In order to design and develop the strategic plans for reconfiguration and implement them effectively, the following model of constraints and equalities are presented:
Equalities/objective functions
Number of switching operations
If λ
_{1} (S) defines the operations of the number of switches,
$$\lambda_{1} (s) = \sum\limits_{i = 1}^{{N_{sw} }} {X_{i} }$$
(1)
then, here, X_{i} is switch state vector given by [S_{1}, S_{2}, S_{3} …, S_{NSW}], N_{sw} = The total switches that can be operated in the network under consideration, X_{i} = status of the switch. The conditions for the switch status are: X_{i} = 1, if switch is opened from closed position or vice versa, X_{i} = 0, if status of switch is not changed. Minimum number of switching operations indicates that the system will be more stability.
Maximum loading among backup feeders
The maximum loading, λ
_{2} (S) among supported feeder is given by Eq. (2):
$$\lambda_{2} (S) = Max(I_{{FD_{i} }} ),\quad i = 1,2, \ldots ,N_{FD}$$
(2)
\({\text{I}}_{{{\text{FD}}_{\text{i}} }}\) represents the current over the supported feeder FD_{i} after switching operations. N_{FD} defines the number of supported feeders. To meet the constraints criteria, λ_{2} (s) shall be minimised. This objective function will give the most loaded backup feeder and by this we can have the remaining marginal load.
Maximum loading among backup laterals
Like loading criteria for feeders the supported laterals shall also meet the load criteria. This objective function will give the most loaded backup laterals. A lesser value of λ_{3} (s) is preferred.
λ_{3} (s) is the capacity of supported laterals and LAT_{i} is the load current over the laterals after switching operation and N_{
LAT
} is the number of lateral branches. For technoeconomic operation the λ_{3} (S), Eq. (3) is desired to be minimized:
$${\lambda _3}(S) = Max({I_{LA{T_i}}}),\quad i = 1,2, \ldots ,{N_{LAT}}$$
(3)
where λ_{3} (S) defines the supported laterals for maximum loading and \(I_{{LAT_{i} }}\) defines current over of the supported lateral LAT_{i} after switching operation. N_{LAT} defines the number of laterals in the distribution network. The load on the laterals should be minimum for the best operating conditions during restoration.
Unbalanced loading of feeders
The feeders as well as laterals shall have the balanced loading of feeders and laterals. It is an important feature for line loss reduction and voltage stability criteria. Thus, the load unbalancing index of feeders and laterals can be computed using Eqs. (4) and (5) respectively.
$$\lambda_{4} (S) = \sqrt {\sum\limits_{i = 1}^{{N_{FD} }} {(LV_{{FD_{i} }}  LV_{FD} )^{2} } }$$
(4)
where, \({\text{LV}}_{{{\text{FD}}_{\text{i}} }}\) is percentage load level of feeder FD_{i} and LV_{FD} is percentage refrence load level which is given by Eq. (5)
$$LV_{FD} = \frac{{\sum\nolimits_{i = 1}^{{N_{FD} }} {I_{{FD_{i} }} } }}{{\sum\nolimits_{i = 1}^{{N_{FD} }} {IR_{{FD_{i} }} } }}*100$$
(5)
In the above equation \({\text{I}}_{{{\text{FD}}_{\text{i}} }}\) and \({\text{IR}}_{{{\text{FD}}_{\text{i}} }}\) represents the load current and rated load current of feeder. In order to improve the performance of the system the unbalancing loading index shall be as minimised.
Unbalanced loading of laterals
Similarly, the lateral branches unbalance load index λ_{5} (s) can be computed using equation:
$$\lambda_{5} (S) = \sqrt {\sum\limits_{i = 1}^{{N_{LAT} }} {(LV_{{LAT_{i} }}  LV_{LAT} )^{2} } }$$
(6)
where, \({\text{LV}}_{{{\text{LAT}}_{\text{i}} }}\) is percentage load level of lateral LAT_{i} and LV_{LAT} is percentage reference load level which is given by Eq. (7) as:
$$LV_{LAT} = \frac{{\sum\nolimits_{i = 1}^{{N_{LAT} }} {I_{{LAT_{i} }} } }}{{\sum\nolimits_{{}}^{{}} {IR_{{LAT_{i} }} } }}*100$$
(7)
In the above equation, \({\text{I}}_{{{\text{LAT}}_{\text{i}} }}\)
_{and}
\({\text{IR}}_{{{\text{LAT}}_{\text{i}} }}\) represents the load current and rated load current of lateral respectively. This objective function is used to determine the degree of unbalanced loading of laterals, therefore, less value of λ_{5} (s) is preferred.
Maximum loading among backup transformer
Transformer is the main source of power supply to feeders and laterals. Its maximum loading capacity and unbalanced loading index after the isolation of the fault needs to be computed and checked. These shall be as minimum as possible. The minimization of maximum loading of transformer due to supported feeders and laterals is desirable. Maximum loading of transformer, λ_{6} (s) is computed by Eq. (8) as:
$$\lambda_{6} (S) = Max(I_{{TRS_{i} }} ),\quad i = 1,2, \ldots ,N_{TRS}$$
(8)
Unbalanced loading of transformer
The unbalanced loading index of transformer, λ_{7} (s) is given by Eq. (9), where,
$$\lambda_{7} (s) = \sqrt {\sum\limits_{i = 1}^{{N_{TRS} }} {(LV_{{TRS_{i} }}  LV_{TRS} )^{2} } }$$
(9)
where, LV_{TRSi} is percentage load level of transformer TRS_{i} and LV_{TRS} is percentage reference load level which is given by Eq. (10)
$$LV_{TRS} = \frac{{\sum\nolimits_{i = 1}^{{N_{TRS} }} {I_{{TRS_{i} }} } }}{{\sum\nolimits_{i = 1}^{{N_{TRS} }} {IR_{{TRS_{i} }} } }}*100$$
(10)
In the above equation \({\text{I}}_{{{\text{TRS}}_{\text{i}} }}\)
_{and}
\({\text{IR}}_{{{\text{TRS}}_{\text{i}} }}\) represents the load current and rated load current of transformer respectively. This gives the degree of unbalance loading of transformer for the backup and the value of this function should be minimum.
Constraints
To further optimize the switching operation for the reconfiguration of distribution system, the following constraints shall have to be met:

1.
Open switch operation have been complemented by closed switch operation.

2.
I_{LATmin} < I_{j} < I_{LATmax}

3.
I_{FEEDERmin} < I_{j} < I_{FEEDERmax}

4.
I_{TRSmin} < I_{j} < I_{TRSmax}