4.1. Defining the Cost Functions
Prior to establishing its operation strategy, an MGO has to define its operating cost functions. The MGO’s operating cost in grid-connected mode can be defined as shown below:
The reserve supply cost term
consists of the band purchase cost and the imbalance penalty cost imposed according to Rule II; i.e.,
The imbalance penalty cost
is determined according to a prearranged mutual agreement among the market participants. The penalty cost function was defined in this study as the product of the penalty price and the band violation capacity, as shown below [
21].
Similarly, an MGO’s operating cost in the islanded mode is expressed as follows:
The reconnection cost
represents the total cost that the MGO has to pay during the island operation, except for the energy supply costs, which consist of the control cost for stabilization and reconnection, the wear cost associated with switching, etc. The values of the energy supply cost terms
and
in Equations (4) and (7) are determined using a procedure similar to the economic dispatch. An MGO can also arrive at its energy bidding (
) and generation schedule (
and
) by solving the following sub-optimization problems:
where the sum of
and
, as well as the sum of
and
, should be equal to the forecasted demand for the microgrid,
. Assuming that load shedding does not occur in the grid-connected mode (i.e., assuming that the price of load shedding is much higher than the price of other generation resources), the load shedding cost term
is not considered in Equation (8).
4.2. Microgrid Islanding Model
Islanding Rules I and III in
Section 3 generalize the triggering conditions for microgrid islanding. Because this generalized island operation can be affected by both the reserve band contract and the system condition, the triggering condition expressed by Equation (1) is not sufficient in this case. To resolve this problem, a conditional probability function,
, which represents the islanding occurrence probability according to
, was newly defined in this study. An MGO can formulate the triggering condition using this conditional probability function, expressed as shown below:
where
A in Equation (11) is a set of islanding events;
is a function of
and
, which are the basic variables of reserve operation in the PXFC market environment; and
reflects the system conditions that can affect microgrid islanding, which may be various types of functions, depending on the grid condition. In general, the value of
is close to zero when
is smaller than or similar to
, and it increases as
increases. The following sigmoid-type function is an example of
:
where
,
, and
are constant coefficients and can represent the grid condition. For example, two coefficients
and
may indirectly reflect the remains of spinning reserve or ISO’s load shedding operation scheme, and
can represent the line fault probability. The dashed line in
Figure 1 is an example of Equation (12). This figure shows that an islanding event occurs stochastically even if an MGO violates its reserve band contract, and the occurrence probability increases as the imbalance generated by the MGO increases. A comparison of Equations (1) and (10) shows that the shape of the conditional probability function
in Equation (1) can be derived as the solid line in
Figure 1. This quantum-well-shaped probability function is consistent with the fact that, according to [
14], an islanding event occurs unconditionally when an MGO violates its reserve band contract.
The microgrid islanding model of Equation (10) with the conditional probability function clarifies the triggering condition for an islanding event and plays a key role in quantitative analysis of the calculated MIP. In other words, proper modeling of is essential to evaluate the risk of island operation. However, the main subject of this paper is to present an optimal operation strategy for a microgrid, taking into consideration the stochastic island operation. Therefore, it is assumed that the islanding model is given to the MGO as the demand uncertainty model is given.
4.3. Formulating the Objective Function and MIP
The MIP value of the ith stage, , is the ratio of the expected number of steps in which the islanded state is maintained in the ith stage to the total number of steps in that stage. The expression of can be derived by examining the form of the expected incurred cost during the ith stage. To obtain this form, two cases are considered at the beginning of the ith stage: (grid-connected) and (islanded).
For the grid-connected mode (
), the state may persist until the end of the
ith stage with no islanding event (Case G1), or the state may transfer during the stage with an islanding event (Case G2). The expected cost of Case G1 can be calculated as follows:
The pi-product of
in the integration indicates that no islanding event is triggered during the stage.
corresponds to the operating cost in that case. For Case G2, if an islanding event occurs during the
jth step of the
ith stage, the expected cost can be calculated as follows:
The sum
corresponds to the incurred cost during the stage, and its components can be calculated as follows.
where
is the operating cost before the islanding event occurs, and
is the operating cost after the islanding event occurs. The pi-product in Equation (14) represents the event-triggering probability of Case G2. Because an islanding event may occur during any step in the
ith stage, the total expected cost of Case G2 can be represented as the summation of Equation (14):
For the islanded mode (
), the calculation of the expected cost must reflect the situation in which the islanded microgrid attempts to synchronize and reconnect with the main grid during the
ith stage. Because this trial results in success or failure (Case I1/I2), the expected values of the operating costs for these two cases are represented as follows:
where
in Equations (18) and (19) indicates the success probability of reconnection in the
ith stage. As mentioned in
Section 3, this probability depends on the grid condition and the synchronization capability of the MGO. Assuming that the synchronization error is a function of the time that is used for synchronizing,
can be represented as shown below:
where index
i and
s represent the reconnection attempt stage and the islanding trigger stage, respectively. The value of
will be different, even if two identical microgrids tried to reconnect to the main grid at the same time, if the duration of island operation was different for the two microgrids. For example, if the success probability is inversely proportional to the synchronization time, or
, the following can serve as a possible reconnection attempt model.
The number in the probability step, , which has a value of 3 in Equation (21), can also be a part of this reconnection model. This means that this MGO is able to recover its grid connection in less than stages.
Consequently, the expected value of the overall operating cost for all stages is given by the following:
The MIP formulation can be derived by comparing Equation (22) to Equation (3). After substituting Equations (13) and (17)–(19) into each
in Equation (22), and after some arrangement, MIP
can be expressed as follows:
According to Rules III and IV,
and
in Equation (23) can be represented by the following recurrence equation:
The first term on the right-hand side of Equation (25) represents the situation in which there is no islanding event during the (i − 1)th stage, as in Equation (13), and is maintained until the beginning of the ith stage. The second term represents all of the situations in which the MGO recovers from the islanded state during the (i − 1)th stage and can start the ith stage in a grid-connected state. The summation in the second term indicates that a microgrid starts in the grid-connected state at the beginning of the (i – k + 1)th stage. This microgrid enters the islanded state during this stage and recovers its grid connection after k reconnection attempts.