# Two-objective optimization

The multiple objectives prioritized cost (MOPC) of resource *R* in a two-objective optimization problem is:

Here, *a* is the weighting factor for the first objective function.

In order to prioritize power plants for any weighting factor, it is enough to identify the order of prioritized cost at that weighting factor. However. Equation (8) shows that the MOPC has a linear relationship with the weighting factor *a.* Therefore, a plot of MOPC vs, weighting factor would be a straight line for all available energy sources. This also means that the prioritising sequence would vary with the value of the weighting factor. By plotting the MOPC vs. weighting factor lines for all possible new power plants, the changes in prioritising sequence with weighting factor can directly be identified. The fact that the relationship between MOPC and weighting factor is linear pror ides additional advantages and an opportunity to develop an entirely graphical solution space.

Since two straight lines can intersect only once, a system of *N* new energy sources can generate a maximum of (*') intersection points. The prioritising sequence can only change at the weighting factors corresponding to these intersection points, given the linear nature of the graphs. In the interval between intersection points, the MOPC of various power plants follow a fixed order, ensuring that the prioritising sequence in each interval is a constant. This gives rise to a maximum of (,' ) + 1 possible prioritising sequences. Figure 2 illustrates how the prioritising sequence for three new energy sources can be identified using this method. The prioritising sequences are represented in square brackets.

In Figure 2, a system with three new power plants is considered. To identify the prioritizing sequence, consider that the energy sources are numbered in increasing order of emission factor (i.e., *q _{{} < q_{2} < q_{}}). *The maximum number of possible intersection points is (,), indicated by weighting factors a, b and c in Figure 2. Consider the region where the weighting factor is less than a. Here, R,, the energy source with the lowest emission factor, also has the lowest prioritised cost. This means that adding Rj to the energy mix is optimal both for reducing emission and minimising the cost function. Therefore, in this region, Rj alone can be part of the optimal energy mix. However, beyond the weighting factor ‘a’, the energy source R, has a lower MOPC than R

_{(}. In this region, R, also has a lower MOPC that R,. This means that its MOPC is now lower than that of all energy sources with a lower emission factor. Therefore, in this region, both Rj and R. are part of the prioritising sequence.

This trend continues till weighting factor ‘b where the MOPC of energy source R, becomes lower than that of R, and, therefore, much like before. R, becomes part of the prioritizing sequence. Beyond ‘c the MOPC of R, is higher than R, and, therefore, no longer part of the prioritizing sequence. This example illustrates a case in which three new energy sources generate four or *('j* + 1 possible prioritising sequences for the entire range of weighting factors: [RJ, [Rj-RJ, [Rj-R.-RJ, and [R -RJ. This may not always be the case. For example, if the MOPC of any given new energy source is consistently higher

Figure 2. MOPC vs. weighting factor for two-objective optimization (Krishna Priya, G.S. and Bandyopadhyay, S. 2017a. Multiple objectives pinch analysis. Resources, Conservation and Recycling 119: 128-141).

than the МОРС of R_{[} in this example, it would have never been part of the prioritising sequence. Such observations obtained from the MOPC vs. weighting factor plot can further reduce the number of possible solutions. These plots help make the two-objective optimization process efficient by eliminating unviable combinations and identifying the optimal prioritising sequences for various values of weighting factors

This method can effectively reduce the total number of combinations from 2'V^{1} to a maximum of (•'>■) + 1. This reduction in the number of possible solutions is significant. In a system of just 6 possible new energy sources, this method reduces the number of possibilities from 32 (2‘V^{1}) to 16 (('V) + 1). In reality, a national level power system planning problem is likely to have a significantly larger number of energy sources. Therefore, incorporating the MOPA solution method can significantly reduce the complexity of large-scale emission constrained power system planning problems. After identifying the pinch point, the MOPC plots can be obtained and a prioritising sequence can be identified for any given value of weighting factor. The new power plants are then added according to the prioritising sequence.

# Three-objective optimisation

In a three-objective optimization problem, let the three-objective functions associated with the ?^{th} power plant be *Ф **,* Ф „ and Ф,. It is assumed that the relation between energy flow and each of these objective functions is linear. Let C_{fl>} C,, and *C _{t}}* be the cost coefficients associated with each of these objective functions. As described earlier, weighting factors are introduced to combine the various cost functions. Two weighting factors

*a*and

*/3*are introduced corresponding to cost functions

*Ф*

*,*and

*Ф*

_{;У}**Given that the stmt of weighting factors is one, the overall objective function can be written as:**

As per Equation (6),

The equation of prioritized cost for a three-objective optimization problem is that of a plane. If and when the prioritizing sequence of any two new energy sources become equal, the two corresponding planes of MOPC will intersect to form a line, which will be termed as Prioritised Cost Intersection Lines (PCIL) in this study. Similar to the two-objective example, the prioritizing sequence can only change along these lines. These lines pror ide the prioritizing sequence for any weighting factor as the prioritizing sequence changes only along these lines.

In order to identify the prioritizing sequence, it is enough to understand the variations in PCILs for the entire solution space. A simple example consisting of just two energy sources, Rj and R, (such that q_{tl}< q_{r},), can be used to illustrate this point. The prioritizing cost of these new energy sources can be obtained using Equation (10). The prioritized costs of R, and R_{:} will be equal at the PCIL where the prioritized cost planes of R_{t} and R, intersect. Using this condition, it is possible to obtain the relationship between weighting factors *a* and */3.* This line of one weighting factor as a function of the other is the PCIL ofR, and R, and a sample of such a PCIL is shown in Figure 3.

As the sum of weighting factors cannot be higher than one, half the region in the plot is infeasible. Of the remaining feasible region, the PCIL divides the solution space into two distinct parts, each with a different prioritizing sequence: one where R, alone is part of the solutions space as R_{t} has a lower prioritized cost, and another were both R, and R, are part of the energy mix. In some cases, if the planes of Rj and R, do not intersect for any value of weighting factors, then R, alone would constitute the optimal energy mix in this example.

In a system with three energy sources, the three planes of prioritized cost can, at best, generate three PCILs which will then determine the solution space. Figure 4 show a generic example of such a system. The shaded region represents the infeasible solution space and the arrows indicate the direction in which a given energy source is viable. It is interesting to note that the three PCILs will intersect at a common point. Since along line R^R,. *p _{Rl} = p* and along line R,-R

_{3},

*P*=

_{Rl}*p*at then intersection point

_{Rr}**Figure 3. **Prioritising sequences for two energy sources.

Figure 4. Prioritizing sequences for three energy sources (Krishna Pnya, G.S. and Bandyopadhyay, S. 2017b. Multi- objective pinch analysis for power system planning. Applied Energy 202: 335-347).

*p _{R2} = p_{R}}.* This necessitates that the PCIL for R,-R, passes through the same point as well. Therefore, it is safe to conclude that any three energy sources will generate three PCILs and one unique node in the solution space. Since these are straight lines intersecting, the solution space can be divided into a maximum of six regions.

In Figure 4, R_{[} has the lowest MOPC, therefore, it alone will be part of the prioritised sequence. Hence, in this region, the prioritizing sequence is [RJ. In Region II, *p _{R}, > p_{R1} > p_{Ry}* R, now has a lower MOPC than all other energy sources with a lower emission factor. Therefore, the prioritising sequence in this region is modified to [Rj-R,]. Interestingly, in Region III,

*p*

_{Rl}>P_{R2}Rr

Here, it should be noted that since*p*R, cannot be part of the prioritising sequence though it has a lower MOPC than R,. However, Region IV,

_{n}< p_{Rr}*p*giving rise to the sequence [R.-R.-R.]. In Region V,

_{R1}> p_{R2}> p_{R}}*p*changing the prioritising sequence to [Rj-R.] and in Region VI,

_{Rl}< p_{R}, < p_{gy }*p*making R, alone part of the optimal prioritising sequence. Therefore, in this example, the entire feasible solution space is divided between three possible prioritising sequences. Once the prioritising sequences are identified, the optimal energy mix can be identified.

_{R1}< P_{R2}< p_{Rr}The interaction of each of these energy sources with the energy source with the lowest emission factor (Rj) is of significance. In the given example, in Region I, R_{[} alone needs to be considered, and all other interactions between energy sources can be ignored. Similarly, in Region II, R_{1} and R, are alone part of the solution space, making it unnecessary to study the relationship between R. and various energy sources. However, in Region IV, both R, and R, are cheaper than R_{r} making two prioritizing sequences likely. If the MOPC of R, is lower than that of R,. then the solution would be [Rj-R,]. and if not, it would be [Rj-R.-R.]. A case study involving the Indian power sector is considered in order to demonstrate the applicability of this method.