Problem Formulation

The main objective of the Planning of Distribution Systems (PDS) is to minimize the cost of substations, transformers, MV feeders and LV conductors while the bus voltage

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and feeder current are maintained within acceptable ranges. To accommodate these, the objective function (OF) as the net present value of total cost is defined as:

DP ) r 1 ( C C C C OF Y 1 y y L I M & O CAP ₊ + + + + =

∑

where CCAP is the total capital cost, CO&M is the total operation and maintenance cost, CI is the interruption cost, CL is the loss cost, r is the discount rate, Y is the number of years in the study timeframe, and DP is the penalty factor.

The interruption cost is calculated using two components − the cost related to the duration of interruptions and that related to the number of interruptions. The summation of these two costs is taken as the interruption cost. The duration based interruption cost is the multiplication of the cost for the average interruption duration in a year (in terms of minutes) and the average interruption duration. The average interruption duration can be found using the multiplication of SAIDI, as a reliability index, and the number of customers. Similarly, the number based interruption cost is found by the multiplication of SAIFI, the cost of average interruption number per customer, and the number of customers. The cost of average interruption duration and number per customer is provided by the local electrical company. Based on the above description, the total cost of interruption is calculated using (3.2).

CI = WSAIDI × SAIDI + WSAIFI × SAIFI (3.2)

WSAIDI = NC × CID (3.3)

WSAIFI = NC × CIN (3.4)

where WSAIDI and WSAIFI are the reliability weight factors, CID and CIN are the cost of average interruption number per customer ($/interruption) and the cost of average

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interruption duration per customer ($/minute), respectively. NC is the number of customers served.

The loss cost is expressed in (3.5). In this, the loss cost has two parts − the energy loss cost which is proportional to the cost per MWh and the peak power cost which is proportional to the cost saving per MW reduction in the peak power.

CL = PLOSS×( kPL+ kL× 8760×lsf) (3.5)

where PLOSS is the loss power, kPL is the saving per MW reduction in the peak power, kL is the cost per MWh, and lsf is the loss load factor. The constraints include bus voltages and feeder currents. The bus voltage (Vbus) should be maintained within the standard level.

Vmin ≤ Vbus≤ Vmax (3.6)

The feeder current (If_i) should be less than the feeder rated current ( rated fi I ) in the ith feeder. rated f fi I i I ≤ (3.7)

The Death Penalty method is a simple and popular method to handle constrained optimization problems for including constraints. In this method, the constraints are incorporated in the objective function with a penalty factor, called DP. If all constraints are satisfied, DP will be zero. Otherwise, DP is set as a large number and is added to the objective function to exclude the relevant solution from the search space [121].

3.3. Methodology

The proposed methodology is to plan both LV and MV networks sequentially. For this purpose, the planning procedure starts by dividing the planning area into the regions

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where the loads density is relatively uniform. Each region is composed of several MV zones and LV zones. An LV zone contains an MV/LV transformer along with a number of LV loads supplied by this transformer. The MV zone includes an HV/MV transformer together with several LV zones supplied by this transformer. In this research it is assumed that both of zones, MV and LV, and regions are rectangular.

As variables, the dimensions of LV zones along with the placement and rating of MV/LV transformers and the route and type of LV feeders are optimized using the loads’ powers and configuration in the LV zone planning. The dimensions of MV zones along with the placement and rating of HV/MV transformers and the route and type of MV feeders are optimized in the MV zone planning. These zones are defined in the following sub-sections.

3.3.1. LV Network

In this methodology, each customer is assumed to occupy a rectangular block, called load block, with a specific power demand. The dimensions of these blocks and their power consumption are related to the average load density of the region. Subsequently, a rectangular service area, composed of these load blocks that are supplied by a distribution transformer, is formed. This service area, called the LV zone, is shaped in an arrangement as shown in Figure 3.1. In this figure, a distribution transformer “T” supplies several customers. The white blocks are the customers and the grey parts are the streets. The length and the width of each load block are denoted by LLB and LWB, respectively and WS indicates the width of streets. It should be noted that the three- phase distribution line is located in all streets.

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Figure 3.1. Typical distribution transformer service area (LV Zone)

The aim is to find the length and width of the LV zones along with the LV feeders’ types and routes and the transformer size and location as the variables to minimize the total cost per load block (or per unit area). The objective function for LV planning problem is the cumulative cost of the transformer, LV and MV feeders, and line loss. Note that the length and thus the cost of the MV feeders are partly determined by the dimensions of the LV zone. The reliability cost is not incorporated into the LV zone planning since the cost benefit obtained from reducing outages for some loads in an LV zone is usually much lower than the cost of required switches.

3.3.2. MV Network

After finding the dimensions of the LV zones and corresponding transformer size, a rectangular zone is allocated to a distribution substation for optimization of the MV system. This rectangular zone, called MV zone, is composed of LV zones. Figure 3.2

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shows an MV zone when all LV zones belong to a single load density region. In this figure, TLB and TWB are the length and the width of each LV zone. Transformers and substations are shown by “T” and “SS”, respectively.

Figure 3.2. Typical distribution substation service area (MV Zone)

The values of TLB and TWB are known since they are the output of the LV zone planning program.

TLB = LLB × HNLB (3.8)

TWB = (LWB + 0.5 × WS) × VNLB (3.9)

where HNLB and VNLB are the number of load blocks supplied by each distribution transformer in the horizontal and vertical axes which have been optimized in the LV zone planning section, respectively.

The location and size of substations and MV feeders’ types and routes in addition to the length and width of the MV zones are as the variables in the MV zone planning procedure. The objective function is composed of the loss cost, the reliability cost as

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well as the capital cost for HV/MV transformers and MV feeders per unit area. Here only the SAIDI and SAIFI contribution from the feeder faults is considered. The bus voltage level and the feeder current constraints should be satisfied in both LV and MV zones.