6.4 Cost-Reliability Approach
6.4.1 Problem Description
Investment Cost
The investment cost is the installation cost of new interconnections and the expansion cost of the existing ones. The cost of installation of new interconnections involves the cost of the transmission line, which depends on the capacity and the distance between the connected substations in both sides, and the cost of the interconnecting converters, which depends on the capacity of the interconnection. The cost of expansion of existing interconnections
Chapter 6. Capacity Expansion of Interconnecting Converters
depends on the expansion capacity. Therefore, the investment cost can be formulated as
cinv= X (i , j )∉E ¡cL i jui jDi j+ cICi jui j¢ + X (i , j )∈E ci jE (6.11)
where i and j are the index for substations in the ERPS and the public grid sides, respectively. The pair of substations (i , j ) is a candidate for locating a new interconnecting converter or expanding the capacity of the existing converter. Note thatE is the set of substation pairs where an interconnecting converter is already existing. In the first term of (6.11), the binary variable ui j represents the state of the interconnection between substations i and j . After
solving the optimization problem, if ui j = 1, there will be a new interconnection between
these substations. The parameter Di jis the distance between substations. The optimization
variables ci jL and ci jICrepresent the transmission line and the interconnecting converter costs, respectively. Here in (6.12) and (6.13) these cost variables are modeled as a linear function of the rated power (size) of the interconnection.
ci jL = AL+ BLpi j (6.12)
ci jIC= BICpi j (6.13)
where AL, BLand BICare the linear cost coefficients corresponding to transmission lines and interconnecting converters, respectively. The optimization variable pi jis the rated power of
the interconnection in the candidate substation pair (i , j ).
The cost of capacity expansion for the existing interconnections cEi , j: (i , j ) ∈ E can be formu- lated as a linear function of the size of the capacity expansion (pi , jE ) as follows:
ci jE = BEpEi j (6.14)
where the value of the expansion cost coefficient BE, is slightly lower than the value of BIC. In general, it is difficult to estimate the cost of the components. The estimations made in this work are based on the historical financial data or market data in Switzerland.
Finally, it should be noted that the presented investment cost formulation in (6.11) is nonlinear as it includes the production of binary and continuous variables (e.g., ui j and ci jL). In this
respect, a linearization technique as described in the Appendix B is applied to tackle this non-linearity.
6.4. Cost-Reliability Approach
Figure 6.6 – Power flows associated with the bus j of ERPS
Technical constraints
The main technical constraints of the problem are, the nodal power balance, voltage level constraint of the candidate substations and power flow limits of the transmission lines.
Power Balance The power balance at each substation of ERPS as presented in figure 6.6 is formulated in (6.15). X i fi j+ X j0 fj0j+ gj= Lj, ∀j (6.15)
where fi jand fj0jare the power flow from the public grid substation i and the ERPS substation j0to the substation j . The variable gjand the parameter Ljare the generated power and the
power demand at bus j . Here in this problem, the generated power is simply constrained by the rated power of the generator (Gj).
0 ≤ gj≤ Gj, ∀j (6.16)
Power Flow Limit The optimal location and capacity of the interconnections must ensure the transmission of energy to the load points, avoiding any line congestion during a normal op- eration state. This implies that the power flow through each line of ERPS must be constrained by the maximum transmission capacity of the line (6.17). Also, the power flow between the two grids at each interconnection is limited to the capacity of the interconnection (6.18)-(6.19).
Chapter 6. Capacity Expansion of Interconnecting Converters
where Fjmax0j is the capacity of the transmission line between buses j and j0.
−pi j≤ fi j≤ pi j, ∀(i , j ) ∉ E (6.18)
−(pi jE + P
E
i j) ≤ fj0j≤ pEi j+ Pi jE, ∀(i , j ) ∈ E (6.19)
where the parameter Pi jE represents the capacity of the existing converters.
Voltage Level Constraint The high voltage network of the public grids includes several volt- age levels (e.g., 380 KV and 220 KV). In fact, it is not possible to connect a low power converter to a very high voltage substation and vice versa. Hence, the capacity of the interconnection is also constrained by the voltage level of the substation in the public grid side. Here the public grid substations are partitioned into three groups depending on their appropriate interconnection capacity range. Since each group has its allowed capacity range, we have
PV1≤ pi j≤ PV1, i ∈ SV1 (6.20)
PV2≤ pi j≤ PV2, i ∈ SV2 (6.21)
PV3≤ pi j≤ PV3, i ∈ SV3 (6.22)
where PVkand PVkare the lower and the upper bound of the appropriate capacity range for the set of public grid substationSVk(k = 1,2,3).
Reliability Criteria
There are two approaches to include the reliability criteria in the optimization model. The first approach introduces a new constraint in the problem formulation that sets a minimum acceptable reliability level. This level can be defined using both of the reliability indexes. However, the ELNS provides a better idea of the system reliability since it considers the loss of load probability resulting from the different outages and the amount of load curtailed as a
6.4. Cost-Reliability Approach
consequence of those outages.
E LN S ≤ ELN Sdesired (6.23)
The second approach presents a way to find a balance between the investment cost and the desired reliability level. This balance can be achieved considering both of the issues in the objective function: a cost is associated to each expected megawatt not served and this cost is added to the investment cost in the updated objective function.
min cinv+ cELNS (6.24)
In order to be able to compare both of the costs, the cELNShas to be defined as an operating cost, i.e. it must be evaluated over the amortization time (T ) of the new interconnecting converters: cELNS= T X t =1 E LN S(t ) ·UC (6.25)
Where UC is the estimated unitary cost of load not served in [C H F /MW ]. It is important to remark that the UC must consider not only the energy price but also the consequences that a lack of supply has on the demand.
Like the previous approach, here it is possible to incorporate the demand features of the power system (i.e. the sensitivity of the demand against a loss of load) in the formulation, by modifying the value of the UC . The advantage of this approach is that it evaluates the two main issues of the problem and finds the best solution that balances both of the costs.