DAE Variables, or variables for short, are unknowns to be solved using numerical or analytical methods. A variable stores values, equation values, and addresses in the DAE array. The base class for variables is BaseVar. In this subsection, BaseVar is used to represent any subclass of VarBase list in the table below.

BaseVar([name, tex_name, info, unit, v_str, ...])

Base variable class.

ExtVar(model, src[, indexer, allow_none, ...])

Externally defined algebraic variable

State([name, tex_name, info, unit, v_str, ...])

Differential variable class, an alias of the BaseVar.

Algeb([name, tex_name, info, unit, v_str, ...])

Algebraic variable class, an alias of the BaseVar.

ExtState(model, src[, indexer, allow_none, ...])

External state variable type.

ExtAlgeb(model, src[, indexer, allow_none, ...])

External algebraic variable type.

AliasState(var, **kwargs)

Alias state variable.

AliasAlgeb(var, **kwargs)

Alias algebraic variable.

Note that equations associated with state variables are in the form of \(\mathbf{M} \dot{x} = \mathbf{f(x, y)}\), where \(\mathbf{x}\) are the differential variables, \(\mathbf{y}\) are the algebraic variables, and \(\mathbf{M}\) is the mass matrix, and \(\mathbf{f}\) are the right-hand side of differential equations. Equations associated with algebraic variables take the form of \(0 = \mathbf{g}\), where \(\mathbf{g}\) are the equation right-hand side

BaseVar has two types: the differential variable type State and the algebraic variable type Algeb. State variables are described by differential equations, whereas algebraic variables are described by algebraic equations. State variables can only change continuously, while algebraic variables can be discontinuous.

Based on the model the variable is defined, variables can be internal or external. Most variables are internal and only appear in equations in the same model. Some models have "public" variables that can be accessed by other models. For example, a Bus defines v for the voltage magnitude. Each device attached to a particular bus needs to access the value and impose the reactive power injection. It can be done with ExtAlgeb or ExtState, which links with an existing variable from a model or a group.

Variable, Equation and Address#

Subclasses of BaseVar are value providers and equation providers. Each BaseVar has member attributes v and e for variable values and equation values, respectively. The initial value of v is set by the initialization routine, and the initial value of e is set to zero. In the process of power flow calculation or time domain simulation, v is not directly modifiable by models but rather updated after solving non-linear equations. e is updated by the models and summed up before solving equations.

Each BaseVar also stores addresses of this variable, for all devices, in its member attribute a. The addresses are 0-based indices into the numerical DAE array, f or g, based on the variable type.

For example, Bus has self.a = Algeb() as the voltage phase angle variable. For a 5-bus system, Bus.a.a stores the addresses of the a variable for all the five Bus devices. Conventionally, Bus.a.a will be assigned np.array([0, 1, 2, 3, 4]).

Value and Equation Strings#

The most important feature of the symbolic framework is allowing to define equations using strings. There are three types of strings for a variable, stored in the following member attributes, respectively:

  • v_str: equation string for explicit initialization in the form of v = v_str(x, y).

  • v_iter: equation string for implicit initialization in the form of v_iter(x, y) = 0

  • e_str: equation string for (full or part of) the differential or algebraic equation.

The difference between v_str and v_iter should be clearly noted. v_str evaluates directly into the initial value, while all v_iter equations are solved numerically using the Newton-Krylov iterative method.

Values Between DAE and Models#

ANDES adopts a decentralized architecture which provides each model a copy of variable values before equation evaluation. This architecture allows to parallelize the equation evaluation (in theory, or in practice if one works round the Python GIL). However, this architecture requires a coherent protocol for updating the DAE arrays and the BaseVar arrays. More specifically, how the variable and equations values from model VarBase should be summed up or forcefully set at the DAE arrays needs to be defined.

The protocol is relevant when a model defines subclasses of BaseVar that are supposed to be "public". Other models share this variable with ExtAlgeb or ExtState.

By default, all v and e at the same address are summed up. This is the most common case, such as a Bus connected by multiple devices: power injections from devices should be summed up.

In addition, BaseVar provides two flags, v_setter and e_setter, for cases when one VarBase needs to overwrite the variable or equation values.

Flags for Value Overwriting#

BaseVar have special flags for handling value initialization and equation values. This is only relevant for public or external variables. The v_setter is used to indicate whether a particular BaseVar instance sets the initial value. The e_setter flag indicates whether the equation associated with a BaseVar sets the equation value.

The v_setter flag is checked when collecting data from models to the numerical DAE array. If v_setter is False, variable values of the same address will be added. If one of the variable or external variable has v_setter is True, it will, at the end, set the values in the DAE array to its value. Only one BaseVar of the same address is allowed to have v_setter == True.

A v_setter Example#

A Bus is allowed to default the initial voltage magnitude to 1 and the voltage phase angle to 0. If a PV device is connected to a Bus device, the PV should be allowed to override the voltage initial value with the voltage set point.

In Bus.__init__(), one has

self.v = Algeb(v_str='1')

In PV.__init__, one can use

self.v0 = Param()
self.bus = IdxParam(model='Bus')

self.v = ExtAlgeb(src='v',

where an ExtAlgeb is defined to access Bus.v using indexer self.bus. The v_str line sets the initial value to v0. In the variable initialization phase for PV, PV.v.v is set to v0.

During the value collection into DAE.y by the System class, PV.v, as a final v_setter, will overwrite the voltage magnitude for Bus devices with the indices provided in PV.bus.