API Reference¶
Functions¶
- class Solverz.sym_algebra.functions.sin(*args)¶
The sine function.
- class Solverz.sym_algebra.functions.cos(*args)¶
The cosine function.
- class Solverz.sym_algebra.functions.exp(*args)¶
The exponential function, \(e^x\).
- class Solverz.sym_algebra.functions.Abs(*args)¶
The element-wise absolute value function:
\[\operatorname{Abs}(x)=|x|\]with derivative
\[\frac{\mathrm{d}}{\mathrm{d}x}{\operatorname{Abs}}(x)=\operatorname{Sign}(x).\]
- class Solverz.sym_algebra.functions.Sign(*args)¶
The element-wise indication of the sign of a number
\[\begin{split}\operatorname{Sign}(x)= \begin{cases} 1&x> 0\\ 0& x==0\\ -1&x< 0 \end{cases}\end{split}\]
- class Solverz.sym_algebra.functions.AntiWindUp(u, umin, umax, e)¶
For PI controller
\[\begin{split}\begin{aligned} \dot{z}(t) & =e(t) \\ u_{d e s}(t) & =K_p e(t)+K_i z(t) \end{aligned},\end{split}\]we limit the integrator output by setting, in the \(K_i>0\) case,
\[\begin{split}\dot{z}(t)= \begin{cases} 0 & \text { if } u_{\text {des }}(t) \geq u_{\max } \text { and } e(t) \geq 0 \\ 0 & \text { if } u_{\text {des }}(t) \leq u_{\min } \text { and } e(t) \leq 0 \\ e(t) & \text { otherwise } \end{cases}.\end{split}\]So the AntiWindUp function reads
\[\dot{z}(t)= \operatorname{AntiWindUp}(u_{\text {des }}(t), u_{\min }, u_{\max }, e(t)).\]
- class Solverz.sym_algebra.functions.Min(x, y)¶
The element-wise minimum of two numbers
\[\begin{split}\begin{split}\min(x,y)= \begin{cases} x&x\leq y\\ y& x>y \end{cases}\end{split}\end{split}\]
- class Solverz.sym_algebra.functions.Saturation(v, vmin, vmax)¶
The element-wise saturation a number
\[\begin{split}\operatorname{Saturation}(v, v_\min, v_\max)= \begin{cases} v_\max&v> v_\max\\ v& v_\min\leq v\leq v_\max\\ v_\min&v< v_\min \end{cases}\end{split}\]
- class Solverz.sym_algebra.functions.heaviside(*args)¶
The heaviside step function
\[\begin{split}\operatorname{Heaviside}(x)= \begin{cases} 1 & x >= 0\\ 0 & x < 0\\ \end{cases}\end{split}\]which should be distinguished from sympy.Heaviside
- class Solverz.sym_algebra.functions.ln(*args)¶
The ln function, \(ln(x)\).
Solvers¶
AE solver¶
- Solverz.solvers.nlaesolver.nr.nr_method(eqn: nAE, y: ndarray, opt: Opt = None)¶
The Newton-Raphson method.
- Parameters:
eqn : nAE
Numerical AE object.
y : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- ite_tol: 1e-5(default)|float
The iteration error tolerance.
- Returns:
sol : aesol
The aesol object.
References
- Solverz.solvers.nlaesolver.cnr.continuous_nr(eqn: nAE, y: ndarray, opt: Opt = None)¶
The Continuous Newton-Raphson method.
- Parameters:
eqn : nAE
Numerical AE object.
y : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- ite_tol: 1e-5(default)|float
The iteration error tolerance.
- Returns:
sol : aesol
The aesol object.
References
[R2]F. Milano, ‘Continuous Newton’s Method for Power Flow Analysis’, IEEE Transactions on Power Systems, vol. 24, no. 1, pp. 50–57, Feb. 2009, doi: 10.1109/TPWRS.2008.2004820.
- Solverz.solvers.nlaesolver.lm.lm(eqn: nAE, y: ndarray, opt: Opt = None)¶
The Levenberg-Marquardt method by minpack, wrapped in scipy.
Warning
Note that this function uses only dense Jacobian.
- Parameters:
eqn : nAE
Numerical AE object.
y : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- ite_tol: 1e-5(default)|float
The iteration error tolerance.
- Returns:
sol : aesol
The aesol object.
References
[R3]More, Jorge J., Burton S. Garbow, and Kenneth E. Hillstrom. 1980. User Guide for MINPACK-1.
- Solverz.solvers.nlaesolver.sicnm.sicnm(ae: nAE, y0: ndarray, opt: Opt = None)¶
The semi-implicit continuous Newton method [R5]. The philosophy is to rewrite the algebraic equation
\[0=g(y)\]as the differential algebraic equations
\[\begin{split}\begin{aligned} \dot{y}&=z \\ 0&=J(y)z+g(y) \end{aligned}\end{split}\]with \(y_0\) being the initial value guess and \(z_0=-J(y_0)^{-1}g(y_0)\), where \(z\) is an intermediate variable introduced. Then the DAEs are solved by Rodas. SICNM is found to be more robust than the Newton’s method, for which the theoretical proof can be found in my paper [R5]. In addition, the non-iterative nature of Rodas guarantees the efficiency.
One can change the rodas scheme according to the ones implemented in the DAE version of Rodas.
SICNM used the Hessian-vector product (HVP) interface of the algebraic equation models, the make_hvp flag should be set to True when printing numerical modules.
An illustrative example of SICNM can be found in the power flow section of Solverz’ cookbook.
- Parameters:
ae : nAE
Numerical AE object.
y0 : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- ite_tol: 1e-5(default)|float
The iteration error tolerance.
- Returns:
sol : aesol
The aesol object.
References
FDAE solver¶
- Solverz.solvers.fdesolver.fdae_solver(fdae: nFDAE, tspan: List | ndarray, u0: ndarray, opt: Opt = None)¶
The general solver of FDAE.
- Parameters:
fdae : nFDAE
Numerical FDAE object.
tspan : List | np.ndarray
An array specifying t0 and tend
u0 : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- step_size: 1e-3(default)|float
The step size
- ite_tol: 1e-8(default)|float
The error tolerance of inner Newton iterations.
- Returns:
sol : daesol
The daesol object.
DAE solver¶
- Solverz.solvers.daesolver.beuler.backward_euler(dae: nDAE, tspan: List | ndarray, y0: ndarray, opt: Opt = None)¶
The fixed step backward Euler (implicit Euler) method.
- Parameters:
dae : nDAE
Numerical DAE object.
tspan : List | np.ndarray
An array specifying t0 and tend
y0 : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- step_size: 1e-3(default)|float
The step size
- Returns:
sol : daesol
The daesol object.
References
- Solverz.solvers.daesolver.trapezoidal.implicit_trapezoid(dae: nDAE, tspan: List | ndarray, y0: ndarray, opt: Opt = None)¶
The fixed step implicit trapezoidal method.
- Parameters:
dae : nDAE
Numerical DAE object.
tspan : List | np.ndarray
An array specifying t0 and tend
y0 : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- step_size: 1e-3(default)|float
The step size
- Returns:
sol : daesol
The daesol object.
References
- Solverz.solvers.daesolver.rodas.Rodas(dae: nDAE, tspan: List | ndarray, y0: ndarray, opt: Opt = None)¶
The stiffly accurate Rosenbrock methods including Rodas4 [R8], Rodasp [R9], Rodas5p [R10].
- Parameters:
dae : nDAE
Numerical DAE object.
tspan : List | np.ndarray
Either - a list specifying [t0, tend], or - a np.ndarray specifying the time nodes that you are concerned about
y0 : np.ndarray
The initial values of variables
opt : Opt
The solver options, including:
- scheme: ‘rodas4’(default)|’rodasp’|’rodas5p’
The rodas scheme
- rtol: 1e-3(default)|float
The relative error tolerance
- atol: 1e-6(default)|float
The absolute error tolerance
- event: Callable
The simulation events, with \(t\) and \(y\) being args, and value, is_terminal and direction being outputs
- fixed_h: False(default)|bool
To use fixed step size
- hinit: None(default)|float
Initial step size
- hmax: None(default)|float
Maximum step size.
- Returns:
sol : daesol
The daesol object.
References