system.py

"""
System class for power system data and methods
"""

#  [ANDES] (C)2015-2021 Hantao Cui
#
#  This program is free software; you can redistribute it and/or modify
#  it under the terms of the GNU General Public License as published by
#  the Free Software Foundation; either version 3 of the License, or
#  (at your option) any later version.
#
#  File name: system.py

import configparser
import importlib
import logging
import os
import sys
import inspect
import time

import dill

from collections import OrderedDict
from typing import Dict, Tuple, Union, Optional

import andes.io
from andes.models import file_classes
from andes.models.group import GroupBase
from andes.variables import FileMan, DAE
from andes.routines import all_routines
from andes.utils.tab import Tab
from andes.utils.misc import elapsed
from andes.utils.paths import confirm_overwrite
from andes.utils.paths import get_config_path, andes_root
from andes.utils.paths import get_pycode_path, get_pkl_path
from andes.core import Config, Model, AntiWindup
from andes.io.streaming import Streaming

from andes.shared import np, jac_names, dilled_vars
from andes.shared import matrix, spmatrix, sparse, Pool, Process


logger = logging.getLogger(__name__)
dill.settings['recurse'] = True


class ExistingModels:
    """
    Storage class for existing models
    """

    def __init__(self):
        self.pflow = OrderedDict()
        self.tds = OrderedDict()   # if a model needs to be initialized before TDS, set `flags.tds = True`
        self.pflow_tds = OrderedDict()


class System:
    """
    System contains models and routines for modeling and simulation.

    System contains a several special `OrderedDict` member attributes for housekeeping.
    These attributes include `models`, `groups`, `routines` and `calls` for loaded models, groups,
    analysis routines, and generated numerical function calls, respectively.

    Parameters
    ----------
    no_undill : bool, optional
        True to disable the call to ``System.undill()`` at the end of object creation.
        False by default.

    Notes
    -----
    System stores model and routine instances as attributes.
    Model and routine attribute names are the same as their class names.
    For example, `Bus` is stored at ``system.Bus``, the power flow calculation routine is at
    ``system.PFlow``, and the numerical DAE instance is at ``system.dae``. See attributes for the list of
    attributes.

    Attributes
    ----------
    dae : andes.variables.dae.DAE
        Numerical DAE storage
    files : andes.variables.fileman.FileMan
        File path storage
    config : andes.core.Config
        System config storage
    models : OrderedDict
        model name and instance pairs
    groups : OrderedDict
        group name and instance pairs
    routines : OrderedDict
        routine name and instance pairs
    """

    def __init__(self,
                 case: Optional[str] = None,
                 name: Optional[str] = None,
                 config: Optional[Dict] = None,
                 config_path: Optional[str] = None,
                 default_config: Optional[bool] = False,
                 options: Optional[Dict] = None,
                 no_undill: Optional[bool] = False,
                 **kwargs
                 ):
        self.name = name
        self.options = {}
        if options is not None:
            self.options.update(options)
        if kwargs:
            self.options.update(kwargs)
        self.calls = OrderedDict()         # a dictionary with model names (keys) and their ``calls`` instance
        self.models = OrderedDict()        # model names and instances
        self.groups = OrderedDict()        # group names and instances
        self.routines = OrderedDict()      # routine names and instances
        self.switch_times = np.array([])   # an array of ordered event switching times
        self.switch_dict = OrderedDict()   # time: OrderedDict of associated models
        self.with_calls = False            # if generated function calls have been loaded
        self.n_switches = 0                # number of elements in `self.switch_times`
        self.exit_code = 0                 # command-line exit code, 0 - normal, others - error.

        # get and load default config file
        self._config_path = get_config_path()
        if config_path is not None:
            self._config_path = config_path
        if default_config is True:
            self._config_path = None

        self._config_object = self.load_config(self._config_path)
        self.config = Config(self.__class__.__name__, dct=config)
        self.config.load(self._config_object)

        # custom configuration for system goes after this line
        self.config.add(OrderedDict((('freq', 60),
                                     ('mva', 100),
                                     ('ipadd', 1),
                                     ('seed', 'None'),
                                     ('diag_eps', 1e-8),
                                     ('warn_limits', 1),
                                     ('warn_abnormal', 1),
                                     ('dime_enabled', 0),
                                     ('dime_name', 'andes'),
                                     ('dime_address', 'ipc:///tmp/dime2'),
                                     ('numba', 0),
                                     ('numba_parallel', 0),
                                     ('numba_nopython', 0),
                                     ('yapf_pycode', 0),
                                     ('np_divide', 'warn'),
                                     ('np_invalid', 'warn'),
                                     ('pickle_path', get_pkl_path())
                                     )))
        self.config.add_extra("_help",
                              freq='base frequency [Hz]',
                              mva='system base MVA',
                              ipadd='use spmatrix.ipadd if available',
                              seed='seed (or None) for random number generator',
                              diag_eps='small value for Jacobian diagonals',
                              warn_limits='warn variables initialized at limits',
                              warn_abnormal='warn initialization out of normal values',
                              numba='use numba for JIT compilation',
                              numba_parallel='enable parallel for numba.jit',
                              numba_nopython='nopython mode for numba',
                              yapf_pycode='format generated code with yapf',
                              np_divide='treatment for division by zero',
                              np_invalid='treatment for invalid floating-point ops.',
                              pickle_path='path models should be (un)dilled to/from',
                              )
        self.config.add_extra("_alt",
                              freq="float",
                              mva="float",
                              ipadd=(0, 1),
                              seed='int or None',
                              warn_limits=(0, 1),
                              warn_abnormal=(0, 1),
                              numba=(0, 1),
                              numba_parallel=(0, 1),
                              numba_nopython=(0, 1),
                              yapf_pycode=(0, 1),
                              np_divide={'ignore', 'warn', 'raise', 'call', 'print', 'log'},
                              np_invalid={'ignore', 'warn', 'raise', 'call', 'print', 'log'},
                              )
        self.config.check()
        self._set_numpy()

        self.exist = ExistingModels()

        self.files = FileMan(case=case, **self.options)    # file path manager
        self.dae = DAE(system=self)                        # numerical DAE storage
        self.streaming = Streaming(self)                   # Dime2 streaming

        # dynamic imports of groups, models and routines
        self.import_groups()
        self.import_models()
        self.import_routines()  # routine imports come after models

        self._getters = dict(f=list(), g=list(), x=list(), y=list())
        self._adders = dict(f=list(), g=list(), x=list(), y=list())
        self._setters = dict(f=list(), g=list(), x=list(), y=list())
        self.antiwindups = list()
        self.no_check_init = list()  # states for which initialization check is omitted

        # internal flags
        self.is_setup = False        # if system has been setup

        if not no_undill:
            self.undill()

    def _set_numpy(self):
        """
        Configure NumPy based on Config.
        """
        # set up numpy random seed
        if isinstance(self.config.seed, int):
            np.random.seed(self.config.seed)
            logger.debug("Random seed set to <%d>.", self.config.seed)

        np.seterr(divide=self.config.np_divide,
                  invalid=self.config.np_invalid,
                  )

    def reload(self, case, **kwargs):
        """
        Reload a new case in the same System object.
        """

        self.options.update(kwargs)
        self.files.set(case=case, **kwargs)
        # TODO: clear all flags and empty data
        andes.io.parse(self)
        self.setup()

    def _clear_adder_setter(self):
        """
        Clear adders and setters storage
        """
        self._getters = dict(f=list(), g=list(), x=list(), y=list())
        self._adders = dict(f=list(), g=list(), x=list(), y=list())
        self._setters = dict(f=list(), g=list(), x=list(), y=list())

    def prepare(self, quick=False, incremental=False, models=None, nomp=False, ncpu=os.cpu_count()):
        """
        Generate numerical functions from symbolically defined models.

        All procedures in this function must be independent of test case.

        Parameters
        ----------
        quick : bool, optional
            True to skip pretty-print generation to reduce code generation time.
        incremental : bool, optional
            True to generate only for modified models, incrementally.
        models : list, OrderedDict, None
            List or OrderedList of models to prepare
        nomp : bool
            True to disable multiprocessing

        Notes
        -----
        Option ``incremental`` compares the md5 checksum of all var and
        service strings, and only regenerate for updated models.

        Examples
        --------
        If one needs to print out LaTeX-formatted equations in a Jupyter Notebook, one need to generate such
        equations with ::

            import andes
            sys = andes.prepare()

        Alternatively, one can explicitly create a System and generate the code ::

            import andes
            sys = andes.System()
            sys.prepare()

        Warnings
        --------
        Generated lambda functions will be serialized to file, but pretty prints (SymPy objects) can only exist in
        the System instance on which prepare is called.
        """
        # info
        if incremental is True:
            mode_text = 'rapid incremental mode'
        elif quick is True:
            mode_text = 'quick mode'
        else:
            mode_text = 'full mode'

        logger.info('Numerical code generation (%s) started...', mode_text)

        t0, _ = elapsed()

        # consistency check for group parameters and variables
        self._check_group_common()

        # get `pycode` folder path without automatic creation
        pycode_path = get_pycode_path(self.options.get("pycode_path"), mkdir=False)

        # determine which models to prepare based on mode and `models` list.
        if incremental and models is None:
            if not self.with_calls:
                self._load_calls()
            models = self._find_stale_models()
        elif not incremental and models is None:
            models = self.models
        else:
            models = self._get_models(models)

        total = len(models)
        width = len(str(total))

        if nomp is False:
            print(f"Generating code for {total} models on {ncpu} processes.")
            self._mp_prepare(models, quick, pycode_path, ncpu=ncpu)

        else:
            for idx, (name, model) in enumerate(models.items()):
                print(f"\r\x1b[K Generating code for {name} ({idx+1:>{width}}/{total:>{width}}).",
                      end='\r', flush=True)
                model.prepare(quick=quick, pycode_path=pycode_path)

        if len(models) > 0:
            self._finalize_pycode(pycode_path)
            self._store_calls(models)
            self.dill()

        _, s = elapsed(t0)
        logger.info('Generated numerical code for %d models in %s.', len(models), s)

    def _mp_prepare(self, models, quick, pycode_path, ncpu):
        """
        Wrapper function for multiprocess prepare.
        """

        # create empty models without dependency
        if len(models) == 0:
            return

        model_names = list(models.keys())
        model_list = list()

        for file, cls_list in file_classes.items():
            for model_name in cls_list:
                if model_name not in model_names:
                    continue
                the_module = importlib.import_module('andes.models.' + file)
                the_class = getattr(the_module, model_name)
                model_list.append(the_class(system=None, config=self._config_object))

        yapf_pycode = self.config.yapf_pycode

        def _prep_model(model: Model, ):
            """
            Wrapper function to call prepare on a model.
            """

            model.prepare(quick=quick,
                          pycode_path=pycode_path,
                          yapf_pycode=yapf_pycode
                          )
        Pool(ncpu).map(_prep_model, model_list)

    def _finalize_pycode(self, pycode_path):
        """
        Helper function for finalizing pycode generation by
        writing ``__init__.py`` and reloading ``pycode`` package.
        """

        init_path = os.path.join(pycode_path, '__init__.py')
        with open(init_path, 'w') as f:
            f.write(f"__version__ = '{andes.__version__}'\n\n")

            for name in self.models.keys():
                f.write(f"from . import {name:20s}  # NOQA\n")
            f.write('\n')

        logger.info('Saved generated pycode to "%s"', pycode_path)

        # RELOAD REQUIRED as the generated Jacobian arguments may be in a different order
        self._call_from_pycode()

    def _find_stale_models(self):
        """
        Find models whose ModelCall are stale using md5 checksum.
        """
        out = OrderedDict()
        for model in self.models.values():
            calls_md5 = getattr(model.calls, 'md5', None)
            if calls_md5 != model.get_md5():
                out[model.class_name] = model

        return out

    def _to_orddct(self, model_list):
        """
        Helper function to convert a list of model names to OrderedDict
        with name as keys and model instances as values.
        """
        if isinstance(model_list, OrderedDict):
            return model_list
        if isinstance(model_list, list):
            out = OrderedDict()
            for name in model_list:
                if name not in self.models:
                    logger.error("Model <%s> does not exist. Check your inputs.", name)
                    continue
                out[name] = self.models[name]
            return out
        else:
            raise TypeError("Type %s not recognized" % type(model_list))

    def setup(self):
        """
        Set up system for studies.

        This function is to be called after adding all device data.
        """
        ret = True
        t0, _ = elapsed()

        if self.is_setup:
            logger.warning('System has been setup. Calling setup twice is not allowed.')
            ret = False
            return ret

        self.collect_ref()
        self._list2array()     # `list2array` must come before `link_ext_param`
        if not self.link_ext_param():
            ret = False

        self.find_devices()    # find or add required devices

        # === no device addition or removal after this point ===
        self.calc_pu_coeff()   # calculate parameters in system per units
        self.store_existing()  # store models with routine flags

        # assign address at the end before adding devices and processing parameters
        self.set_address(self.exist.pflow)
        self.set_dae_names(self.exist.pflow)        # needs perf. optimization
        self.store_sparse_pattern(self.exist.pflow)
        self.store_adder_setter(self.exist.pflow)

        if ret is True:
            self.is_setup = True  # set `is_setup` if no error occurred
        else:
            logger.error("System setup failed. Please resolve the reported issue(s).")
            self.exit_code += 1

        _, s = elapsed(t0)
        logger.info('System internal structure set up in %s.', s)

        return ret

    def store_existing(self):
        """
        Store existing models in `System.existing`.

        TODO: Models with `TimerParam` will need to be stored anyway.
        This will allow adding switches on the fly.
        """
        self.exist.pflow = self.find_models('pflow')
        self.exist.tds = self.find_models('tds')
        self.exist.pflow_tds = self.find_models(('tds', 'pflow'))

    def reset(self, force=False):
        """
        Reset to the state after reading data and setup (before power flow).

        Warnings
        --------
        If TDS is initialized, reset will lead to unpredictable state.
        """
        if self.TDS.initialized is True and not force:
            logger.error('Reset failed because TDS is initialized. \nPlease reload the test case to start over.')
            return
        self.dae.reset()
        self.call_models('a_reset', models=self.models)
        self.e_clear(models=self.models)
        self._p_restore()
        self.is_setup = False
        self.setup()

    def add(self, model, param_dict=None, **kwargs):
        """
        Add a device instance for an existing model.

        This methods calls the ``add`` method of `model` and registers the device `idx` to group.
        """
        if model not in self.models:
            logger.warning("<%s> is not an existing model.", model)
            return

        if self.is_setup:
            raise NotImplementedError("Adding devices are not allowed after setup.")
        group_name = self.__dict__[model].group
        group = self.groups[group_name]

        if param_dict is None:
            param_dict = {}
        if kwargs is not None:
            param_dict.update(kwargs)

        idx = param_dict.pop('idx', None)
        if idx is np.nan:
            idx = None

        idx = group.get_next_idx(idx=idx, model_name=model)
        self.__dict__[model].add(idx=idx, **param_dict)
        group.add(idx=idx, model=self.__dict__[model])

        return idx

    def find_devices(self):
        """
        Add dependent devices for all model based on `DeviceFinder`.
        """
        for mdl in self.models.values():
            if len(mdl.services_fnd) == 0:
                continue

            for fnd in mdl.services_fnd.values():
                fnd.find_or_add(self)

    def set_address(self, models):
        """
        Set addresses for differential and algebraic variables.
        """
        for mdl in models.values():
            if mdl.flags.address is True:
                logger.debug('%s internal address exists', mdl.class_name)
                continue
            if mdl.n == 0:
                continue

            # set internal variable addresses
            logger.debug('Setting internal address for %s', mdl.class_name)
            n = mdl.n
            m0 = self.dae.m
            n0 = self.dae.n
            m_end = m0 + len(mdl.algebs) * n
            n_end = n0 + len(mdl.states) * n
            collate = mdl.flags.collate

            if not collate:
                for idx, item in enumerate(mdl.algebs.values()):
                    item.set_address(np.arange(m0 + idx * n, m0 + (idx + 1) * n), contiguous=True)
                for idx, item in enumerate(mdl.states.values()):
                    item.set_address(np.arange(n0 + idx * n, n0 + (idx + 1) * n), contiguous=True)
            else:
                for idx, item in enumerate(mdl.algebs.values()):
                    item.set_address(np.arange(m0 + idx, m_end, len(mdl.algebs)), contiguous=False)
                for idx, item in enumerate(mdl.states.values()):
                    item.set_address(np.arange(n0 + idx, n_end, len(mdl.states)), contiguous=False)

            self.dae.m = m_end
            self.dae.n = n_end

        # set external variable addresses
        for mdl in models.values():
            # handle external groups
            for name, instance in mdl.cache.vars_ext.items():
                ext_name = instance.model
                try:
                    ext_model = self.__dict__[ext_name]
                except KeyError:
                    raise KeyError('<%s> is not a model or group name.' % ext_name)

                try:
                    instance.link_external(ext_model)
                except (IndexError, KeyError) as e:
                    logger.error('Error: <%s> cannot retrieve <%s> from <%s> using <%s>:\n  %s',
                                 mdl.class_name, instance.name, instance.model,
                                 instance.indexer.name, repr(e))

        # set external variable RHS addresses
        for mdl in models.values():
            if mdl.flags.address is True:
                logger.debug('%s RHS address exists', mdl.class_name)
                continue
            if mdl.n == 0:
                continue

            for item in mdl.states_ext.values():
                # skip if no equation, i.e., no RHS value
                if item.e_str is None:
                    continue
                item.set_address(np.arange(self.dae.p, self.dae.p + item.n))
                self.dae.p += item.n
            for item in mdl.algebs_ext.values():
                if item.e_str is None:
                    continue
                item.set_address(np.arange(self.dae.q, self.dae.q + item.n))
                self.dae.q += item.n

            mdl.flags.address = True

        # allocate memory for DAE arrays
        self.dae.resize_arrays()

        # set `v` and `e` in variables
        self.set_var_arrays(models=models)

        self.dae.alloc_or_extend_names()

    def set_dae_names(self, models):
        """
        Set variable names for differential and algebraic variables,
        right-hand side of external equations, and discrete flags.
        """

        for mdl in models.values():
            _set_xy_name(mdl, mdl.states, (self.dae.x_name, self.dae.x_tex_name))
            _set_xy_name(mdl, mdl.algebs, (self.dae.y_name, self.dae.y_tex_name))

            _set_hi_name(mdl, mdl.states_ext, (self.dae.h_name, self.dae.h_tex_name))
            _set_hi_name(mdl, mdl.algebs_ext, (self.dae.i_name, self.dae.i_tex_name))

            # add discrete flag names
            if self.TDS.config.store_z == 1:
                _set_z_name(mdl, self.dae, (self.dae.z_name, self.dae.z_tex_name))

    def set_var_arrays(self, models, inplace=True, alloc=True):
        """
        Set arrays (`v` and `e`) for internal variables to access
        dae arrays in place.

        This function needs to be called after de-serializing a System object,
        where the internal variables are incorrectly assigned new memory.

        Parameters
        ----------
        models : OrderedDict, list, Model, optional
            Models to execute.
        inplace : bool
            True to retrieve arrays that share memory with dae
        alloc : bool
            True to allocate for arrays internally
        """

        for mdl in models.values():
            if mdl.n == 0:
                continue

            for var in mdl.cache.vars_int.values():
                var.set_arrays(self.dae, inplace=inplace, alloc=alloc)

            for var in mdl.cache.vars_ext.values():
                var.set_arrays(self.dae, inplace=inplace, alloc=alloc)

    def _init_numba(self, models: OrderedDict):
        """
        Helper function to compile all functions with Numba before init.
        """

        if not self.config.numba:
            return

        use_parallel = bool(self.config.numba_parallel)
        nopython = bool(self.config.numba_nopython)

        logger.info("Numba compilation initiated with caching. Parallel=%s.",
                    use_parallel)

        for mdl in models.values():
            mdl.numba_jitify(parallel=use_parallel,
                             nopython=nopython,
                             )

    def precompile(self,
                   models: Union[OrderedDict, None] = None,
                   nomp: bool = False,
                   ncpu: int = os.cpu_count()):
        """
        Trigger precompilation for the given models.

        Arguments are the same as ``prepare``.
        """

        t0, _ = elapsed()

        if models is None:
            models = self.models
        else:
            models = self._get_models(models)

        # turn on numba for precompilation
        self.config.numba = 1

        self.setup()
        self._init_numba(models)

        def _precompile_model(model: Model):
            model.precompile()

        logger.info("Compilation in progress. This might take a minute...")

        if nomp is True:
            for name, mdl in models.items():
                _precompile_model(mdl)
                logger.debug("Model <%s> compiled.", name)

        # multi-processed implementation. `Pool.map` runs very slow somehow.
        else:
            jobs = []
            for idx, (name, mdl) in enumerate(models.items()):
                job = Process(
                    name='Process {0:d}'.format(idx),
                    target=_precompile_model,
                    args=(mdl,),
                    )
                jobs.append(job)
                job.start()

                if (idx % ncpu == ncpu - 1) or (idx == len(models) - 1):
                    time.sleep(0.02)
                    for job in jobs:
                        job.join()
                    jobs = []

        _, s = elapsed(t0)
        logger.info('Numba compiled %d model%s in %s.',
                    len(models),
                    '' if len(models) == 1 else 's',
                    s)

    def init(self, models: OrderedDict, routine: str):
        """
        Initialize the variables for each of the specified models.

        For each model, the initialization procedure is:

        - Get values for all `ExtService`.
        - Call the model `init()` method, which initializes internal variables.
        - Copy variables to DAE and then back to the model.
        """

        self._init_numba(models)

        for mdl in models.values():
            # link externals services first
            for instance in mdl.services_ext.values():
                ext_name = instance.model
                try:
                    ext_model = self.__dict__[ext_name]
                except KeyError:
                    raise KeyError('<%s> is not a model or group name.' % ext_name)

                try:
                    instance.link_external(ext_model)
                except (IndexError, KeyError) as e:
                    logger.error('Error: <%s> cannot retrieve <%s> from <%s> using <%s>:\n  %s',
                                 mdl.class_name, instance.name, instance.model,
                                 instance.indexer.name, repr(e))

            # initialize variables second
            mdl.init(routine=routine)

            self.vars_to_dae(mdl)
            self.vars_to_models()

        self.s_update_post(models)

        # store the inverse of time constants
        self._store_tf(models)

    def store_adder_setter(self, models):
        """
        Store non-inplace adders and setters for variables and equations.
        """
        self._clear_adder_setter()

        for mdl in models.values():
            # Note:
            #   We assume that a Model with no device is not addressed and, therefore,
            #   contains no value in each variable.
            #   It is always true for the current architecture.
            if not mdl.n:
                continue

            # Fixes an issue if the cache was manually built but stale
            # after assigning addresses for simulation
            # Assigning memory will affect the cache of `v_adders` and `e_adders`.

            mdl.cache.refresh()

            # ``getters` that retrieve variable values from DAE
            for var in mdl.cache.v_getters.values():
                self._getters[var.v_code].append(var)

            # ``adders`` that add variable values to the DAE array
            for var in mdl.cache.v_adders.values():
                self._adders[var.v_code].append(var)
            for var in mdl.cache.e_adders.values():
                self._adders[var.e_code].append(var)

            # ``setters`` that force set variable values in the DAE array
            for var in mdl.cache.v_setters.values():
                self._setters[var.v_code].append(var)
            for var in mdl.cache.e_setters.values():
                self._setters[var.e_code].append(var)

            # ``antiwindups`` stores all AntiWindup instances
            for item in mdl.discrete.values():
                if isinstance(item, AntiWindup):
                    self.antiwindups.append(item)

        return

    def store_no_check_init(self, models):
        """
        Store differential variables with ``check_init == False``.
        """
        self.no_check_init = list()
        for mdl in models.values():
            if mdl.n == 0:
                continue

            for var in mdl.states.values():
                if var.check_init is False:
                    self.no_check_init.extend(var.a)

    def link_ext_param(self, model=None):
        """
        Retrieve values for ``ExtParam`` for the given models.
        """
        if model is None:
            models = self.models
        else:
            models = self._get_models(model)

        ret = True
        for model in models.values():
            # get external parameters with `link_external` and then calculate the pu coeff
            for instance in model.params_ext.values():
                ext_name = instance.model
                ext_model = self.__dict__[ext_name]

                try:
                    instance.link_external(ext_model)
                except (IndexError, KeyError) as e:
                    logger.error('Error: <%s> cannot retrieve <%s> from <%s> using <%s>:\n  %s',
                                 model.class_name, instance.name, instance.model,
                                 instance.indexer.name, repr(e))
                    ret = False
        return ret

    def calc_pu_coeff(self):
        """
        Perform per unit value conversion.

        This function calculates the per unit conversion factors, stores input parameters to `vin`, and perform
        the conversion.
        """
        Sb = self.config.mva

        for mdl in self.models.values():
            # default Sn to Sb if not provided. Some controllers might not have Sn or Vn.
            if 'Sn' in mdl.__dict__:
                Sn = mdl.Sn.v
            else:
                Sn = Sb

            # If both Vn and Vn1 are not provided, default to Vn = Vb = 1
            # test if is shunt-connected or series-connected to bus, or unconnected to bus
            Vb, Vn = 1, 1
            if 'bus' in mdl.__dict__:
                Vb = self.Bus.get(src='Vn', idx=mdl.bus.v, attr='v')
                Vn = mdl.Vn.v if 'Vn' in mdl.__dict__ else Vb
            elif 'bus1' in mdl.__dict__:
                Vb = self.Bus.get(src='Vn', idx=mdl.bus1.v, attr='v')
                Vn = mdl.Vn1.v if 'Vn1' in mdl.__dict__ else Vb

            Zn = Vn ** 2 / Sn
            Zb = Vb ** 2 / Sb

            # process dc parameter pu conversion
            Vdcb, Vdcn, Idcn = 1, 1, 1
            if 'node' in mdl.__dict__:
                Vdcb = self.Node.get(src='Vdcn', idx=mdl.node.v, attr='v')
                Vdcn = mdl.Vdcn.v if 'Vdcn' in mdl.__dict__ else Vdcb
                Idcn = mdl.Idcn.v if 'Idcn' in mdl.__dict__ else (Sb / Vdcb)
            elif 'node1' in mdl.__dict__:
                Vdcb = self.Node.get(src='Vdcn', idx=mdl.node1.v, attr='v')
                Vdcn = mdl.Vdcn1.v if 'Vdcn1' in mdl.__dict__ else Vdcb
                Idcn = mdl.Idcn.v if 'Idcn' in mdl.__dict__ else (Sb / Vdcb)
            Idcb = Sb / Vdcb
            Rb = Vdcb / Idcb
            Rn = Vdcn / Idcn

            coeffs = {'voltage': Vn / Vb,
                      'power': Sn / Sb,
                      'ipower': Sb / Sn,
                      'current': (Sn / Vn) / (Sb / Vb),
                      'z': Zn / Zb,
                      'y': Zb / Zn,
                      'dc_voltage': Vdcn / Vdcb,
                      'dc_current': Idcn / Idcb,
                      'r': Rn / Rb,
                      'g': Rb / Rn,
                      }

            for prop, coeff in coeffs.items():
                for p in mdl.find_param(prop).values():
                    p.set_pu_coeff(coeff)

    def l_update_var(self, models: OrderedDict, niter=None, err=None):
        """
        Update variable-based limiter discrete states by calling ``l_update_var`` of models.

        This function is must be called before any equation evaluation.
        """
        self.call_models('l_update_var', models,
                         dae_t=self.dae.t, niter=niter, err=err)

    def l_update_eq(self, models:  OrderedDict):
        """
        Update equation-dependent limiter discrete components by calling ``l_check_eq`` of models.
        Force set equations after evaluating equations.

        This function is must be called after differential equation updates.
        """
        self.call_models('l_check_eq', models)

    def s_update_var(self, models: OrderedDict):
        """
        Update variable services by calling ``s_update_var`` of models.

        This function is must be called before any equation evaluation after
        limiter update function `l_update_var`.
        """
        self.call_models('s_update_var', models)

    def s_update_post(self, models: OrderedDict):
        """
        Update variable services by calling ``s_update_post`` of models.

        This function is called at the end of `System.init()`.
        """
        self.call_models('s_update_post', models)

    def fg_to_dae(self):
        """
        Collect equation values into the DAE arrays.

        Additionally, the function resets the differential equations associated with variables pegged by
        anti-windup limiters.
        """
        self._e_to_dae(('f', 'g'))

        # reset mismatches for islanded buses
        self.g_islands()

        # update variable values set by anti-windup limiters
        for item in self.antiwindups:
            if len(item.x_set) > 0:
                for key, val, _ in item.x_set:
                    np.put(self.dae.x, key, val)

    def f_update(self, models: OrderedDict):
        """
        Call the differential equation update method for models in sequence.

        Notes
        -----
        Updated equation values remain in models and have not been collected into DAE at the end of this step.
        """
        try:
            self.call_models('f_update', models)
        except TypeError as e:
            logger.error("f_update failed. Have you run `andes prepare -i` after updating?")
            raise e

    def g_update(self, models: OrderedDict):
        """
        Call the algebraic equation update method for models in sequence.

        Notes
        -----
        Like `f_update`, updated values have not collected into DAE at the end of the step.
        """
        try:
            self.call_models('g_update', models)
        except TypeError as e:
            logger.error("g_update failed. Have you run `andes prepare -i` after updating?")
            raise e

    def g_islands(self):
        """
        Reset algebraic mismatches for islanded buses.
        """
        if self.Bus.n_islanded_buses == 0:
            return

        self.dae.g[self.Bus.islanded_a] = 0.0
        self.dae.g[self.Bus.islanded_v] = 0.0

    def j_update(self, models: OrderedDict, info=None):
        """
        Call the Jacobian update method for models in sequence.

        The procedure is
        - Restore the sparsity pattern with :py:func:`andes.variables.dae.DAE.restore_sparse`
        - For each sparse matrix in (fx, fy, gx, gy), evaluate the Jacobian function calls and add values.

        Notes
        -----
        Updated Jacobians are immediately reflected in the DAE sparse matrices (fx, fy, gx, gy).
        """
        self.call_models('j_update', models)

        self.dae.restore_sparse()
        # collect sparse values into sparse structures
        for j_name in jac_names:
            j_size = self.dae.get_size(j_name)

            for mdl in models.values():
                for rows, cols, vals in mdl.triplets.zip_ijv(j_name):
                    try:
                        if self.config.ipadd:
                            self.dae.__dict__[j_name].ipadd(vals, rows, cols)
                        else:
                            self.dae.__dict__[j_name] += spmatrix(vals, rows, cols, j_size, 'd')
                    except TypeError as e:
                        logger.error("Error adding Jacobian triplets to existing sparsity pattern.")
                        logger.error(f'{mdl.class_name}: j_name {j_name}, row={rows}, col={cols}, val={vals}, '
                                     f'j_size={j_size}')
                        raise e

        self.j_islands()

        if info:
            logger.debug("Jacobian updated at t=%.6f: %s.", self.dae.t, info)
        else:
            logger.debug("Jacobian updated at t=%.6f.", self.dae.t)

    def j_islands(self):
        """
        Set gy diagonals to eps for `a` and `v` variables of islanded buses.
        """
        if self.Bus.n_islanded_buses == 0:
            return

        aidx = self.Bus.islanded_a
        vidx = self.Bus.islanded_v

        if self.config.ipadd:
            self.dae.gy.ipset(self.config.diag_eps, aidx, aidx)
            self.dae.gy.ipset(self.config.diag_eps, vidx, vidx)
        else:
            avals = [-self.dae.gy[int(idx), int(idx)] + self.config.diag_eps for idx in aidx]
            vvals = [-self.dae.gy[int(idx), int(idx)] + self.config.diag_eps for idx in vidx]

            self.dae.gy += spmatrix(avals, aidx, aidx, self.dae.gy.size, 'd')
            self.dae.gy += spmatrix(vvals, vidx, vidx, self.dae.gy.size, 'd')

    def store_sparse_pattern(self, models: OrderedDict):
        """
        Collect and store the sparsity pattern of Jacobian matrices.

        This is a runtime function specific to cases.

        Notes
        -----
        For `gy` matrix, always make sure the diagonal is reserved.
        It is a safeguard if the modeling user omitted the diagonal
        term in the equations.
        """
        self.call_models('store_sparse_pattern', models)

        # add variable jacobian values
        for jname in jac_names:
            ii, jj, vv = list(), list(), list()

            # for `gy`, reserve memory for the main diagonal
            if jname == 'gy':
                ii.extend(np.arange(self.dae.m))
                jj.extend(np.arange(self.dae.m))
                vv.extend(np.zeros(self.dae.m))

            for mdl in models.values():
                for row, col, val in mdl.triplets.zip_ijv(jname):
                    ii.extend(row)
                    jj.extend(col)
                    vv.extend(np.zeros_like(row))
                for row, col, val in mdl.triplets.zip_ijv(jname + 'c'):
                    # process constant Jacobians separately
                    ii.extend(row)
                    jj.extend(col)
                    vv.extend(val * np.ones_like(row))

            if len(ii) > 0:
                ii = np.array(ii, dtype=int)
                jj = np.array(jj, dtype=int)
                vv = np.array(vv, dtype=float)

            self.dae.store_sparse_ijv(jname, ii, jj, vv)
            self.dae.build_pattern(jname)

    def vars_to_dae(self, model):
        """
        Copy variables values from models to `System.dae`.

        This function clears `DAE.x` and `DAE.y` and collects values from models.
        """
        self._v_to_dae('x', model)
        self._v_to_dae('y', model)

    def vars_to_models(self):
        """
        Copy variable values from `System.dae` to models.
        """

        for var in self._getters['y']:
            if var.n > 0:
                var.v[:] = self.dae.y[var.a]

        for var in self._getters['x']:
            if var.n > 0:
                var.v[:] = self.dae.x[var.a]

    def connectivity(self, info=True):
        """
        Perform connectivity check for system.

        Parameters
        ----------
        info : bool
            True to log connectivity summary.
        """
        logger.debug("Entering connectivity check.")

        self.Bus.n_islanded_buses = 0
        self.Bus.islanded_buses = list()
        self.Bus.island_sets = list()
        self.Bus.nosw_island = list()
        self.Bus.msw_island = list()
        self.Bus.islands = list()

        n = self.Bus.n

        # collect from-bus and to-bus indices
        fr, to, u = list(), list(), list()

        # collect from Line
        fr.extend(self.Line.a1.a.tolist())
        to.extend(self.Line.a2.a.tolist())
        u.extend(self.Line.u.v.tolist())
        os = [0] * len(u)

        # find islanded buses
        diag = list(matrix(spmatrix(u, to, os, (n, 1), 'd') +
                           spmatrix(u, fr, os, (n, 1), 'd')))

        nib = self.Bus.n_islanded_buses = diag.count(0)
        for idx in range(n):
            if diag[idx] == 0:
                self.Bus.islanded_buses.append(idx)

        # store `a` and `v` indices for zeroing out residuals
        self.Bus.islanded_a = np.array(self.Bus.islanded_buses)
        self.Bus.islanded_v = self.Bus.n + self.Bus.islanded_a

        # find islanded areas - Goderya's algorithm
        temp = spmatrix(list(u) * 4,
                        fr + to + fr + to,
                        to + fr + fr + to,
                        (n, n),
                        'd')

        cons = temp[0, :]
        nelm = len(cons.J)
        conn = spmatrix([], [], [], (1, n), 'd')
        enum = idx = islands = 0

        while True:
            while True:
                cons = cons * temp
                cons = sparse(cons)  # remove zero values
                new_nelm = len(cons.J)
                if new_nelm == nelm:
                    break
                nelm = new_nelm

            # started with an islanded bus
            if len(conn.J) == 0:
                enum += 1
            # all buses are interconnected
            elif len(cons.J) == n:
                break

            self.Bus.island_sets.append(list(cons.J))
            conn += cons
            islands += 1
            nconn = len(conn.J)
            if nconn >= (n - nib):
                self.Bus.island_sets = [i for i in self.Bus.island_sets if len(i) > 0]
                break

            for element in conn.J[idx:]:
                if not diag[idx]:
                    enum += 1  # skip islanded buses
                if element <= enum:
                    idx += 1
                    enum += 1
                else:
                    break

            cons = temp[enum, :]

        # --- check if all areas have a slack generator ---
        if len(self.Bus.island_sets) > 0:
            for idx, island in enumerate(self.Bus.island_sets):
                nosw = 1
                slack_bus_uid = self.Bus.idx2uid(self.Slack.bus.v)
                slack_u = self.Slack.u.v
                for u, item in zip(slack_u, slack_bus_uid):
                    if (u == 1) and (item in island):
                        nosw -= 1
                if nosw == 1:
                    self.Bus.nosw_island.append(idx)
                elif nosw < 0:
                    self.Bus.msw_island.append(idx)

        # --- Post processing ---
        # extend islanded buses, each in a list
        if len(self.Bus.islanded_buses) > 0:
            self.Bus.islands.extend([[item] for item in self.Bus.islanded_buses])

        if len(self.Bus.island_sets) == 0:
            self.Bus.islands.append(list(range(n)))
        else:
            self.Bus.islands.extend(self.Bus.island_sets)

        if info is True:
            self.summary()

    def to_ipysheet(self, model: str, vin: bool = False):
        """
        Return an ipysheet object for editing in Jupyter Notebook.
        """

        from ipysheet import from_dataframe

        return from_dataframe(self.models[model].as_df(vin=vin))

    def from_ipysheet(self, model: str, sheet, vin: bool = False):
        """
        Set an ipysheet object back to model.
        """

        from ipysheet import to_dataframe

        df = to_dataframe(sheet)
        self.models[model].update_from_df(df, vin=vin)

    def summary(self):
        """
        Print out system summary.
        """

        island_sets = self.Bus.island_sets
        nosw_island = self.Bus.nosw_island
        msw_island = self.Bus.msw_island
        n_islanded_buses = self.Bus.n_islanded_buses

        logger.info("-> System connectivity check results:")
        if n_islanded_buses == 0:
            logger.info("  No islanded bus detected.")
        else:
            logger.info("  %d islanded bus detected.", n_islanded_buses)
            logger.debug("  Islanded Bus indices (0-based): %s", self.Bus.islanded_buses)

        if len(island_sets) == 0:
            logger.info("  No island detected.")
        else:
            logger.info("  A total of %d island(s) detected.", len(island_sets))
            logger.debug("  Bus indices in islanded areas (0-based): %s", island_sets)

        if len(nosw_island) > 0:
            logger.warning('  Slack generator is not defined/enabled for %d island(s).',
                           len(nosw_island))
            logger.debug("  Bus indices in no-Slack areas (0-based): %s",
                         [island_sets[item] for item in nosw_island])

        if len(msw_island) > 0:
            logger.warning('  Multiple slack generators are defined/enabled for %d island(s).',
                           len(msw_island))
            logger.debug("  Bus indices in multiple-Slack areas (0-based): %s",
                         [island_sets[item] for item in msw_island])

        if len(self.Bus.nosw_island) == 0 and len(self.Bus.msw_island) == 0:
            logger.info('  Each island has a slack bus correctly defined and enabled.')

    def _v_to_dae(self, v_code, model):
        """
        Helper function for collecting variable values into ``dae``
        structures `x` and `y`.

        This function must be called with ``dae.x`` and ``dae.y``
        both being zeros.
        Otherwise, adders will be summed again, causing an error.

        Parameters
        ----------
        v_code : 'x' or 'y'
            Variable type name
        """
        if model.n == 0:
            return
        if model.flags.initialized is False:
            return

        for var in model.cache.v_adders.values():
            if var.v_code != v_code:
                continue
            np.add.at(self.dae.__dict__[v_code], var.a, var.v)

        for var in self._setters[v_code]:
            if var.owner.flags.initialized is False:
                continue
            if var.n > 0:
                np.put(self.dae.__dict__[v_code], var.a, var.v)

    def _e_to_dae(self, eq_name: Union[str, Tuple] = ('f', 'g')):
        """
        Helper function for collecting equation values into `System.dae.f` and `System.dae.g`.

        Parameters
        ----------
        eq_name : 'x' or 'y' or tuple
            Equation type name
        """
        if isinstance(eq_name, str):
            eq_name = [eq_name]

        for name in eq_name:
            for var in self._adders[name]:
                np.add.at(self.dae.__dict__[name], var.a, var.e)
            for var in self._setters[name]:
                np.put(self.dae.__dict__[name], var.a, var.e)

    def get_z(self, models: OrderedDict):
        """
        Get all discrete status flags in a numpy array.
        Values are written to ``dae.z`` in place.

        Returns
        -------
        numpy.array
        """
        if self.TDS.config.store_z != 1:
            return None

        if len(self.dae.z) != self.dae.o:
            self.dae.z = np.zeros(self.dae.o, dtype=float)

        ii = 0
        for mdl in models.values():
            if mdl.n == 0 or len(mdl._input_z) == 0:
                continue
            for zz in mdl._input_z.values():
                self.dae.z[ii:ii + mdl.n] = zz
                ii += mdl.n

        return self.dae.z

    def find_models(self, flag: Optional[Union[str, Tuple]], skip_zero: bool = True):
        """
        Find models with at least one of the flags as True.

        Warnings
        --------
        Checking the number of devices has been centralized into this function.
        ``models`` passed to most System calls must be retrieved from here.

        Parameters
        ----------
        flag : list, str
            Flags to find

        skip_zero : bool
            Skip models with zero devices

        Returns
        -------
        OrderedDict
            model name : model instance

        """
        if isinstance(flag, str):
            flag = [flag]

        out = OrderedDict()
        for name, mdl in self.models.items():

            if skip_zero is True:
                if (mdl.n == 0) or (mdl.in_use is False):
                    continue

            for f in flag:
                if mdl.flags.__dict__[f] is True:
                    out[name] = mdl
                    break
        return out

    def dill(self):
        """
        Serialize generated numerical functions in ``System.calls`` with package ``dill``.

        The serialized file will be stored to ``~/.andes/calls.pkl``, where `~` is the home directory path.

        Notes
        -----
        This function sets `dill.settings['recurse'] = True` to serialize the function calls recursively.

        """
        np_ver = np.__version__.split('.')
        # Read only first two elements. Last one may contain 'rcxx'
        np_ver = tuple([int(i) for i in np_ver[:2]])
        if np_ver < (1, 20):
            logger.debug("Dumping calls to calls.pkl with dill")
            dill.settings['recurse'] = True

            with open(self.config.pickle_path, 'wb') as f:
                dill.dump(self.calls, f)
        else:
            logger.debug("Dumping calls to calls.pkl is not supported with NumPy 1.2+")
            logger.debug("ANDES is fully functional with generated Python code.")

    def undill(self):
        """
        Deserialize the function calls from ``~/.andes/calls.pkl`` with ``dill``.

        If no change is made to models, future calls to ``prepare()`` can be replaced with ``undill()`` for
        acceleration.
        """

        # load equations and jacobian from saved code
        loaded = self._load_calls()
        stale_models = self._find_stale_models()

        if loaded is False:
            self.prepare(quick=True, incremental=False)
            loaded = True
        elif len(stale_models) > 0:
            logger.info("Generated code for <%s> is stale.", ', '.join(stale_models.keys()))
            self.prepare(quick=True, incremental=True, models=stale_models)
            loaded = True

        return loaded

    def _load_calls(self):
        """
        Helper function for loading calls from pkl or pycode module.
        """
        loaded = self._call_from_pycode()

        if loaded is False:
            logger.debug("Pycode not found. Trying to load from `calls.pkl`.")
            loaded = self._call_from_pkl()

        self.with_calls = loaded

        return loaded

    def _load_pkl(self):
        """
        Helper function to open and load dill-pickled functions.
        """

        loaded_calls = None

        if os.path.isfile(self.config.pickle_path):
            with open(self.config.pickle_path, 'rb') as f:

                try:
                    loaded_calls = dill.load(f)
                    logger.info('Loaded generated code from pkl file "%s"', self.config.pickle_path)
                except (IOError, EOFError, AttributeError):
                    logger.debug('Cannot open pkl file at "%s"', self.config.pickle_path)

        return loaded_calls

    def _call_from_pkl(self):
        """
        Helper function for loading ModelCall from pickle file.
        """

        loaded = False
        any_calls = self._load_pkl()

        if any_calls is not None:
            self.calls = any_calls

            for name, model_call in self.calls.items():
                if name in self.__dict__:
                    self.__dict__[name].calls = model_call

            loaded = True

        return loaded

    def _call_from_pycode(self):
        """
        Helper function to import generated pycode.

        ``pycode`` is imported in the following sequence:

        - a user-provided path from CLI
        - ``~/.andes/pycode``
        - ``<andes_root>/pycode``

        """

        loaded = False

        # below are executed serially because of priority
        pycode = reload_submodules('pycode')
        if not pycode:
            pycode_path = get_pycode_path(self.options.get("pycode_path"), mkdir=False)
            pycode = load_pycode_from_path(pycode_path)
        if not pycode:
            pycode = reload_submodules('andes.pycode')
        if not pycode:
            pycode = load_pycode_from_path(os.path.join(andes_root(), 'pycode'))

        # DO NOT USE `elif` here since below depends on the above.
        if pycode:
            try:
                self._expand_pycode(pycode)
                loaded = True
            except KeyError:
                logger.error("Your generated pycode is broken. Run `andes prep` to re-generate. ")

        return loaded

    def _expand_pycode(self, pycode_module):
        """
        Expand imported ``pycode`` module to model calls.

        Parameters
        ----------
        pycode : module
            The module for generated code for models.
        """
        for name, model in self.models.items():
            if name not in pycode_module.__dict__:
                logger.debug("Model %s does not exist in pycode", name)
                continue

            pycode_model = pycode_module.__dict__[model.class_name]

            # md5
            model.calls.md5 = getattr(pycode_model, 'md5', None)

            # reload stored variables
            for item in dilled_vars:
                model.calls.__dict__[item] = pycode_model.__dict__[item]

            # equations
            model.calls.f = pycode_model.__dict__.get("f_update")
            model.calls.g = pycode_model.__dict__.get("g_update")

            # services
            for instance in model.services.values():
                if instance.v_str is not None:
                    sv_name = f'{instance.name}_svc'
                    model.calls.s[instance.name] = pycode_model.__dict__[sv_name]

            # load initialization; assignment
            for instance in model.cache.all_vars.values():
                if instance.v_str is not None:
                    ia_name = f'{instance.name}_ia'
                    model.calls.ia[instance.name] = pycode_model.__dict__[ia_name]

            # load initialization: iterative
            for item in model.calls.init_seq:
                if isinstance(item, list):
                    name_concat = '_'.join(item)
                    model.calls.ii[name_concat] = pycode_model.__dict__[name_concat + '_ii']
                    model.calls.ij[name_concat] = pycode_model.__dict__[name_concat + '_ij']

            # load Jacobian functions
            for jname in model.calls.j_names:
                model.calls.j[jname] = pycode_model.__dict__.get(f'{jname}_update')

    def _get_models(self, models):
        """
        Helper function for sanitizing the ``models`` input.

        The output is an OrderedDict of model names and instances.
        """
        out = OrderedDict()

        if isinstance(models, OrderedDict):
            out.update(models)

        elif models is None:
            out.update(self.exist.pflow)

        elif isinstance(models, str):
            out[models] = self.__dict__[models]

        elif isinstance(models, Model):
            out[models.class_name] = models

        elif isinstance(models, list):
            for item in models:
                if isinstance(item, Model):
                    out[item.class_name] = item
                elif isinstance(item, str):
                    out[item] = self.__dict__[item]
                else:
                    raise TypeError(f'Unknown type {type(item)}')

        return out

    def _store_tf(self, models):
        """
        Store the inverse time constant associated with equations.
        """
        for mdl in models.values():
            for var in mdl.cache.states_and_ext.values():
                if var.t_const is not None:
                    np.put(self.dae.Tf, var.a, var.t_const.v)

    def call_models(self, method: str, models: OrderedDict, *args, **kwargs):
        """
        Call methods on the given models.

        Parameters
        ----------
        method : str
            Name of the model method to be called
        models : OrderedDict, list, str
            Models on which the method will be called
        args
            Positional arguments to be passed to the model method
        kwargs
            Keyword arguments to be passed to the model method

        Returns
        -------
        The return value of the models in an OrderedDict

        """
        ret = OrderedDict()
        for name, mdl in models.items():
            ret[name] = getattr(mdl, method)(*args, **kwargs)

        return ret

    def _check_group_common(self):
        """
        Check if all group common variables and parameters are met.

        This function is called at the end of code generation by `prepare`.

        Raises
        ------
        KeyError if any parameter or value is not provided
        """
        for group in self.groups.values():
            for item in group.common_params:
                for model in group.models.values():
                    # the below includes all of BaseParam (NumParam, DataParam and ExtParam)
                    if item not in model.__dict__:
                        raise KeyError(f'Group <{group.class_name}> common param <{item}> does not exist '
                                       f'in model <{model.class_name}>')
            for item in group.common_vars:
                for model in group.models.values():
                    if item not in model.cache.all_vars:
                        raise KeyError(f'Group <{group.class_name}> common param <{item}> does not exist '
                                       f'in model <{model.class_name}>')

    def collect_ref(self):
        """
        Collect indices into `BackRef` for all models.
        """
        models_and_groups = list(self.models.values()) + list(self.groups.values())

        # create an empty list of lists for all `BackRef` instances
        for model in models_and_groups:
            for ref in model.services_ref.values():
                ref.v = [list() for _ in range(model.n)]

        # `model` is the model who stores `IdxParam`s to other models
        # `BackRef` is declared at other models specified by the `model` parameter
        # of `IdxParam`s.

        for model in models_and_groups:
            if model.n == 0:
                continue

            # skip: a group is not allowed to link to other groups
            if not hasattr(model, "idx_params"):
                continue

            for idxp in model.idx_params.values():
                if (idxp.model not in self.models) and (idxp.model not in self.groups):
                    continue
                dest = self.__dict__[idxp.model]

                if dest.n == 0:
                    continue

                for name in (model.class_name, model.group):
                    # `BackRef` not requested by the linked models or groups
                    if name not in dest.services_ref:
                        continue

                    for model_idx, dest_idx in zip(model.idx.v, idxp.v):
                        if dest_idx not in dest.uid:
                            continue

                        dest.set_backref(name,
                                         from_idx=model_idx,
                                         to_idx=dest_idx)

            # set model ``in_use`` flag
            if isinstance(model, Model):
                model.set_in_use()

    def import_groups(self):
        """
        Import all groups classes defined in ``devices/group.py``.

        Groups will be stored as instances with the name as class names.
        All groups will be stored to dictionary ``System.groups``.
        """
        module = importlib.import_module('andes.models.group')

        for m in inspect.getmembers(module, inspect.isclass):

            name, cls = m
            if name == 'GroupBase':
                continue
            elif not issubclass(cls, GroupBase):
                # skip other imported classes such as `OrderedDict`
                continue

            self.__dict__[name] = cls()
            self.groups[name] = self.__dict__[name]

    def import_models(self):
        """
        Import and instantiate models as System member attributes.

        Models defined in ``models/__init__.py`` will be instantiated `sequentially` as attributes with the same
        name as the class name.
        In addition, all models will be stored in dictionary ``System.models`` with model names as
        keys and the corresponding instances as values.

        Examples
        --------
        ``system.Bus`` stores the `Bus` object, and ``system.GENCLS`` stores the classical
        generator object,

        ``system.models['Bus']`` points the same instance as ``system.Bus``.
        """
        for file, cls_list in file_classes.items():
            for model_name in cls_list:
                the_module = importlib.import_module('andes.models.' + file)
                the_class = getattr(the_module, model_name)
                self.__dict__[model_name] = the_class(system=self, config=self._config_object)
                self.models[model_name] = self.__dict__[model_name]
                self.models[model_name].config.check()

                # link to the group
                group_name = self.__dict__[model_name].group
                self.__dict__[group_name].add_model(model_name, self.__dict__[model_name])

    def import_routines(self):
        """
        Import routines as defined in ``routines/__init__.py``.

        Routines will be stored as instances with the name as class names.
        All groups will be stored to dictionary ``System.groups``.

        Examples
        --------
        ``System.PFlow`` is the power flow routine instance, and ``System.TDS`` and ``System.EIG`` are
        time-domain analysis and eigenvalue analysis routines, respectively.
        """
        for file, cls_list in all_routines.items():
            for cls_name in cls_list:
                file = importlib.import_module('andes.routines.' + file)
                the_class = getattr(file, cls_name)
                attr_name = cls_name
                self.__dict__[attr_name] = the_class(system=self, config=self._config_object)
                self.routines[attr_name] = self.__dict__[attr_name]
                self.routines[attr_name].config.check()

    def store_switch_times(self, models, eps=1e-4):
        """
        Store event switching time in a sorted Numpy array in ``System.switch_times``
        and an OrderedDict ``System.switch_dict``.

        ``System.switch_dict`` has keys as event times and values as the OrderedDict
        of model names and instances associated with the event.

        Parameters
        ----------
        models : OrderedDict
            model name : model instance
        eps : float
            The small time step size to use immediately before
            and after the event

        Returns
        -------
        array-like
            self.switch_times
        """
        out = np.array([], dtype=float)

        if self.options.get('flat') is True:
            return out

        names = []
        for instance in models.values():
            times = np.array(instance.get_times()).ravel()
            out = np.append(out, times)
            out = np.append(out, times - eps)
            out = np.append(out, times + eps)
            names.extend([instance.class_name] * (3 * len(times)))

        # sort
        sort_idx = np.argsort(out).astype(int)
        out = out[sort_idx]
        names = [names[i] for i in sort_idx]

        # select t > current time
        ltzero_idx = np.where(out >= self.dae.t)[0]
        out = out[ltzero_idx]
        names = [names[i] for i in ltzero_idx]

        # make into an OrderedDict with unique keys and model names combined
        for i, j in zip(out, names):
            if i not in self.switch_dict:
                self.switch_dict[i] = {j: self.models[j]}
            else:
                self.switch_dict[i].update({j: self.models[j]})

        self.switch_times = np.array(list(self.switch_dict.keys()))

        # self.switch_times = out
        self.n_switches = len(self.switch_times)
        return self.switch_times

    def switch_action(self, models: OrderedDict):
        """
        Invoke the actions associated with switch times.

        Switch actions will be disabled if `flat=True` is passed to system.
        """
        for instance in models.values():
            instance.switch_action(self.dae.t)

    def _p_restore(self):
        """
        Restore parameters stored in `pin`.
        """
        for model in self.models.values():
            for param in model.num_params.values():
                param.restore()

    def e_clear(self, models: OrderedDict):
        """
        Clear equation arrays in DAE and model variables.

        This step must be called before calling `f_update` or `g_update` to flush existing values.
        """
        self.dae.clear_fg()
        self.call_models('e_clear', models)

    def remove_pycapsule(self):
        """
        Remove PyCapsule objects in solvers.
        """
        for r in self.routines.values():
            r.solver.clear()

    def _store_calls(self, models: OrderedDict):
        """
        Collect and store model calls into system.
        """
        logger.debug("Collecting Model.calls into System.")

        self.calls['__version__'] = andes.__version__

        for name, mdl in models.items():
            self.calls[name] = mdl.calls

    def _list2array(self):
        """
        Helper function to call models' ``list2array`` method, which
        usually performs memory preallocation.
        """
        self.call_models('list2array', self.models)

    def set_config(self, config=None):
        """
        Set configuration for the System object.

        Config for models are routines are passed directly to their constructors.
        """
        if config is not None:
            # set config for system
            if self.__class__.__name__ in config:
                self.config.add(config[self.__class__.__name__])
                logger.debug("Config: set for System")

    def get_config(self):
        """
        Collect config data from models.

        Returns
        -------
        dict
            a dict containing the config from devices; class names are keys and configs in a dict are values.
        """
        config_dict = configparser.ConfigParser()
        config_dict[self.__class__.__name__] = self.config.as_dict()

        all_with_config = OrderedDict(list(self.routines.items()) +
                                      list(self.models.items()))

        for name, instance in all_with_config.items():
            cfg = instance.config.as_dict()
            if len(cfg) > 0:
                config_dict[name] = cfg
        return config_dict

    @staticmethod
    def load_config(conf_path=None):
        """
        Load config from an rc-formatted file.

        Parameters
        ----------
        conf_path : None or str
            Path to the config file. If is `None`, the function body
            will not run.

        Returns
        -------
        configparse.ConfigParser
        """
        if conf_path is None:
            return

        conf = configparser.ConfigParser()
        conf.read(conf_path)
        logger.info('Loaded config from file "%s"', conf_path)
        return conf

    def save_config(self, file_path=None, overwrite=False):
        """
        Save all system, model, and routine configurations to an rc-formatted file.

        Parameters
        ----------
        file_path : str, optional
            path to the configuration file default to `~/andes/andes.rc`.
        overwrite : bool, optional
            If file exists, True to overwrite without confirmation.
            Otherwise prompt for confirmation.

        Warnings
        --------
        Saved config is loaded back and populated *at system instance creation time*.
        Configs from the config file takes precedence over default config values.
        """
        if file_path is None:
            andes_path = os.path.join(os.path.expanduser('~'), '.andes')
            os.makedirs(andes_path, exist_ok=True)
            file_path = os.path.join(andes_path, 'andes.rc')

        elif os.path.isfile(file_path):
            if not confirm_overwrite(file_path, overwrite=overwrite):
                return

        conf = self.get_config()
        with open(file_path, 'w') as f:
            conf.write(f)

        logger.info('Config written to "%s"', file_path)
        return file_path

    def supported_models(self, export='plain'):
        """
        Return the support group names and model names in a table.

        Returns
        -------
        str
            A table-formatted string for the groups and models
        """

        pairs = list()
        for g in self.groups:
            models = list()
            for m in self.groups[g].models:
                models.append(m)
            if len(models) > 0:
                pairs.append((g, ', '.join(models)))

        tab = Tab(title='Supported Groups and Models',
                  header=['Group', 'Models'],
                  data=pairs,
                  export=export,
                  )

        return tab.draw()

    def as_dict(self, vin=False, skip_empty=True):
        """
        Return system data as a dict where the keys are model names
        and values are dicts. Each dict has parameter names as keys
        and corresponding data in an array as values.

        Returns
        -------
        OrderedDict

        """
        out = OrderedDict()

        for name, instance in self.models.items():
            if skip_empty and instance.n == 0:
                continue
            out[name] = instance.as_dict(vin=vin)

        return out

    def fix_address(self):
        """
        Fixes addressing issues after loading a snapshot.

        This function properly sets ``v`` and ``e`` of internal
        variables as views of the corresponding DAE arrays.

        Inputs will be refreshed for each model.
        """

        self.set_var_arrays(self.models)

        for model in self.models.values():
            model.get_inputs(refresh=True)

        return True


def load_pycode_from_path(pycode_path):
    """
    Helper function to load pycode from ``.andes``.
    """

    MODULE_PATH = os.path.join(pycode_path, '__init__.py')
    MODULE_NAME = 'pycode'

    pycode = None
    if os.path.isfile(MODULE_PATH):
        try:
            spec = importlib.util.spec_from_file_location(MODULE_NAME, MODULE_PATH)
            pycode = importlib.util.module_from_spec(spec)  # NOQA
            sys.modules[spec.name] = pycode
            spec.loader.exec_module(pycode)
            logger.info('Loaded generated Python code in "%s".', pycode_path)
        except ImportError:
            logger.debug('Failed loading generated Python code in "%s".', pycode_path)

    return pycode


def reload_submodules(module_name):
    """
    Helper function for reloading an existing module and its submodules.

    It is used to reload the ``pycode`` module after regenerating code.
    """

    if module_name in sys.modules:
        pycode = sys.modules[module_name]
        for _, m in inspect.getmembers(pycode, inspect.ismodule):
            importlib.reload(m)

        logger.info('Reloaded generated Python code of module "%s".', module_name)
        return pycode

    return None


def _append_model_name(model_name, idx):
    """
    Helper function for appending ``idx`` to model names.
    Removes duplicate model name strings.
    """

    out = ''
    if isinstance(idx, str) and (model_name in idx):
        out = idx
    else:
        out = f'{model_name} {idx}'

    # replaces `_` with space for LaTeX to continue
    out = out.replace('_', ' ')
    return out


def _set_xy_name(mdl, vars_dict, dests):
    """
    Helper function for setting algebraic and state variable names.
    """

    mdl_name = mdl.class_name
    idx = mdl.idx
    for name, item in vars_dict.items():
        for idx_item, addr in zip(idx.v, item.a):
            dests[0][addr] = f'{name} {_append_model_name(mdl_name, idx_item)}'
            dests[1][addr] = rf'${item.tex_name}$ {_append_model_name(mdl_name, idx_item)}'


def _set_hi_name(mdl, vars_dict, dests):
    """
    Helper function for setting names of external equations.
    """

    mdl_name = mdl.class_name
    idx = mdl.idx
    for item in vars_dict.values():
        if len(item.r) != len(idx.v):
            idxall = item.indexer.v
        else:
            idxall = idx.v

        for idx_item, addr in zip(idxall, item.r):
            dests[0][addr] = f'{item.ename} {_append_model_name(mdl_name, idx_item)}'
            dests[1][addr] = rf'${item.tex_ename}$ {_append_model_name(mdl_name, idx_item)}'


def _set_z_name(mdl, dae, dests):
    """
    Helper function for addng and setting discrete flag names.
    """

    for item in mdl.discrete.values():
        if mdl.flags.initialized:
            continue
        mdl_name = mdl.class_name

        for name, tex_name in zip(item.get_names(), item.get_tex_names()):
            for idx_item in mdl.idx.v:
                dests[0].append(f'{name} {_append_model_name(mdl_name, idx_item)}')
                dests[1].append(rf'${item.tex_name}$ {_append_model_name(mdl_name, idx_item)}')
                dae.o += 1