第三节牛顿-拉夫逊法潮流计算

4-3牛顿—拉夫逊法潮流计算第三节牛顿—拉夫逊法潮流计算•功率方程与修正方程组•潮流计算的基本步骤•算例4-3牛顿—拉夫逊法潮流计算•已知参数和求解变量•节点注入功率-电压实数方程组(极坐标形式)•n节点网络的功率方程组•修正方程组一、功率方程与修正方程组4-3牛顿—拉夫逊法潮流计算•已知参数(n节点网络)平衡节点1个(编号为s):给定电压PQ节点m-1个(编号为1、2、…m):给定节点注入P、QPV节点n-m个(编号为m+1、m+2、…n):给定节点注入P、U已知参数和求解变量0Ð=ssUU&共2(m-1)+(n-m)=n+m-2个变量，则需n+m-2个独立方程ijiieUUd=&•求解变量•各节点电压(第一求解对象):PQ节点：Ui,diPV节点：di•节点功率和支路功率(第二求解对象)4-3牛顿—拉夫逊法潮流计算节点注入功率—电压实数方程组(极坐标形式)å=÷øöçèæ=+=njjijiiiiUYUjQPS1**~&()å=--=njjjijijjiijieUjBGeUS1~dd()å=-=njjjijijiiijeUjBGUS1~d()()å=+-=njijijjijijiijUjBGUS1sincos~dd对节点i：jijjjjiiijijijeUUeUUjBGYdd==+=&&，，令极坐标形式jiijddd-=ijijjjeijdddsincos+=(4-35)4-3牛顿—拉夫逊法潮流计算()()å=+-=+=njijijjijijiiiijUjBGUjQPS1sincos~dd()()åå==d-d=d+d=njijijijijjiinjijijijijjiiBGUUQBGUUP11cossinsincos实、虚部分开•左端为节点注入功率(已知量),右端为节点输出功率(电压为待求量)节点注入功率—电压实数方程组(极坐标形式)(4-43a)(4-43b)4-3牛顿—拉夫逊法潮流计算N节点网络的功率方程组PQ节点(m-1个)PV节点(n-m个)共2(m-1)个方程共n-m个方程总共：2(m-1)+(n-m)=n+m-2个方程,求解n+m-2个变量……()()()()ïïîïïíì-=+=ïïîïïíì-=+=åååå====njjjjjjnjjjjjjnjjjjjjnjjjjjjBGUUQBGUUPBGUUQBGUUP1222222122222211111111111111cossinsincoscossinsincosdddddddd……()()åå=++++++=+++++++=+=njjmjmjmjmjmmnjjmjmjmjmjmmBGUUPBGUUP1)2()2()2()2(221)1()1()1()1(11sincossincosdddd4-3牛顿—拉夫逊法潮流计算节点电压修正方程()å=d+d=njijijijijjiiBGUUP1sincos()å=d-d=njijijijijjiiBGUUQ1cossinúûùêëéDDúúúúûùêêêêë鶶¶¶¶¶¶¶=úûùêëéDDiiijjijijjijiiiUUUUQQUUPPQP/ddd电压修正方程Df=JDx()å=d-d-=DnjijijijijjiiiBGUUQQ1cossin)δ,U(()å=d+d-=DnjijijijijjiiiBGUUPP1sincos)δ,U((U,δ)不是真解(4-45a)(4-45b)产生功率不平衡量⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡ΔΔΔΔΔΔ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⋅⋅⋅⋅⋅⋅∂∂∂∂⋅⋅⋅∂∂∂∂∂Δ∂∂∂⋅⋅⋅∂∂∂∂⋅⋅⋅∂∂∂∂∂∂∂∂⋅⋅⋅⋅⋅⋅∂∂∂∂⋅⋅⋅∂∂∂∂∂∂∂∂⋅⋅⋅∂∂∂∂⋅⋅⋅∂∂∂∂∂∂∂∂⋅⋅⋅∂∂∂∂⋅⋅⋅∂∂∂∂∂∂∂∂⋅⋅⋅∂∂∂∂⋅⋅⋅∂∂∂∂∂∂∂∂=⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡ΔΔΔΔΔΔ+++++++++++++++++++++++++++MMMM2122211122122222211212211122121111111212222221121212122222211212111221211111111112212111111212211//mmmmmmmmmmmmmmmmmmmmmmnmmmmmUUUUPPUUPPUUPPPPUUPPUUPPQQUUQQUUQQPPUUPPUUPPQQUUQQUUQQPPUUPPUUPPPPQPQPdddddddddddddddddddddddddddd(n+m-2)×1(n+m-2)×(n+m-2)(n+m-2)×1节点电压修正方程(展开式)PQPVPQPV4-3牛顿—拉夫逊法潮流计算Jacobian矩阵元素•Jacobian矩阵的元素分四类：jiijPHd¶¶=jjiijUUPN¶¶=jiijQJd¶¶=jjiijUUQL¶¶=(4-46)•Jacobian矩阵的元素是节点i的注入功率对另一个节点j的电压Uj或dj的偏导数,故只有Uj或dj是变量,其余参数均视为常量()()()()1111111111111212313131313121212cocosssinsincossincossin...njijijijijjPUUGBUUGUBBBUGGdddddddd==+=+++⎡⎣++⎦++⎤∑仅U2是变量)sincos(121212122122112d+d=========¶¶=BGUUUUPN•节点14-3牛顿—拉夫逊法潮流计算,cossinijjijddd-=¶¶,sincosijiijddd-=¶¶ijiijdddcossin=¶¶Jacobian矩阵中三角函数的导数同理：•对dj求偏导,仅dj为变量,故dij＝di-dj是变量ijijjjijijijjijijddddddddddddddsin)1)(sin()()()cos()cos(cos=--=¶-¶-¶-¶=¶-¶=¶¶4-3牛顿—拉夫逊法潮流计算•非对角元Hij、Nij(i≠j)的计算)cossin(ijijijijjijiijBGUUPHddd-=¶¶=)sincos(ijijijijjijjiijBGUUUUPNdd+=¶¶=Jacobian矩阵非对角元的计算()å=+=njijijijijjiiBGUUP1sincosdd仅一个节点j的电压Uj或dj是变量,仅涉及一个大项4-3牛顿—拉夫逊法潮流计算()å=-=njijijijijjiiBGUUQ1cossindd)sincos(ijijijijjijiijBGUUQJddd+-=¶¶=)cossin(ijijijijjijjiijBGUUUUQLdd-=¶¶=Jacobian矩阵非对角元的计算•非对角元Jij、Lij(i≠j)的计算仅一个节点j的电压Uj或dj是变量,仅涉及一个大项4-3牛顿—拉夫逊法潮流计算iiinijjijijijijjiiiiiiGUBGUUUUPN212)sincos(++=¶¶=å¹=ddJacobian矩阵对角元的计算•对角元Hii、Nii的计算()()iiinijjijijijijjinjijijijijjiiGUBGUUBGUUP211sincossincos++=+=åå¹==ddddå¹=--=¶¶=nijjijijijijjiiiiiBGUUPH1)cossin(ddd仅节点i的电压Uj或dj是变量,但涉及各项4-3牛顿—拉夫逊法潮流计算()()()()1111111111111212121313121231313cossincossincossincossin...njijijijijjPUUGBUUGBUGBUGBdddddddd==+=++++⎡⎣+++⎤⎦∑()()1111212121212131313131312sincossincos...(sincos)njijijijijjPHUUGBUGBUUGBddddddd=∂==-+⎡⎣∂+-++⎤⎦=--∑例如：Jacobian矩阵对角元的计算4-3牛顿—拉夫逊法潮流计算()åå¹==--=-=nijjijiijijijijjinjijijijijjiiBUBGUUBGUUQ121)cossin(cossinddddå¹=+=¶¶=nijjijijijijjiiiiiBGUUQJ1)sincos(dddå¹=--=¶¶=nijjiiiijijijijjiiiiiiBUBGUUUUQL122)cossin(ddJacobian矩阵对角元的计算•对角元Jii、Lii的计算仅节点i的电压Uj或dj是变量,但涉及各项4-3牛顿—拉夫逊法潮流计算Jacobian矩阵的特点•n节点网络,平衡节点1个,PQ节点m-1个,PV节点n-m个,则雅可比矩阵为n+m-2阶非奇异方阵•Jacobian矩阵的元素是节点电压Ui、di的函数•稀疏矩阵•非对称矩阵•分块Jacobian矩阵与节点导纳矩阵YB具有相同的结构4-3牛顿—拉夫逊法潮流计算二、N-R法潮流计算的基本步骤；节点)0()0(,iiUPQd)0(jPVd节点()å=+=njijijijijjiiBGUUP1)0()0()0()0()0(sincosdd)0()0()0()0()0()0(,jjjiiiiiiPPPPVQQQPPPPQ-=D-=D-=D节点：节点：(1)形成节点导纳矩阵(2)设定各节点电压初解：(3)计算各节点注入功率与输出功率的不平衡量：()å=-=njijijijijjiiBGUUQ1)0()0()0()0()0(cossindd0,1)0()0(=d=iiU一般设4-3牛顿—拉夫逊法潮流计算二、N-R法潮流计算的基本步骤(4)形成Jacobian矩阵(5)解电压修正方程组úûùêëéDD+úûùêëé=úûùêëé++UUU)()()1()1(dddkkkk(6)修正电压求电压新值：edeDD,U若eeDDiiQP,若(7)收敛判断：或者：则结束,否则转第3步则结束,否则转第3步úûùêëéDDúûùêëé=úûùêëéDDULJNHQPdúûùêëéDDUd计算修正量4-3牛顿—拉夫逊法潮流计算二、N-R法潮流计算的基本步骤(9)计算平衡节点注入功率：()()åå==-=+=njsjsjsjsjjssnjsjsjsjsjjssBGUUQBGUUP11cossinsincosdddd))((~***0***ijjiiiiijiijyUUyUUIUS-+==&&))((~***0***ijijjjjjijjiyUUyUUIUS-+==&&(8)计算线路两端功率：yijyi0yj0ijIij&Iji&Sij~Sji~I1&I2&4-3牛顿—拉夫逊法潮流计算jiijjiSSS~~~+=D二、N-R法潮流计算的基本步骤(10)计算线路功率损耗：(11)计算电压降落、电压偏移等yijyi0yj0ijIij&Iji&Sij~Sji~I1&I2&4-3牛顿—拉夫逊法潮流计算例:对如图系统进行潮流计算.基准功率100MVA,电压、电抗均为标么值,计算收敛精度0.010.02+j0.061-30.06+j0.182-30.08+j0.241-2阻抗Zij支路i-j线路参数三、N-R法算例GG60+j2512350+j20Z12Z13Z234-3牛顿—拉夫逊法潮流计算60500P负荷功率（MVA）δ节点类型发电机注入功率（MVA）电压PQ25003PV20201.032平衡001.051QQPU节点节点数据三、N-R法算例4-3牛顿—拉夫逊法潮流计算GG1230.08+j0.240.06+j0.180.02+j0.0675.325.124.008.0111212jjZy-=+==1550.56667.11323jyjy-=-=úúúûùêêêëé-+-+-+--+-+-+--=206667.60.56667.11550.56667.175.89167.27532511557532517518256Yjj