北京交通大学内点法最优潮流*BEUINGJIAOTONGUNIVERSITYLogarithmic barrierO(x) = -Z1og(-f(x)dom0(x)= (x I fi(x) <0, : : : ,fm(x)<0)i=1: convex (follows from composition rules) twice continuously differentiable, with derivativesm1V0(x) =Vf(x)-f.(x)i=lm1m1ZV?0(x) =Vf(x)Vf(x) +ZV? f (x)-f(x)f (x)i=1i=l12
内点法最优潮流* Logarithmic barrier m Θ(x) = −∑log(−fi (x)) i=1 dom Θ(x) = {x | f1(x) < 0, . . . , fm (x) < 0} • convex (follows from composition rules) • twice continuously differentiable, with derivatives 12
北京交通大学内点法最优潮流*BEUINGJIAOTONGUNIVERSITYCentral path? for t > O, define x*(t) as the solution ofmin imizetfo(x)+0(x)subject to Ax =b(for now, assume x*(t) exists and is unique for each t > 0)· central path is (x*(t) I t > 0)example: central path for an LPcTxminimizesubject to a' x <b ,i =l,..,6*(10)2hyperplane cTx = cTx*(t) is tangent tolevel curve of @ through x*(t)13
内点法最优潮流* Central path • for t > 0, define x*(t) as the solution of min imize tf0 (x) + Θ(x) subject to Ax = b (for now, assume x*(t) exists and is unique for each t > 0) • central path is {x*(t) | t > 0} example: central path for an LP minimize cT x subject to aT x ≤ b ,i =1,.,6 i i 13
北京交通大学内点法最优潮流*BEUING JIAOTONG UNIVERSITYBarrier methodgiven strictly feasible x, t := t(0) > 0, u > 1, tolerance > 0repeat1. Centering step. Compute x *(t) by minimizing tfo + O, subject to Ax = b2. Update. x := x *(t).3. Stopping criterion. quit if m/ t < &.4. Increase t.t := ut.-terminates with fo (x) - p* ≤ (stopping criterion follows fromfo(x*(t) - p*≤m/t)-centering usually done using Newton's method, starting at current x·choice of μ involves a trade-off: large u means fewer outer iterations.more inner (Newton) iterations; typical values: μ = 10-20: several heuristics for choice of t(0)14
内点法最优潮流* Barrier method •terminates with f0 (x) − p* ≤ Ɛ (stopping criterion followsfrom f0 (x*(t)) − p*≤m/t) •centering usually done using Newton’s method, starting at current x •choice of μ involves a trade-off: large μ means fewer outer iterations, more inner (Newton) iterations; typical values: μ = 10–20 • several heuristics for choice of t(0) 14
京文通大学内点法最优潮流*BEUINGJIAOTONGUNIVERSITY内点法是一种直接求解KKT条件的解法牛顿法：将含等式约束的二次可微目标函数简化为一系列线性等式约束问题求解内点法：将含不等式和等式约束的优化问题简化成一系列线性等式约束问题求解。15
内点法最优潮流* 内点法是一种直接求解KKT条件的解法 牛顿法：将含等式约束的二次可微目标函数简 化为一系列线性等式约束问题求解。 内点法：将含不等式和等式约束的优化问题简 化成一系列线性等式约束问题求解。 15
北京交通大学(OPF)7.1最优潮流BEUING JIAOTONG UNIVERSITY·最优潮流的定位潮流问题是潮流调整问题是考虑约束非线性系统非线性系统发电自由度和网络安全的的解不唯一的求解问题约束非线性系统求解问题为了确定唯一的约潮流调整束非线型系统解必问题是带目标的约束非线性须为控制变量自由最优潮流系统是一个优化系统度指定一个效益目问题标函数1、最优潮流问题的第一自标是安全优化目标只是副产品16
• 最优潮流的定位 潮流问题是 非线性系统 的求解问题 潮流调整问题是考虑 发电自由度和网络安全的 约束非线性系统求解问题 约束非线性系统 的解不唯一 为了确定唯一的约 束非线型系统解必 须为控制变量自由 度指定一个效益目 标函数 带目标的约束非线性 系统是一个优化系统 潮流调整 问题是 最优潮流 问题 1、最优潮流问题的第一目标是安全 优化目标只是副产品 7.1 最优潮流（OPF） 16