addition, the supply and demand data for each node in the transport pipeline and the specification data for each pipeline are input to the processing unit, and the cost function of each pipeline in the pipeline analysis to obtain the pressure at each contact and the flow rate in each pipeline is In the one-input, one-output network minimum cost flow analysis method as expressed by the increasing function,
A source that is a virtual supply source, each supply point, a sink that is a virtual consumption point, and each demand point are connected by a virtual pipeline to obtain an expansion pipeline, and the flow rate of all pipelines is set to 0 for the expansion pipeline And setting a function value obtained by approximating a given incremental cost function as a step function as a cost function of each pipeline, and searching a minimum cost path tree from the source to the sink based on the cost function In step 2 and on the minimum cost path from the source to sink obtained from the minimum cost path tree, the maximum increaseable flow rate obtained from the section width of the flow rate section corresponding to the cost function of each pipe line is determined as the current flow rate of each pipe line. Step 3 for adding to the flow rate of Step 4, Step 4 for determining whether or not the flow rate of the virtual pipeline from all demand points to the sink is equal to the demand amount, and when the determination in Step 4 is equal, the flow rate calculation is terminated. Stee When the determination in step 4 is not equal to that in step 5, for the pipeline whose flow rate matches the segment division point of the current flow segment according to the calculation in step 3, the segment division point of the adjacent flow segment and the new cost function 6, storing the data of the remaining tree structure from the minimum cost path tree except for the pipe to the sink subjected to the processing of step 6, and the processing of step 7 from the sink Step 8 for updating the minimum cost path tree by searching for the minimum cost path to any terminal node of the tree structure part stored in step S3 and connecting to the tree structure part, and the processing from Step 3 to Step 8 above Is repeated until the determination in step 4 is equal to obtain the minimum cost flow on the network.
The step 6 in the one-input one-output network minimum cost flow analysis method includes
The first flow rate width based on the flow rate tolerance and the pressure zone width based on the pressure tolerance are determined in advance, and the second flow rate width corresponding to the pressure zone width is set to the first for each pipe line. A boundary flow value that is smaller than the flow rate interval width is obtained in advance and stored, and in a zone where the absolute value is smaller than the boundary flow value, the first flow rate width is adopted, and the absolute value is greater than the boundary flow value. In the section, by obtaining a second flow section width, a section dividing point of the adjacent flow section is obtained, and a function value of a cost function for the flow rate at a midpoint of the adjacent flow section is obtained. The cost function is used.
In addition, in a pipe network management system comprising a demand distribution prediction unit, a pipe network data creation unit, a pipe network analysis unit comprising an analysis operation unit and a tree structure storage unit, a supply plan creation unit, and an analysis result output unit,
The analysis calculation unit inputs the supply and demand data of each node and the specification data of each pipe in the pipe network that transports fluid from the pipe network data creation unit, and the connection state of the tree structure part of the pipe network and the tree structure part The demand amount of each node is stored in the tree structure storage unit, and the total amount of demand amounts of all the nodes constituting the tree structure portion is allocated to the node that is the root of each tree structure portion. The pipe network is contracted by representing the portion with one representative node, and for the new contracted network, the simultaneous equation solution consisting of the flow balance equation at each node and the pressure balance equation at each pipeline is solved. By asking for Flow rate The pressure distribution is obtained, and each tree structure portion becomes the root of the tree structure portion based on the flow rate of each pipe and the specification data of each pipe determined uniquely based on the information stored in the tree structure storage unit. The flow pressure distribution is obtained by sequentially obtaining the pressure at each node from the pressure at the nodes to the end of the tree structure portion, and the flow pressure distribution of the entire pipe network is output.
[0008]
[Action]
The analysis process can be effectively simplified by storing the tree structure portion and reducing the pipe network. And an efficient pipe network management system can be obtained by applying this analysis processing.
Further, processing efficiency can be improved by processing the analysis process as a minimum cost flow problem and storing a tree structure portion at this time. By applying this, the general network minimum cost flow problem can be efficiently processed.
[0009]
【Example】
<Example 1>
A pipe network management system in a gas business will be taken up as an embodiment of the present invention and will be described below with reference to the drawings.
FIG. 1 is a diagram showing an overall configuration of a gas pipe network management system to which the present invention is applied. The purpose of this system is to plan gas supply plans and equipment plans for transportation facilities, and gas production and supply, governors (pressure suppressors), and valves are controlled according to the planned plans. It is necessary to make an appropriate plan in order not to supply a gas having a sufficient pressure or to prevent the gas pipe from bursting due to excessive pressure.
In this system, planning is made through the following process.
First, based on the demand distribution obtained by the demand distribution prediction unit 106, the supply plan creation unit 108 creates a gas supply plan.
The supply plan is determined by determining information on (1) the amount and supply pressure of gas in each manufacturing factory and each holder, (2) pressure in each governor (pressure regulator), (3) valve opening and closing, etc. is there.
Next, based on the demand distribution and the supply plan, the pipe network data creation unit 107 creates pipe network data to be input for pipe network analysis calculation.
Pipe network data refers to supply and demand data for nodes, that is, demand at each demand point of the network, supply quantity or pressure at each supply point, and specification data for pipes, that is, connection status, resistance coefficient, pipe length, pipe Diameter, etc.
Finally, the pipe network analysis unit 102 performs pipe network analysis based on the pipe network data, and displays the analysis result on the analysis result output unit 109.
At the same time, the analysis result is transferred to the supply plan creation unit 108, and the supply plan creation unit 108 determines whether the analysis result (pressure distribution, flow rate distribution, etc.) is appropriate. . An appropriate plan is made by repeating the above processing until an appropriate analysis result is obtained.
[0010]
As described above, a supply plan based on a certain demand distribution is drawn up. However, the demand distribution is constantly changing, and in order to provide more detailed services to consumers, it is necessary to make a supply plan at shorter intervals and to achieve a stable gas supply accordingly.
In this system, the pipe network analysis method according to the present invention is applied to the pipe network analysis calculation in the pipe network analysis unit 102, so that the pipe network analysis can be executed in a short time and a real-time plan can be made.
A feature of the pipe network analysis method according to the present invention is that an appropriate tree structure portion of the pipe network is stored in the calculation process of the pipe network analysis, and the calculation efficiency is realized by using it.
Therefore, the pipe network analysis unit 102 includes an analysis calculation unit 104 and a tree structure storage unit 105. The analysis operation is executed by the analysis operation unit 104, and the tree structure storage unit 105 is used to store or read the data of the tree structure part. Hereinafter, a pipe network analysis method performed in the pipe network analysis unit 102 will be described.
[0011]
The pipe network analysis problem is a problem of obtaining the pressure at each node and the flow rate at each pipe when the pipe network data, that is, the supply and demand data at each node and the specification data of each pipe are given as boundary conditions. . At this time, the following flow rate balance condition is satisfied at each node due to the property of the steady flow of the pipeline network.
[0012]
[Expression 1]

[0013]
here,
N: set of all nodes
N _IN : Set of supply points
N _OUT : A set of demand points (a branch point is considered a demand point with zero demand)
x _j : Flow rate in line j (m ^Three / h)
w _i : Supply amount at supply point i (m ^Three / h)
y _i : Demand volume at demand point i (m ^Three / h)
A ₊ (I): Set of pipes starting from node i
A _- (I): A set of pipelines with node i as the end point
When the node is a complete supply point, the first term on the left side of Equation 1 is “0”, and when the node is a complete demand point, the second term on the left side of Equation 1 is “0”. When the node is a branch point not related to supply and demand, the right side of Equation 1 is “0”.
Furthermore, the following pressure equilibrium conditions are established in each pipeline.
[0014]
[Expression 2]

[0015]
here,
B: Set of all pipelines
p _i : Pressure at node i (kgf / cm ² )
s (j): Start point of pipeline j
e (j): End point of pipeline j
R _j : Resistance coefficient of pipe line j
sgn (x): sign of x
It is.
In the case of gas, r = 2 and R _j According to Weymouth's empirical formula,
[0016]
[Equation 3]

[0017]
[Expression 4]

[0018]
here,
s: Specific gravity of gas
L _j : Pipe length of pipe j (m)
D _j : Pipe diameter of pipe j (cm)
It is.
When the number of nodes is n and the number of pipes is m, the number of equations is (Equation 1) and (Equation 2), and the total is m + n. On the other hand, the number of unknowns is m in number of pipeline flow rate, n at the supply amount or pressure at the supply point, and the pressure at the demand point, for a total of m + n. Therefore, by solving the simultaneous equations, the flow rate and pressure distribution on the pipe network can be obtained.
[0019]
As the size of the pipe network increases and the number of equations increases, it becomes very difficult to find a solution to the above equations. In the present invention, by storing the data of the part of the pipe network having a tree structure, the pipe network is effectively reduced, and the number of equations to be coupled is greatly reduced to increase the calculation efficiency. .
In general, the flow rate can be uniquely determined from the demand amount without considering the pressure of a portion that does not include a loop on the pipe network, that is, a tree structure portion. Therefore, as long as the pressure of the node serving as the root of the tree structure portion is obtained, the pressure of each node can be obtained sequentially toward the end using (Equation 2), and it is not necessary to solve the simultaneous equations. Focusing on this, the pipe network is contracted by representing the tree structure of the pipe network by the root node.
The data of the tree structure portion is stored in the tree structure storage unit 105 and deleted from the original pipe network, and the root node is the total amount of demand of all nodes constituting the tree structure portion. Is assigned. By solving the equations expressed by (Equation 1) and (Equation 2) for the reduced pipe network, the pressure of the root node is obtained, and the tree deleted from this pressure and the data in the tree structure storage unit 105 The flow rate and pressure of the structural part can also be calculated. This method does not cause any error due to the contraction.
[0020]
FIG. 3 shows an example of contraction. The actual pipe network can be divided into a loop structure portion 314 composed of nodes 301 and 302 and a conduit 308 and a tree structure portion 315 composed of nodes 303 to 307 and conduits 309 to 313. By adding the total demand of the tree structure unit 315 to the demand of the node 302 that is the root of the tree structure unit 315 and deleting the tree structure unit, the original pipe network is a tube made up of only the loop structure unit. Can be reduced to the net.
[0021]
In order to find and reduce the tree structure portion from the pipe network, for example, the processing shown in the flowchart of FIG. 4 may be used. First, a node whose supply amount or demand amount is known is set as a contractible node (a node whose supply pressure is already known is excluded) (401).
Next, it is the only node of pipe j and the other node i of pipe j ₂ Contract execution node i such that is a contractible node ₁ (402).
If the above reduction execution node i ₁ If no exists, the pipe network reduction process is terminated (403).
If present, the contraction execution node i ₁ And pipe j are removed from the pipe network and node i ₂ Demand q ₂ And contraction execution node i ₁ Demand q ₁ Q ₁ + Q ₂ A new node i ₂ (404). However, supply is considered negative demand.
Subsequently, the number of the deleted pipeline and the number of the contraction execution node and the demand amount at that time (that is, the demand amount after correction) are sequentially stored in the tree structure storage unit 105 (405). Returning to step 402 again, the above processing is repeated.
[0022]
The pipe network model shown in (a) of FIG. 3 is reduced to the pipe network model shown in (b). It is assumed that any node of the pipe network in (a) is a contractible node.
The demand amount of the nodes 302 to 307 is y ₂ ~ Y ₇ Then, in the reduced pipe network of (b), the demand amount of the node 302 is y ₂ (Demand amount of the node 302 itself different from the flow rate flowing through the pipe 309) y ₂ + Y _Three + Y _Four + Y _Five + Y ₆ + Y ₇ become. If the reduced demand is negative, the node is considered a supply point. The data stored in the tree structure storage unit 105 is a set of a deleted node and a pipe number, and a demand amount at that time of the deleted node. This is stored in the order of deletion. As a result of the contraction shown in FIG. 3, (node 303, pipe 310: y _Three ), (Node 306, pipe 312: y ₆ ), (Node 307, pipe 313: y ₇ ), (Node 305, pipeline 311: y _Five + Y ₇ ), (Node 304, pipe 309: y _Three + Y _Four + Y _Five + Y ₆ + Y ₇ ) y _Four Is the demand amount of the node 304 itself.
The demand amount stored here also means the flow rate in the deleted pipeline. Contrary to the stored time, the pressure at the deleted node is obtained by solving the pressure balance equation in each of the pipes 309, 311, 313, 312 and 310 in this order. Can be sought.
It has been described above that the pipe network is effectively reduced by storing the data of the tree structure according to the present invention. 8 and 9 show examples of actual gas pipe network data before contraction and after contraction. In this example, the number of nodes is reduced from 283 to 104 due to contraction, the number of pipes is reduced from 296 to 116, and the scale becomes about 1/3.
[0023]
Next, a method for solving the simultaneous equations described in (Equation 1) and (Equation 2) for the reduced pipe network will be described.
Among these methods, the calculation efficiency can be further increased by applying a method of storing the tree structure portion. This equation results in a conventional minimum cost flow problem (in the gas network analysis problem, cost can be thought of as a square loss of pressure, ie, difficulty of flow corresponds to cost). Can be solved by.
First, the formulation will be described. First, a “virtual supply point” (source) and a “virtual demand point” (sink), and a pipeline connecting the source and supply point and the demand point and sink are newly set in the pipeline network. At this time, the cost function u per unit flow rate of each pipeline j _j (x _j ) Is set as follows.
[0024]
[Equation 5]

[0025]
here,
B _IN1 : A set of pipes connecting the source and the specified pressure supply point
B _IN2 : A set of pipelines that connect a source and a supply point with a specified supply amount
B _OUT : A set of pipelines connecting demand points and sinks
It is.
In addition, it is assumed that the specified supply amount is a capacity for the pipeline connecting the source and the specified supply amount supply point, and that the demand amount is a capacity for the pipeline connecting the demand point and the sink. The capacity of other pipes is unlimited, and the capacity a of each pipe j _j Is set as follows.
[0026]
[Formula 6]

[0027]
At this time, the pipe network analysis problem is formulated into the following minimum cost flow problem.
Restrictions:
[0028]
[Expression 7]

[0029]
[Equation 8]

[0030]
Objective function:
[0031]
[Equation 9]

[0032]
Where A ₊ (i), A _- (i) is a set including virtual pipelines flowing into and out of the source and sink. In the conventional method, pressure is specified for all supply points. However, as shown in (Equation 5) and (Equation 6), the cost function is sufficiently small (-∝) and the capacity is supplied for supply points for which supply amount is specified. It can be formulated by making it equal to the quantity.
[0033]
The minimum cost flow problem has a nonlinearly variable cost function, as shown in (Equation 5). By applying a step function approximation to this, the Primal-Dual method can be applied. This approximation method will be described. In FIG. 2, 201 indicates a cost function u of the pipeline jεB. _j (X _j ) And the cost function is an increasing function. Set the flow axis to multiple sections [a _j (K), a _j (K + 1)], and the cost function 202 is approximated by the step function 202.
When determining the section width, if the flow rate axis is equally divided, the cost error increases when the flow rate increases. However, if the section width is determined so that the error in the vicinity of the maximum flow rate falls within the allowable range, the division becomes unnecessary fine in the vicinity of 0, and the amount of calculation increases. If the cost axis is always equally divided, the flow rate error increases when the flow rate decreases, and if the section width is determined so that the error in the vicinity of 0 falls within the allowable range, an unnecessary fine division is made near the maximum flow rate. Will do.
Therefore, in order to guarantee the accuracy and speed up the calculation, the section width is determined by combining both. Section width δ sufficient to guarantee accuracy for flow axis and cost axis ₁ , Δ ₂ And set the starting point of the division to x _j Take = 0. Dividing point x _j = A _j When (k) is found, the next division point x in the positive direction _j = A _j (K + 1) is obtained by the following equation.
[0034]
[Expression 10]

[0035]
However, Δx _j Is the solution of
[0036]
[Expression 11]

[0037]
The same applies to the division in the negative direction. At this time, the section [a _j (K), a _j U in (k + 1)] _j (X _j ) Is approximated by the following equation.
[0038]
[Expression 12]

[0039]
According to this method, as shown by the step function 202 in FIG. 2, when the flow rate is a small value, the flow rate axis has a width δ. ₁ When the flow rate becomes a large value, the cost axis is δ ₂ Divided by. Therefore, the boundary flow rate value α that becomes the boundary point of the two types of division _j ≧ 0 is obtained in advance, and the interval [−α _j , Α _j ], Set the flow axis to the width δ ₁ The cost axis is divided by the width δ for other sections. ₂ Then, the next division point is determined. α _j Satisfies the following equation simultaneously and δ ₁ It is uniquely determined on the condition that it is an integer multiple of.
[0040]
[Formula 13]

[0041]
[Expression 14]

[0042]
Since r = 2 in the case of gas, α _j Is obtained analytically by the following equation.
[0043]
[Expression 15]

[0044]
Therefore, for each pipeline j, the boundary flow rate value α _j Is stored in the interval [−α _j , Α _j ], Δx according to (Equation 11) _j The calculation of can be omitted. As will be described later, in the solution process of the minimum cost flow problem, every time the flow rate of the pipeline is updated, a calculation for determining the next division point is required, and thus the above method is effective.
[0045]
Based on the flowchart of FIG. 5, the process by the Prime-Dual method which is a solution of the minimum cost flow problem will be described. The feature of this process is that calculation is started from the condition that all pipe flow rates are zero, and the minimum cost path problem and the maximum flow problem are alternately solved repeatedly, and the flow is accumulated to obtain the saturated flow with the minimum cost. It is in.
In the normal Primal-Dual method, it is assumed that the cost function is constant, but the method shown below has been extended so that it can be applied to the case of a cost function that is variable in a step function as described above. .
The flow rate, capacity, and cost function of each pipeline j are each x _j , A _j , U _j The predetermined total demand is v, and the total outflow in the calculation process is q. First, q = 0, that is, the flow rate of each pipeline j is x _j = 0 and the cost function d in the positive direction of each pipeline j _{j +} , Negative cost function d _j- Is set as follows (501).
[0046]
[Expression 16]

[0047]
[Expression 17]

[0048]
However, k = 0.
A minimum cost path from the source to the sink is determined based on the cost function (502). In the search for the minimum cost path, the minimum cost from the source to each node is also calculated. Therefore, a minimum cost path tree from the source to the sink is actually generated.
The flow rate of each pipeline j when q = Q is x _{j (OLD)} , The maximum amount Δq that can be increased along the minimum cost path is calculated by the following equation:
[0049]
[Expression 18]

[0050]
[Equation 19]

[0051]
New flow rate x for each pipeline j _{j (NEW)} The
[0052]
[Expression 20]

[0053]
To q = Q + Δq (503). However,
P ₊ : Set of pipelines included in the forward direction in the minimum cost route
P _- : Set of pipes included in the opposite direction to the minimum cost path
It is.
At this time, it is determined whether the flow is saturated, that is, whether q = v (504). If the flow is saturated, the calculation ends.
If it is not saturated, on the minimum cost route, the pipeline whose changed flow rate coincides with the segment division point is updated with a new segment division point a. _j (K + 1) and cost function U _j (K + 1) is obtained, and the cost function in both positive and negative directions is set again by (Equation 16) and (Equation 17) (505). However, when the flow rate matches the lower limit of the section, k is updated to k−1, and when it matches the upper limit, k is updated to k + 1.
[0054]
The conventional method now returns to step 502 where the minimum cost path tree T based on the updated cost function. _NEW Was newly generated. However, with each iteration, a new minimum cost path tree T _NEW It is computationally expensive to generate Therefore, in the present invention, the previous minimum cost path tree T _OLD Is stored as tree structure data, and the next minimum cost path tree T is stored by using this data. _NEW This saves you the trouble of generating.
Subtree T after the pipe whose cost was changed in step 505 _SUB2 T _OLD Subtree T remaining after removal from _SUB1 The tree structure data is stored (506). And T _SUB1 The minimum path tree T from any suitable node to the sink _SUB3 Searched and remembered T _SUB1 By binding to T _NEW Is generated (507). Here, the process returns to step 503 and is repeated until the condition of step 504 is satisfied.
[0055]
FIG. 6 shows the minimum cost path tree T in the course of the method according to the invention. _OLD New minimum cost path tree T _NEW This shows a specific example in which is generated.
The thick line portion of the network in FIG. 6A indicates the minimum cost path tree T from the previously generated source 601 to the sink 608. _OLD Represents. It is assumed that when the maximum flow is caused to flow through the minimum cost path 610, 613, 616, 620, the end point of a certain section is reached for the first time in the pipe line 616.
At this time, since the cost of the pipeline before the pipeline 616 is not corrected, the subtree T after the pipeline 616 is corrected. _SUB2 (Lines 616, 620) through tree T _OLD Tree structure part T removed from _SUB1 Is the newly generated minimum cost path tree T _NEW It becomes a partial tree.
In the remaining part of the network (nodes 606, 607, 608, pipelines 616, 617, 619, 620, 621), T _SUB1 The minimum path tree T from one of the nodes to the sink 608 _SUB3 (Pipes 617 and 620) are newly generated.
For example, from the sink to the source, T _SUB1 Minimum cost path tree T until the upper node is reached _SUB3 Is generated. T _SUB1 And T _SUB3 By combining, T represented by the bold line portion of the network of FIG. _NEW Can be generated. The minimum cost path tree from the source to the sink is obtained, and the minimum cost path search can be speeded up.
[0056]
By the pipe network analysis method described above, an analysis result can be obtained at a higher speed than the conventional method.
The method according to the present invention is characterized in that a tree structure portion is effectively used. In the system shown in FIG. 1, a tree structure storage unit 105 is provided in the pipe network analysis unit 102 in addition to the analysis calculation unit 104. It is a feature.
On a normal computer, the analysis operation unit 104 is a software module, and the tree structure storage unit 105 can be constructed even if it is a part of the memory, but a dedicated device is more effective.
Since the analysis result can be obtained at high speed, the network management system for the operation plan can shorten the cycle of the plan, and can perform more accurate management. In addition, the efficiency of the calculation method contributes to the cost reduction of the apparatus that performs the analytical calculation.
[0057]
<Example 2>
A pipe network management system in a water supply business will be taken up as a second embodiment of the present invention and will be described below with reference to the drawings.
FIG. 7 is a diagram showing the overall configuration of a pipe network management system for the purpose of monitoring and controlling the flow pressure distribution in the water distribution pipe network. The flow of monitoring control by this system will be described with reference to FIG.
First, based on the demand distribution obtained by the demand distribution prediction unit 706 and the operation command values of pumps and valves output from the pipe network control unit 710, the pipe network data creation unit 707 inputs the pipe network analysis calculation input. Create network data.
Pipe network data is the supply and demand data of nodes, that is, the demand at each demand point of the pipe network, the supply amount and supply pressure at each supply point, and the specification data of the pipeline, that is, connection status, resistance coefficient, pipe length, pipe diameter Etc.
The operation state of the pump, which is the operation command value, and the opening of the valve are reflected in the resistance coefficient of each pipeline. Using the pipe network data as an input, the pipe network analysis unit 102 performs pipe network analysis and outputs an analysis result.
The pipe network control unit 710 determines an operation command value based on the analysis result and transmits the operation command value to the pump 711 and the valve 712. The analysis result output unit 709 creates and displays and prints the flow pressure distribution diagram based on the drawing data output from the drawing data management unit 708 and the analysis result. The output from the analysis result output unit 709 is used for pipe network monitoring.
[0058]
The pipe network analysis problem performed in the pipe network analysis unit 102 is a flow balance condition at each node,
[0059]
[Expression 21]

[0060]
here,
N: set of all nodes
N _IN : Set of supply points
N _OUT : A set of demand points (a branch point is considered a demand point with zero demand)
x _j : Flow rate in line j (m ^Three / h)
w _i : Supply amount at supply point i (m ^Three / h)
y _i : Demand volume at demand point i (m ^Three / h)
A ₊ (I): Set of pipes starting from node i
A _- (I): A set of pipelines with node i as the end point
And pressure equilibrium conditions in each pipeline,
[0061]
[Expression 22]

[0062]
here,
B: Set of all pipelines
p _i : Pressure at node i (kgf / cm ² )
s (j): Start point of pipeline j
e (j): End point of pipeline j
R _j : Resistance coefficient of pipe line j
sgn (x): sign of x
It is a simultaneous equation consisting of
The difference from the gas of Example 1 is that the index of the node pressure on the left side of the pressure equilibrium condition is 1, and the index of the pipe flow rate on the right side is r = 1.85. R _j Is
According to Hazen-Williams' empirical formula:
[0063]
[Expression 23]

[0064]
here,
C _j : Flow velocity coefficient of pipeline j
L _j : Pipe length of pipe j (m)
D _j : Pipe diameter of pipe j (cm)
It is.
This problem can be solved in the same manner as in the first embodiment. However, in the section width determination method when approximating the cost function to the step function, the boundary flow rate value α _j Since r = 1.85 in the case of a water distribution system, it cannot be obtained analytically as in (Equation 15). However, since it can be obtained in advance by repeated calculation such as Newton's method, the same method can be used. Therefore, the pipe network analysis unit 102 executes real-time pipe network analysis, and fine control and monitoring are possible.
[0065]
The solution method of the present invention is described as a solution method for the pipe network analysis problem in the present embodiment, but as is understood from the first and second embodiments, it is originally a solution method for the minimum cost flow problem. Since the pipe network analysis problem can be converted into a minimum cost flow problem as shown in the first embodiment, the present method can be applied. In the network including nodes (nodes) and arcs (pipe lines), It is applicable to all minimum cost flow problems where the cost function of each arc is represented by an increasing function.
"Network Theory" September 27, 1976, published by Nikka Giren, Masao Iri, Takashi Furubayashi (p. 81), introduced several methods for solving the minimum cost flow problem. The method is applied to a certain hitchcock type transportation problem.
The method of the present invention can of course be applied to transportation problems.For example, in order to grasp the power distribution of the distribution pipe network in the power business, if the voltage corresponds to the pressure and the current corresponds to the flow rate, The water and gas pipe network analysis method can be applied as it is.
[0066]
【The invention's effect】
As described above, according to the present invention, it is possible to obtain a pipe network analysis result with sufficient accuracy and in a short time even for a large-scale pipe network, thereby enabling finer pipe network management. Become. Furthermore, it is not necessary to use a high-speed and expensive computer, and downsizing to a low-cost computer is possible.
[Brief description of the drawings]
FIG. 1 is a diagram showing an overall configuration of a gas pipe network management system.
FIG. 2 is a diagram illustrating a step function approximation of a cost function.
FIG. 3 is a diagram showing an example of pipe network contraction.
FIG. 4 is a diagram showing a flowchart of pipe network contraction processing to which the present invention is applied.
FIG. 5 is a diagram showing a flowchart of processing by a primary-dual method to which the present invention is applied.
FIG. 6 is a diagram illustrating a specific example in which a new minimum cost path tree is generated from a minimum cost path tree.
FIG. 7 is a diagram showing an overall configuration diagram of a water distribution pipe network management system.
FIG. 8 is a diagram showing actual gas pipe network data before contraction.
FIG. 9 is a diagram showing actual gas pipe network data after contraction.

Claims (6)

In the pipe network analysis method, the supply and demand data of each node in the pipe network that transports fluid and the specification data of each pipe line are input to the processing device, and the pressure at each contact point and the flow rate in each pipe line are obtained.
Storing the connection state of the tree structure portion of the pipe network and the demand of each node of the tree structure portion in the storage means;
By assigning the total amount of demand of all nodes that make up the tree structure part to the node that is the root of each tree structure part, each tree structure part is represented by one representative node, thereby reducing the pipe network. About
For the new reduced pipe network, find the flow pressure distribution by finding the solution of the simultaneous equation consisting of the flow balance equation at each node and the pressure balance equation at each pipeline,
For each tree structure part, the tree flow is determined from the pressure at the root of the tree structure part based on the flow rate of each pipe and the specification data of each pipe uniquely determined based on the information stored in the storage means. Obtain the flow pressure distribution by obtaining the pressure at each node sequentially toward the end of the structural part,
A pipe network analysis method characterized by outputting a flow pressure distribution of the entire pipe network. In the pipe network analysis method, the supply and demand data of each node in the pipe network that transports fluid and the specification data of each pipe line are input to the processing device, and the pressure at each contact point and the flow rate in each pipe line are obtained.
By connecting a source that is a virtual supply source, each supply point, and a sink that is a virtual consumption point, and each demand point with a virtual pipeline to obtain an expansion pipeline,
The flow rate of all pipelines is set to 0, and the coefficient for the flow rate of the pressure equilibrium equation for each pipeline obtained from the specification data is regarded as a cost function. Step 1 to set as a cost function of
Searching for a minimum cost path tree from source to sink based on the cost function;
On the minimum cost path from source to sink obtained from the minimum cost path tree, the maximum increase flow rate obtained from the section width of the flow section corresponding to the cost function of each pipeline is added to the current flow rate of each pipeline. Step 3,
Step 4 for determining whether the flow rate of the virtual pipeline from all demand points to the sink is equal to the demand amount;
When the determination in step 4 is equal, step 5 ends the flow rate calculation;
When the determination in step 4 is not equal, for a pipeline whose flow rate coincides with the segment division point of the current flow segment by the calculation in step 3, the segment division point of the adjacent flow segment and a new cost function are obtained. When,
Step 7 of storing the data of the remaining tree structure except for the pipeline from the minimum cost path tree to the processing target in Step 6 to the sink;
A step 8 for updating a minimum cost path tree by searching for a minimum cost path from a sink to any terminal node of the tree structure part stored in the process of step 7 and connecting to the tree structure part;
The flow distribution of the entire pipe network is calculated by repeating the processing of step 3 to step 8 until the determination of step 4 becomes equal, and the sink pressure is calculated based on the source pressure using the specification data of each pipe. A method for analyzing a pipe network, which sequentially obtains the pressure at each node toward the center and outputs the flow pressure distribution of the entire pipe network. In the pipe network analysis method according to claim 2,
Step 6 includes
The first flow rate width based on the flow rate tolerance and the pressure zone width based on the pressure tolerance are determined in advance,
A boundary flow rate value in which the second flow rate width corresponding to the pressure zone width is smaller than the first flow rate width is determined and stored in advance for each pipe line,
In the section where the absolute value is smaller than the boundary flow value, the first flow section width is adopted,
In a section where the absolute value is larger than the boundary flow value, a section dividing point of the adjacent flow section is obtained by obtaining a second flow section width, and a cost function for the flow at the midpoint of the adjacent flow section is obtained. A pipe network analysis method, wherein a function value is a cost function of the adjacent flow rate section. Supply and demand data for each node in the transport pipeline and specification data for each pipeline are input to the processor, and the cost function for each pipeline in the pipeline analysis to obtain the pressure at each contact and the flow rate in each pipeline is an increasing function. In the one-input, one-output network minimum cost flow analysis method represented by
By connecting a source that is a virtual supply source, each supply point, and a sink that is a virtual consumption point, and each demand point with a virtual pipeline to obtain an expansion pipeline,
Step 1 for setting the flow rate of all pipelines to 0 and setting a function value obtained by approximating a given increase cost function as a step function as a cost function of each pipeline;
Searching for a minimum cost path tree from source to sink based on the cost function;
On the minimum cost path from source to sink obtained from the minimum cost path tree, the maximum increase flow rate obtained from the section width of the flow section corresponding to the cost function of each pipeline is added to the current flow rate of each pipeline. Step 3,
Step 4 for determining whether the flow rate of the virtual pipeline from all demand points to the sink is equal to the demand amount;
When the determination in step 4 is equal, step 5 ends the flow rate calculation;
When the determination in step 4 is not equal, for a pipeline whose flow rate coincides with the segment division point of the current flow segment by the calculation in step 3, the segment division point of the adjacent flow segment and a new cost function are obtained. When,
Step 7 of storing the data of the remaining tree structure except for the pipeline from the minimum cost path tree to the processing target in Step 6 to the sink;
A step 8 for updating a minimum cost path tree by searching for a minimum cost path from a sink to any terminal node of the tree structure part stored in the process of step 7 and connecting to the tree structure part;
A network minimum cost flow analysis method characterized in that the minimum cost flow on the network is obtained by repeating the processing of step 3 to step 8 until the determination of step 4 becomes equal. The network minimum cost flow analysis method according to claim 4,
Step 6 includes
The first flow rate width based on the flow rate tolerance and the pressure zone width based on the pressure tolerance are determined in advance,
A boundary flow rate value in which the second flow rate width corresponding to the pressure zone width is smaller than the first flow rate width is determined and stored in advance for each pipe line,
In the section where the absolute value is smaller than the boundary flow value, the first flow section width is adopted,
In a section where the absolute value is larger than the boundary flow value, a section dividing point of the adjacent flow section is obtained by obtaining a second flow section width, and a cost function for the flow at the midpoint of the adjacent flow section is obtained. A network minimum cost flow analysis method, wherein a function value is a cost function of the adjacent flow rate section. In a pipe network management system comprising a demand distribution prediction unit, a pipe network data creation unit, a pipe network analysis unit comprising an analysis operation unit and a tree structure storage unit, a supply plan creation unit, and an analysis result output unit,
The analysis calculation unit inputs the supply and demand data of each node and the specification data of each pipe in the pipe network that transports fluid from the pipe network data creation unit, and the connection state of the tree structure part of the pipe network and the tree structure part Store the demand amount of each node in the tree structure storage unit,
By assigning the total amount of demand of all nodes that make up the tree structure part to the node that is the root of each tree structure part, each tree structure part is represented by one representative node, thereby reducing the pipe network. About
For the new reduced pipe network, find the flow pressure distribution by finding the solution of the simultaneous equation consisting of the flow balance equation at each node and the pressure balance equation at each pipeline,
For each tree structure part, the pressure of the node that is the root of the tree structure part is determined by the flow rate of each pipe and the specification data of each pipe uniquely determined based on the information stored in the tree structure storage unit. From the flow rate pressure distribution by obtaining the pressure at each node sequentially from the end of the tree structure part to
A pipe network management system that outputs the flow pressure distribution of the entire pipe network.

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