- Email: [email protected]
Diversification criteria for power systems
Energy Policy 90 (2016) 183–186
Contents lists available at ScienceDirect
Energy Policy journal homepage: www.elsevier.com/locate/enpol
Short Communication
Diversification criteria for power systems Mohammed Kharbach HEC Montreal, Canada and Emirates LNG, United Arab Emirates
H I G H L I G H T S
Ranking power systems, from a diversity perspective, based on one criteria has many shortcomings. Diversity based on energy or capacity could lead to different outcomes in terms of vulnerability to fuel exposure, among others. A Shannon Wiener Index ratio (SWIR) captures better the diversity of a power system compared to the standard SWI.
art ic l e i nf o
a b s t r a c t
Article history: Received 2 June 2015 Received in revised form 29 November 2015 Accepted 21 December 2015 Available online 4 January 2016
Growing power demand, fuel availability and prices, technology changes, the environmental impacts of energy consumption, the changing regulatory environments and the uncertainties around such elements make the planning for optimal power mix a challenging task. The diversity approach is advocated as a most appropriate planning methodology for the optimal energy mix (Hickey et al., 2010). Shannon Wiener Index (SWI), which is the most cited diversity metric has been used to assess power systems diversity mainly from an energy perspective. To our best knowledge, there is no rigorous justification why energy has been the main variable used in diversification exercises rather than other variables such as capacity. We use a stylized power generation framework to show that diversity based on energy or capacity could lead to different outcomes in terms of vulnerability to fuel exposure, among others. We also introduce a Shannon Wiener Index ratio (SWIR) that we believe captures better the diversity of a power system compared to the standard SWI. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Diversity Electricity SWI
1. Introduction Planning for the optimal portfolio of electricity supply has been done mainly through the least cost approach. There is a growing interest, however, in using other planning tools on the basis that they are better in capturing other strategic and risk features. “Most of the literature published on portfolios of electricity supply in recent years has focused on one or more of the following three approaches to capture reliability, security, and flexibility: portfolio theory, real options theory, and diversity (e.g. the Shannon–Wiener Index and Herfindahl–Hirschman Index (HHI))” (Hickey et al., 2010). “A critical comprehensive overview of the most relevant literature concerning the application of portfolio theory in the electricity sector” can be found in Delarue et al. (2011). Contrary to many articles, Delarue et al. (2011) uses a portfolio theory model that “explicitly distinguishes between installed capacity (power), electricity generation (energy) and actual instantaneous power delivery. This way, the variability of wind power and ramp limits of E-mail address: [email protected] http://dx.doi.org/10.1016/j.enpol.2015.12.023 0301-4215/& 2015 Elsevier Ltd. All rights reserved.
conventional power plants are correctly included in the investment optimization”. The application of real options theory in the context of electricity markets has been used to account for optionality and assets flexibility. In Frayer and Uludere (2001), it is argued that, for example, a gas-fired peak plant would lose more than half of its real value when optionality is not considered which is detrimental to the optimality of the supply mix. The adoption of other planning methodologies stem from the uncertainties that surround the long term planning in the power systems. Indeed, regulatory changes and environmental concerns could make certain current energy producing technologies impossible in the future. Technology changes could make certain production methods more economical in the future and the geography and the geopolitical nature of fuel supply could change dramatically both fuel availability and prices assumptions in the power planning exercise. As such, It is argued that diversity is a solution to the problem of ignorance about possible future uncertainties (Stirling, 1994). Diversity in the context of energy supply is described in Stirling (1994). The three main features of diversity as identified in Stirling (1994) are variety, balance and disparity. Variety is the number of
184
M. Kharbach / Energy Policy 90 (2016) 183–186
categories/options available and accessible. In the case of electricity generation, it could refer to the number of generation technologies. Balance refers to the percentage contribution of each option to the total (energy mix). Disparity refers to the extent of independence of the options from each other; for example solar, wind and biomass electricity generation could be considered different (independent). The diversity metric mostly used is the Shannon Wiener Index (SWI). If the energy generation is the variable to be assessed, the SWI is defined as
from all such generators is the same. The SWI based on capacity uses available capacity rather than installed (gross) capacity. The available capacity is a more reasonable variable since it avoids, at least partially, the effect of renewable generators, that have lower capacity factors, on the SWI value. For the fully diversified system defined above, we have n
SWIe =− ∑ 1
n
SWI = − ∑ pi ln (pi ) 1
n
SWIc =− ∑ 1
where pi is the percentage of contribution of the source i to the total energy generation. One of the limitations of using the SWI as a planning metric is that it does not consider costs explicitly. However, it is worth highlighting that using diversity for planning introduces a “diversity premium” in a deliberate way. Indeed, “in a deliberately diversified portfolio, less attractive options are turned to before the possible contributions of preferable options have been exhausted” (Stirling, 1994). The magnitude of this premium calls for a caution when using diversity as a planning criteria. Indeed additional costs due, for example, to the loss of economies of scale should be considered (Costello, 2007,, 2005; Hanser and Graves, 2007). On the other hand, costs of the options within the energy mix could be taken into account as in Stirling (1994); a formulation of a maximization problem whereby performance (the economics side) is added to the diversification component (the diversification index) to derive a “diversity optimal portfolio”. In this note, we use a stylized power generation framework to show that diversity based on energy or capacity could lead to different outcomes in terms of vulnerability to fuel exposure, among others. As such, the question of “what are we diversifying?” is critical for a correct understanding of the ranking of different power systems from a diversity perspective. Besides, diversity metrics that consider different facets of the power system are better in ranking such systems from a diversity perspective. In the following, Section 2 defines a fully diversified power system and shows that diversification based on energy or capacity gives the same SWI value. Section 3 compares a fully diversified system based on capacity to a fully diversified system based on energy. Section 4 uses some numerical examples to illustrate the results from Section 3 and Section 5 concludes.
2. A fully diversified power system A power system is designed so that it can deliver a certain energy and cover a peak demand with a certain reliability. Consider a power system composed of n nonrenewable generation technologies and one renewable one. Each nonrenewable technology is characterized by its capacity (Cinr), its heat rate (hinr) and its capacity factor (CFinr). We consider that the nonrenewable capacities have 100% availability. The renewable technology is characterized by its capacity ( C r ), its capacity factor ( CF r ) and its capacity credit (a* C r ). The parameter a is the availability of the renewable generation capacity during the peak of the system. We define a hypothetical fully diversified system as one whereby all the generators have the same contribution to the energy consumption and the peak demand. The nonrenewable generators in this fully diversified system have the same fuel consumption. The assumption of nonrenewable generators having the same fuel consumption is equivalent to such generators having the same heat rate since by definition, it is assumed that energy
n
SWIf =− ∑ 1
Einr ⎛ Einr ⎞ Eir ⎛ Eir ⎞ ln⎜ ln⎜ ⎟ = ln(n+1) ⎟− ET ⎝ ET ⎠ ET ⎝ ET ⎠
(1)
Cinr ⎛ Cinr ⎞ aCir ⎛ aCir ⎞ ln⎜ ln⎜ ⎟− ⎟ = ln(n+1) CT ⎝ CT ⎠ CT ⎝ CT ⎠
(2)
⎛ f nr ⎞ ln⎜⎜ i ⎟⎟ = ln(n) fT ⎝ fT ⎠
f inr
(3)
where SWIe is the Shannon Wiener Index based on the energy, SWIc is the Shannon Wiener Index based on capacity and, SWIf is the Shannon Wiener Index based on the fuel. Note that the hypothetical system defined above cannot exist in reality unless the conventional generators have the same heat rates and capacity factors. Indeed, if both the energy production and the fuel consumption are the same for all nonrenewable generators, then the heat rates should be the same. Similarly, capacity factors should be the same if capacities and energy generations from conventional generators are the same. The hypothetical system would, however, be used as a benchmark in terms of diversity (maximum diversity in terms of capacity, energy and fuel consumption). Other more realistic systems could be compared to the benchmark system in terms of diversity.
3. Diversity based on a single criteria In this section we also assume that the available technologies can be ordered in terms of their total average costs. Therefore, the first (n 1) technologies would be dispatched as per their optimum capacity factors (Cinr ) while the last one would have a residual capacity factor to balance the energy supply and demand taking into account that the renewable technology is a must run generator. We will compare two systems: the first is a fully diversified system from an (available) capacity perspective and the second is a fully diversified system from an energy perspective. The aim is to show that using a single criteria (capacity or energy) has a “diversity shortcoming” since a fully diversified system from a capacity perspective, for example, could exhibit a higher exposure to the fuel or technology mix vulnerability than a fully diversified system based on energy and vice versa. 3.1. Diversity in terms of generation capacity Maximum diversification occurs when all the (available) capacities are equal. Therefore
Cinr = aC r , i=1,n
(4)
The available capacity during the peak should meet the expected system peak (D) with certain reliability (β). So n
∑ Cinr + aC r =(1+β )D i=1
Rearranging, we get
M. Kharbach / Energy Policy 90 (2016) 183–186
185
Table 1 Diversification based on capacity. Numerical example (assumptions) Peak load – 15,000 MW Reliability factor – 15% n–2 Load factor (LF) – 70%
Technology 1 Technology 2 Renewable
Heat rate (btu/kwh)
Capacity factor (%)
Availalability at the peak (%)
Capacity (MW)
Daily energy (MWh)
7000 10,000 NA
80 43 30
100 100 50
5750 5750 11,500
110,400 58,800 82,800
23,000 1.10
252,000 1.07
Total SWI
Cinr = aC r = D.
1+β , i=1,n n +1
(5)
The installed capacities should provide the required energy. If LF is the “period” load factor, we have n
n
∑ CinrCFinr + C rCF r = D*LF ,
or ∑ CFinr +
i=1
i=1 n
or ∑
CFinr
−
i=1
n+1 CF r , = LF a 1+β
CF r n+1 + LF a 1+β
(6)
The residual capacity factor for the most expensive generator is derived from Eq. (6) when the other capacity factors are introduced exogenously in the numerical example in Section 4. n −1
CFnnr =− ∑ CFinr − i=1
CF r n +1 + LF a 1+β
(7)
3.2. Diversity in terms of energy Maximum diversification occurs when all the generations from the available technologies are equal. Therefore
Einr = E r = E, i=1,n
(8)
If LF is the “period” load factor, we have n
∑ Einr + E r = (n+1)E = D*LF i=1
We also have
Einr
nr
= Ci CFinr
and
E r = C rCF r , i=1,n
Therefore
Cinr =
D*LF (n+1)CFinr
and
Cr =
D*LF , i=1,n (n+1)CF r
(9)
The capacity factors of the dispatchable technologies are such as
Fuel (109 btu/day) 773 588
1361 0.6839
The residual capacity factor for the most expensive generator is derived from Eq. (10) when the other capacity factors are introduced exogenously in the numerical example in Section 4. n −1 ⎡ (1+β )(1+n) 1 1 a ⎤ − nr =− ∑ nr + ⎢ r⎥ ⎣ ⎦ CFn CF LF CF i i=1
(11)
4. Numerical example 4.1. Comparing the system diversified based on capacity to the one diversified based on energy In this section, we use an illustrative example to compare a fully diversified system based on capacity to a similar fully diversified system based on energy. We consider two conventional generators and a must run generator. First, notice that with this supply mix, the SWI (capacity or energy) for a fully diversified system as defined in Section 2 is ln(3) ¼ 1.1. The SWI based on fuel for this system is ln(2) ¼0.693. Technology 1 has a capacity factor of 90% (exogenous value). The must run generator has an availability at the peak of 50% and a capacity factor of 30%. The system in Table 1 is fully diversified based on capacity. The SWI based on capacity is equal to ln(3) ¼1.1 (similar to the fully diversified system as defined in Section 2). For this system the SWI based on energy is equal to 1.07 and the one based on fuel is 0.684. On the other hand, the system in Table 2 is fully diversified based on energy. The SWI based on energy is equal to ln(3) ¼1.1 (similar to the fully diversified system as defined in Section 2). For this system the SWI based on capacity is equal to 1.02 and the one based on fuel is 0.678. From Tables 1 and 2, one is likely to accept that the system diversified based on capacity is “better” than the one based on the energy from a diversity perspective. Indeed, the SWIs of the fuel requirements and the installed capacities are higher for the first system. The SWIs based on energy are “closer” to each other although it is higher for the second system. However, diversification based on energy would have chosen the second system.
n
∑ Cinr + aC r = (1+β )D,
or
4.2. A Shannon Wiener Index ratio
i=1 n
∑ i=1
One could use a simple SWI ratio to compare both systems from a “more global” diversity perspective. The ratio is defined as
D*LF D*LF +a =(1+β )D (n+1)CFinr (n+1)CF r
[SWIc + SWIe + SWIf ] for a diversified system
or n
∑ i=1
1 ⎡ (1+β )(1+n) a ⎤ =⎢ − ⎥ CFinr ⎣ LF CF r ⎦
SWIR= basedon a certain criteria [SWIc + SWIe + SWIf ] for the fully diversified (10)
system as defined in Section 2
(12)
186
M. Kharbach / Energy Policy 90 (2016) 183–186
Table 2 Diversification based on energy. Numerical example (assumptions) Peak – 15,000 MW Reliability factor – 15% n–2 LF – 70%
Technology 1 Technology 2 Renewable
Heat rate (btu/kwh)
Capacity factor (%)
Availalability at the peak (%)
Capacity (MW)
7000 10,000 NA
80 50 30
100 100 50
4375 7042 11,667
Total SWI
From a diversification perspective, the SWIR compares a power system, taking into account three main facets of diversity, i.e. fuel, energy and capacity, to a benchmark system fully diversified based on such three facets. As such, and contrary to the standard SWI, the SWIR shows how close a system to a benchmark system from a diversity perspective. Therefore, it is more useful, in comparing power system from a “more global” diversity perspective, than the standard SWI. The SWIR is always strictly less than unity except when the system is the same as the hypothetical/benchmark one. The values of the SWIRs for different systems could be used to compare them in terms of diversity; the higher the SWIR value the closer is the corresponding system to the benchmark case. This ranking system could be used for example to assess the effect, on diversity, of adding more renewables to the energy mix. In our example and for the system in Table 1 (diversified based on capacity) we have SWIR¼ (1.1 þ1.02 þ0.62)/(2*1.1 þ0.693) ¼0.974. For the system in Table 2 (diversified based on energy), we have SWIR¼ (1.01 þ1.1 þ0.69)/(2*1.1þ 0.693) ¼0.968. Using the SWIR, one would choose the first system, which is diversified based on capacity, while the ranking on the basis of an SWI calculated based on the energy, as is the case for most of the applications of the SWI in diversity studies, would have favored the second system. SWIR like any other diversity metrics has its shortcoming including some of those of the standard SWI (effect of disaggregation). The SWIR defined in Eq. (12) could be calculated as a weighted
23,083 1.02
Daily energy (MWh) 84,000 84,000 84,000
252,000 1.10
Fuel (109 btu/day) 588 840
1428 0.6775
average rather than a simple average ratio to put more weight on the element(s) of higher concern from a diversity perspective.
5. Conclusion and policy implications In this note, it is argued that ranking power systems based on one single criteria has many shortcomings. Indeed, a power system which is diversified based on energy could have less exposure to fuel supply than a system diversified based capacity and vice versa depending on different factors such the demand profile (Load profile as a proxy), the renewable technology effect and the nonrenewable technology effect. We believe that the SWIR defined in this note captures the concept of diversity of a power system in a better way than the standard SWI.
References Costello, K., 2007. Diversity of generation technologies: implications for decisionmaking and public policy. Electr. J. 20 (5), 10–21. Costello, K., 2005. A perspective on fuel diversity. Electr. J. 18 (4), 28–47. Delarue, E., De Jonghe, C., Belmans, R., D’haeseleer, W., 2011. Applying portfolio theory to the electricity sector: energy versus power. Energy Econ. 33 (1), 12–23. Frayer, J., Uludere, N., 2001. What is it worth? application of real options theory to the valuation of generation assets. Electr. J. 14 (10), 40–51. Hickey, A., Carlson, A.E., Lon, J., Loomis, D., 2010. Issues in the determination of the optimal portfolio of electricity supply options. Energy Policy 38 (5), 2198–2207. Hanser, P., Graves, F., 2007. Utility supply portfolio diversity requirements. Electr. J. 20 (5), 22–32. Stirling, Andrew, 1994. Diversity and ignorance in electricity supply investment. Energy Policy 22 (3), 195–216.
Contents lists available at ScienceDirect
Energy Policy journal homepage: www.elsevier.com/locate/enpol
Short Communication
Diversification criteria for power systems Mohammed Kharbach HEC Montreal, Canada and Emirates LNG, United Arab Emirates
H I G H L I G H T S
Ranking power systems, from a diversity perspective, based on one criteria has many shortcomings. Diversity based on energy or capacity could lead to different outcomes in terms of vulnerability to fuel exposure, among others. A Shannon Wiener Index ratio (SWIR) captures better the diversity of a power system compared to the standard SWI.
art ic l e i nf o
a b s t r a c t
Article history: Received 2 June 2015 Received in revised form 29 November 2015 Accepted 21 December 2015 Available online 4 January 2016
Growing power demand, fuel availability and prices, technology changes, the environmental impacts of energy consumption, the changing regulatory environments and the uncertainties around such elements make the planning for optimal power mix a challenging task. The diversity approach is advocated as a most appropriate planning methodology for the optimal energy mix (Hickey et al., 2010). Shannon Wiener Index (SWI), which is the most cited diversity metric has been used to assess power systems diversity mainly from an energy perspective. To our best knowledge, there is no rigorous justification why energy has been the main variable used in diversification exercises rather than other variables such as capacity. We use a stylized power generation framework to show that diversity based on energy or capacity could lead to different outcomes in terms of vulnerability to fuel exposure, among others. We also introduce a Shannon Wiener Index ratio (SWIR) that we believe captures better the diversity of a power system compared to the standard SWI. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Diversity Electricity SWI
1. Introduction Planning for the optimal portfolio of electricity supply has been done mainly through the least cost approach. There is a growing interest, however, in using other planning tools on the basis that they are better in capturing other strategic and risk features. “Most of the literature published on portfolios of electricity supply in recent years has focused on one or more of the following three approaches to capture reliability, security, and flexibility: portfolio theory, real options theory, and diversity (e.g. the Shannon–Wiener Index and Herfindahl–Hirschman Index (HHI))” (Hickey et al., 2010). “A critical comprehensive overview of the most relevant literature concerning the application of portfolio theory in the electricity sector” can be found in Delarue et al. (2011). Contrary to many articles, Delarue et al. (2011) uses a portfolio theory model that “explicitly distinguishes between installed capacity (power), electricity generation (energy) and actual instantaneous power delivery. This way, the variability of wind power and ramp limits of E-mail address: [email protected] http://dx.doi.org/10.1016/j.enpol.2015.12.023 0301-4215/& 2015 Elsevier Ltd. All rights reserved.
conventional power plants are correctly included in the investment optimization”. The application of real options theory in the context of electricity markets has been used to account for optionality and assets flexibility. In Frayer and Uludere (2001), it is argued that, for example, a gas-fired peak plant would lose more than half of its real value when optionality is not considered which is detrimental to the optimality of the supply mix. The adoption of other planning methodologies stem from the uncertainties that surround the long term planning in the power systems. Indeed, regulatory changes and environmental concerns could make certain current energy producing technologies impossible in the future. Technology changes could make certain production methods more economical in the future and the geography and the geopolitical nature of fuel supply could change dramatically both fuel availability and prices assumptions in the power planning exercise. As such, It is argued that diversity is a solution to the problem of ignorance about possible future uncertainties (Stirling, 1994). Diversity in the context of energy supply is described in Stirling (1994). The three main features of diversity as identified in Stirling (1994) are variety, balance and disparity. Variety is the number of
184
M. Kharbach / Energy Policy 90 (2016) 183–186
categories/options available and accessible. In the case of electricity generation, it could refer to the number of generation technologies. Balance refers to the percentage contribution of each option to the total (energy mix). Disparity refers to the extent of independence of the options from each other; for example solar, wind and biomass electricity generation could be considered different (independent). The diversity metric mostly used is the Shannon Wiener Index (SWI). If the energy generation is the variable to be assessed, the SWI is defined as
from all such generators is the same. The SWI based on capacity uses available capacity rather than installed (gross) capacity. The available capacity is a more reasonable variable since it avoids, at least partially, the effect of renewable generators, that have lower capacity factors, on the SWI value. For the fully diversified system defined above, we have n
SWIe =− ∑ 1
n
SWI = − ∑ pi ln (pi ) 1
n
SWIc =− ∑ 1
where pi is the percentage of contribution of the source i to the total energy generation. One of the limitations of using the SWI as a planning metric is that it does not consider costs explicitly. However, it is worth highlighting that using diversity for planning introduces a “diversity premium” in a deliberate way. Indeed, “in a deliberately diversified portfolio, less attractive options are turned to before the possible contributions of preferable options have been exhausted” (Stirling, 1994). The magnitude of this premium calls for a caution when using diversity as a planning criteria. Indeed additional costs due, for example, to the loss of economies of scale should be considered (Costello, 2007,, 2005; Hanser and Graves, 2007). On the other hand, costs of the options within the energy mix could be taken into account as in Stirling (1994); a formulation of a maximization problem whereby performance (the economics side) is added to the diversification component (the diversification index) to derive a “diversity optimal portfolio”. In this note, we use a stylized power generation framework to show that diversity based on energy or capacity could lead to different outcomes in terms of vulnerability to fuel exposure, among others. As such, the question of “what are we diversifying?” is critical for a correct understanding of the ranking of different power systems from a diversity perspective. Besides, diversity metrics that consider different facets of the power system are better in ranking such systems from a diversity perspective. In the following, Section 2 defines a fully diversified power system and shows that diversification based on energy or capacity gives the same SWI value. Section 3 compares a fully diversified system based on capacity to a fully diversified system based on energy. Section 4 uses some numerical examples to illustrate the results from Section 3 and Section 5 concludes.
2. A fully diversified power system A power system is designed so that it can deliver a certain energy and cover a peak demand with a certain reliability. Consider a power system composed of n nonrenewable generation technologies and one renewable one. Each nonrenewable technology is characterized by its capacity (Cinr), its heat rate (hinr) and its capacity factor (CFinr). We consider that the nonrenewable capacities have 100% availability. The renewable technology is characterized by its capacity ( C r ), its capacity factor ( CF r ) and its capacity credit (a* C r ). The parameter a is the availability of the renewable generation capacity during the peak of the system. We define a hypothetical fully diversified system as one whereby all the generators have the same contribution to the energy consumption and the peak demand. The nonrenewable generators in this fully diversified system have the same fuel consumption. The assumption of nonrenewable generators having the same fuel consumption is equivalent to such generators having the same heat rate since by definition, it is assumed that energy
n
SWIf =− ∑ 1
Einr ⎛ Einr ⎞ Eir ⎛ Eir ⎞ ln⎜ ln⎜ ⎟ = ln(n+1) ⎟− ET ⎝ ET ⎠ ET ⎝ ET ⎠
(1)
Cinr ⎛ Cinr ⎞ aCir ⎛ aCir ⎞ ln⎜ ln⎜ ⎟− ⎟ = ln(n+1) CT ⎝ CT ⎠ CT ⎝ CT ⎠
(2)
⎛ f nr ⎞ ln⎜⎜ i ⎟⎟ = ln(n) fT ⎝ fT ⎠
f inr
(3)
where SWIe is the Shannon Wiener Index based on the energy, SWIc is the Shannon Wiener Index based on capacity and, SWIf is the Shannon Wiener Index based on the fuel. Note that the hypothetical system defined above cannot exist in reality unless the conventional generators have the same heat rates and capacity factors. Indeed, if both the energy production and the fuel consumption are the same for all nonrenewable generators, then the heat rates should be the same. Similarly, capacity factors should be the same if capacities and energy generations from conventional generators are the same. The hypothetical system would, however, be used as a benchmark in terms of diversity (maximum diversity in terms of capacity, energy and fuel consumption). Other more realistic systems could be compared to the benchmark system in terms of diversity.
3. Diversity based on a single criteria In this section we also assume that the available technologies can be ordered in terms of their total average costs. Therefore, the first (n 1) technologies would be dispatched as per their optimum capacity factors (Cinr ) while the last one would have a residual capacity factor to balance the energy supply and demand taking into account that the renewable technology is a must run generator. We will compare two systems: the first is a fully diversified system from an (available) capacity perspective and the second is a fully diversified system from an energy perspective. The aim is to show that using a single criteria (capacity or energy) has a “diversity shortcoming” since a fully diversified system from a capacity perspective, for example, could exhibit a higher exposure to the fuel or technology mix vulnerability than a fully diversified system based on energy and vice versa. 3.1. Diversity in terms of generation capacity Maximum diversification occurs when all the (available) capacities are equal. Therefore
Cinr = aC r , i=1,n
(4)
The available capacity during the peak should meet the expected system peak (D) with certain reliability (β). So n
∑ Cinr + aC r =(1+β )D i=1
Rearranging, we get
M. Kharbach / Energy Policy 90 (2016) 183–186
185
Table 1 Diversification based on capacity. Numerical example (assumptions) Peak load – 15,000 MW Reliability factor – 15% n–2 Load factor (LF) – 70%
Technology 1 Technology 2 Renewable
Heat rate (btu/kwh)
Capacity factor (%)
Availalability at the peak (%)
Capacity (MW)
Daily energy (MWh)
7000 10,000 NA
80 43 30
100 100 50
5750 5750 11,500
110,400 58,800 82,800
23,000 1.10
252,000 1.07
Total SWI
Cinr = aC r = D.
1+β , i=1,n n +1
(5)
The installed capacities should provide the required energy. If LF is the “period” load factor, we have n
n
∑ CinrCFinr + C rCF r = D*LF ,
or ∑ CFinr +
i=1
i=1 n
or ∑
CFinr
−
i=1
n+1 CF r , = LF a 1+β
CF r n+1 + LF a 1+β
(6)
The residual capacity factor for the most expensive generator is derived from Eq. (6) when the other capacity factors are introduced exogenously in the numerical example in Section 4. n −1
CFnnr =− ∑ CFinr − i=1
CF r n +1 + LF a 1+β
(7)
3.2. Diversity in terms of energy Maximum diversification occurs when all the generations from the available technologies are equal. Therefore
Einr = E r = E, i=1,n
(8)
If LF is the “period” load factor, we have n
∑ Einr + E r = (n+1)E = D*LF i=1
We also have
Einr
nr
= Ci CFinr
and
E r = C rCF r , i=1,n
Therefore
Cinr =
D*LF (n+1)CFinr
and
Cr =
D*LF , i=1,n (n+1)CF r
(9)
The capacity factors of the dispatchable technologies are such as
Fuel (109 btu/day) 773 588
1361 0.6839
The residual capacity factor for the most expensive generator is derived from Eq. (10) when the other capacity factors are introduced exogenously in the numerical example in Section 4. n −1 ⎡ (1+β )(1+n) 1 1 a ⎤ − nr =− ∑ nr + ⎢ r⎥ ⎣ ⎦ CFn CF LF CF i i=1
(11)
4. Numerical example 4.1. Comparing the system diversified based on capacity to the one diversified based on energy In this section, we use an illustrative example to compare a fully diversified system based on capacity to a similar fully diversified system based on energy. We consider two conventional generators and a must run generator. First, notice that with this supply mix, the SWI (capacity or energy) for a fully diversified system as defined in Section 2 is ln(3) ¼ 1.1. The SWI based on fuel for this system is ln(2) ¼0.693. Technology 1 has a capacity factor of 90% (exogenous value). The must run generator has an availability at the peak of 50% and a capacity factor of 30%. The system in Table 1 is fully diversified based on capacity. The SWI based on capacity is equal to ln(3) ¼1.1 (similar to the fully diversified system as defined in Section 2). For this system the SWI based on energy is equal to 1.07 and the one based on fuel is 0.684. On the other hand, the system in Table 2 is fully diversified based on energy. The SWI based on energy is equal to ln(3) ¼1.1 (similar to the fully diversified system as defined in Section 2). For this system the SWI based on capacity is equal to 1.02 and the one based on fuel is 0.678. From Tables 1 and 2, one is likely to accept that the system diversified based on capacity is “better” than the one based on the energy from a diversity perspective. Indeed, the SWIs of the fuel requirements and the installed capacities are higher for the first system. The SWIs based on energy are “closer” to each other although it is higher for the second system. However, diversification based on energy would have chosen the second system.
n
∑ Cinr + aC r = (1+β )D,
or
4.2. A Shannon Wiener Index ratio
i=1 n
∑ i=1
One could use a simple SWI ratio to compare both systems from a “more global” diversity perspective. The ratio is defined as
D*LF D*LF +a =(1+β )D (n+1)CFinr (n+1)CF r
[SWIc + SWIe + SWIf ] for a diversified system
or n
∑ i=1
1 ⎡ (1+β )(1+n) a ⎤ =⎢ − ⎥ CFinr ⎣ LF CF r ⎦
SWIR= basedon a certain criteria [SWIc + SWIe + SWIf ] for the fully diversified (10)
system as defined in Section 2
(12)
186
M. Kharbach / Energy Policy 90 (2016) 183–186
Table 2 Diversification based on energy. Numerical example (assumptions) Peak – 15,000 MW Reliability factor – 15% n–2 LF – 70%
Technology 1 Technology 2 Renewable
Heat rate (btu/kwh)
Capacity factor (%)
Availalability at the peak (%)
Capacity (MW)
7000 10,000 NA
80 50 30
100 100 50
4375 7042 11,667
Total SWI
From a diversification perspective, the SWIR compares a power system, taking into account three main facets of diversity, i.e. fuel, energy and capacity, to a benchmark system fully diversified based on such three facets. As such, and contrary to the standard SWI, the SWIR shows how close a system to a benchmark system from a diversity perspective. Therefore, it is more useful, in comparing power system from a “more global” diversity perspective, than the standard SWI. The SWIR is always strictly less than unity except when the system is the same as the hypothetical/benchmark one. The values of the SWIRs for different systems could be used to compare them in terms of diversity; the higher the SWIR value the closer is the corresponding system to the benchmark case. This ranking system could be used for example to assess the effect, on diversity, of adding more renewables to the energy mix. In our example and for the system in Table 1 (diversified based on capacity) we have SWIR¼ (1.1 þ1.02 þ0.62)/(2*1.1 þ0.693) ¼0.974. For the system in Table 2 (diversified based on energy), we have SWIR¼ (1.01 þ1.1 þ0.69)/(2*1.1þ 0.693) ¼0.968. Using the SWIR, one would choose the first system, which is diversified based on capacity, while the ranking on the basis of an SWI calculated based on the energy, as is the case for most of the applications of the SWI in diversity studies, would have favored the second system. SWIR like any other diversity metrics has its shortcoming including some of those of the standard SWI (effect of disaggregation). The SWIR defined in Eq. (12) could be calculated as a weighted
23,083 1.02
Daily energy (MWh) 84,000 84,000 84,000
252,000 1.10
Fuel (109 btu/day) 588 840
1428 0.6775
average rather than a simple average ratio to put more weight on the element(s) of higher concern from a diversity perspective.
5. Conclusion and policy implications In this note, it is argued that ranking power systems based on one single criteria has many shortcomings. Indeed, a power system which is diversified based on energy could have less exposure to fuel supply than a system diversified based capacity and vice versa depending on different factors such the demand profile (Load profile as a proxy), the renewable technology effect and the nonrenewable technology effect. We believe that the SWIR defined in this note captures the concept of diversity of a power system in a better way than the standard SWI.
References Costello, K., 2007. Diversity of generation technologies: implications for decisionmaking and public policy. Electr. J. 20 (5), 10–21. Costello, K., 2005. A perspective on fuel diversity. Electr. J. 18 (4), 28–47. Delarue, E., De Jonghe, C., Belmans, R., D’haeseleer, W., 2011. Applying portfolio theory to the electricity sector: energy versus power. Energy Econ. 33 (1), 12–23. Frayer, J., Uludere, N., 2001. What is it worth? application of real options theory to the valuation of generation assets. Electr. J. 14 (10), 40–51. Hickey, A., Carlson, A.E., Lon, J., Loomis, D., 2010. Issues in the determination of the optimal portfolio of electricity supply options. Energy Policy 38 (5), 2198–2207. Hanser, P., Graves, F., 2007. Utility supply portfolio diversity requirements. Electr. J. 20 (5), 22–32. Stirling, Andrew, 1994. Diversity and ignorance in electricity supply investment. Energy Policy 22 (3), 195–216.