1 Introduction

The use of renewable energy sources as a generation resource in electric power systems is growing rapidly due to its advantages that are minimal pollution, non-depletion and low operating cost.

The use of wind turbine generator (WTG) and solar cell generator (SCG) are the fastest growing energy sources on a percentage basis. In the past 20 years, for example, the global wind energy capacity has increased five hundredfold—from 3.5 GW in 1994 to almost 50 GW by the end of 2004 and 1,560 GW by 2013. A major reason for this development is the incentive policies promoted by several countries to reduce the emission of pollutant gases stated by the Kyoto Protocol signed in 1997 in Japan and the Paris Agreement (COP21 UN) in France at Dec. 12, 2015. As a result, it is becoming more important to analyze the best generation mix including WTG and SCG. This is coupled with growing environmental awareness and increasing prospects ratification of the Kyoto Protocol and Paris Agreement (COP21 UN).

This chapter proposes a new approach to solve the long-term generation mix considering renewable energy resources generators namely; WTGs and SCGs with multi- criteria considering CO2, SOx, and NOx emission constraints, under the uncertain circumstances is proposed using linear programming. This chapter focuses on the development tools to analyze the effect of WTGs and SCGs on the best generation mix (BGM) using fuzzy set theory. The method accommodates sensitivity analysis of capacity factors of WTG and SCG [27, 28]. The effectiveness of the proposed approach is demonstrated by applying to solve the flexible generation mix problem on the Korean power system for the year 2030. The test system contains nuclear, coal, LNG, oil, pumped-storage hydro plant, WTG and SCG in multi-years period.

2 The Concept of Flexible Planning

In recently, the uncertainties of external conditions for power system scheduling problem, demand growth, primary energy resource circumstances and reliability of energy supply system and energy security are becoming more and more unmanageable. The uncertainties require more flexible system for preventing power system from the unmanageable collapse. Figure 1 means that the flexible planning and operation for power system is one which, “Although not necessarily gives the optimum solution for the basic forecasted conditions, yet can keep the reasonable scheduling solution from being significantly worsened by any assumed changes in the surrounding situations” [19, 20].

Fig. 1
figure 1

Concept of flexibility

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The useful methods for flexible generators mix problem and generators maintenance flexible scheduling problem using fuzzy LP, DP, IP and search method have been developed by the authors. Some cases of application of fuzzy set theory for the flexible planning and operation of power systems are introduced in this seminar. It is expected that more flexible solution can be obtained with the proposed methods because the fuzzy set theory that can reflect the subjective decision of decision-maker is used in these studies. This chapter uses fuzzy set theory in order to get a flexible long-term generation mix solution in Korea power system considering CO2 emission constraint and renewable energy generators as like as WTG (Wind Turbine Generator) and SCG (Solar Cell Generator).

3 The LP Formulation of Best Generation Mix

3.1 Problem Statement

This BGM problem is to determine the generation mix under the following assumptions:

  1. 1.

    The annual loads are known.

  2. 2.

    The number of generator is not that of units but that of types.

  3. 3.

    Nuclear power plants are able to perform load following.

  4. 4.

    It is assumed that the yearly capacity factors (CFs) of WTG and SCG are given.

For the proposed method, the system can be modeled as shown in Fig. 2. It is considered to be composed of Nuclear = 1, Coal = 2, LNG = 3, Oil = 4, P-G (pumped generator hydro) = 5, WTG = 6, SCG = 7 in convenient in this chapter.

Fig. 2
figure 2

A system model for use by the proposed method

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Because wind speed and solar radiation do not maintain a specified stable level and the resource supply cannot be controlled by operator, the WTG and SCG have resource supply uncertainties rather than generator outage. Figure 3 shows the uncertainties of WTG and SCG plants are different from those of conventional plants. Therefore, the CFs of the WTG and SCG depend on the resource supply entirely with an associated ambiguity. In this chapter, it is assumed that the CFs are available. Therefore, the energy based on the specified CF should be consumed mandatory as is the case with energy limited generator and self-flow hydro generator.

Fig. 3
figure 3

Uncertainties of the renewable energy power and conventional power plants

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In this study, it is assumed that the hydro generator construction is separately planned from that of the other kinds of generating units. In actual systems, the basic resources, which are reserves, of the hydropower plants have limitation in the country. Therefore, the choice of hydro plant construction is not much and non-flexible. Under these assumptions, the best generation mix problem is formulated as follows.

3.2 Objective Functions

3.2.1 The Economic Criterion

The economic criterion in the BGM is to minimize the sum of the construction cost and the fuel cost as

$$ \begin{aligned} {\text{Minimize}}\,Z & = \sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{\text{NG}} {K_{\text{cin}} d_{\text{in}} \alpha_{i} \Delta x_{\text{in}} + } \sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{\text{NG}} {K_{\text{fin}} f_{\text{in}} y_{\text{in}} } } } \\ & = \, F(\Delta x_{\text{in}} ,y_{\text{in}} ) \\ \end{aligned} $$
(1)

where,

i :

Unit type number (1 for nuclear, 2 for coal, 3 for LNG, 4 for oil and 5 for pumped-storage generators are specified in this chapter)

N :

Number of total study stage year

NG:

Number of unit type

K cin = ((1 + e ci )/(1 + r))nΔTk,k−1

K fin = ((1 + e fi )/(1 + r))nΔTk,k−1

eci :

Apparent escalation rate of construction materials of unit

e fi :

Apparent escalation rate of fuel of unit

r :

Discount rate

ΔT k,k−1 :

Step size years of study years from k to k−1

d in :

Construction cost of unit in year n

f in :

Marginal fuel cost of unit in year n (Won/MWh)

α i :

Annual expenses rate of unit

Δ\( x_{\text{in}} \) :

Construction capacity of unit in year n (MW)

y in :

Generation capacity of unit in year n (MWh)

3.2.2 Constraints

  1. 1.

    Installed capacity constraint

where

R n :

Supply reserve rate in n year (p.u)

$$ \sum\limits_{i = 1}^{\text{NG}} {(x_{{{\text{in}} - 1}} + \Delta x_{\text{in}} ) + {\text{HYD}} \ge L_{\text{pn}} (1 + R_{n} )\quad n = 1 \sim N} $$
(2)
HYD n :

Capacity of hydro generator in year n. In this study, it is assumed that the HYD n is given

  1. 2.

    Energy constraint of demand

    $$ \sum\limits_{i = 1}^{\text{NG}} {y_{\text{in}} \ge (L_{\text{pn}} + L_{\text{Bn}} ) \times 8760/2} + V_{n} - {\text{HYD}}_{n} \times 8760 \times {\text{CF}}_{H} $$
    (3)

where

L Pn :

Peak load at n year

L Bn :

Base load at year n

V n :

The added demand energy is caused by pumped-storage generator

CFH :

Average capacity factor of hydro generator

  1. 3.

    Production energy constraint of generation system

    $$ \begin{aligned} y_{\text{in}} & \le (x_{{{\text{in}} - 1}} +\Delta x_{\text{in}} )\, \times \,8760 \times {\text{CF}}_{i} \quad i = 1\sim 5,\quad n\text{ = }1{ \sim }N \\ y_{\text{in}} & = (x_{{{\text{in}} - 1}} + \Delta x_{\text{in}} ) \times 8760 \times {\text{CF}}_{i} \quad i = 6{\text{ and }}7,\quad n = 1{ \sim }N \\ \end{aligned} $$
    (4)

where

CF i :

Average capacity factor of the i-unit

  1. 4.

    Capacity constraint in the initial year

    $$ x_{\text{io}} = {\text{EX}}_{i} \quad i = 1{ \sim }{\text{NG}} $$
    (5)

where

EX i :

Capacity of the existing unit

  1. 5.

    Constraint of mutual relationship between existing generator capacity and new generator capacity (state equation)

    $$ x_{\text{in}} = x_{{{\text{in}} - 1}} + \Delta x_{\text{in}} \quad i = 1{ \sim }{\text{NG}},\,n = 1{ \sim }N $$
    (6)
  2. 6.

    Energy constraint of LNG thermal plant

    $$ y_{3n} \ge {\text{LEP}}_{{\min_{n} }} /\rho_{3} \quad n = 1{ \sim }N $$
    (7)

where

\( {\text{LEP}}_{{\min_{n} }} \) :

LNG thermal generator production energy for LNG minimum due to consumption in n year

  1. 7.

    Constraints of reservoir capacity of pumped-storage generator

    $$ y_{5n} = \eta_{g} \times V_{n} $$
    (8)

where

η g :

Efficiency of pumped-storage generator

  1. 8.

    No load following power constraints of nuclear power plant

    $$ (x_{1n} - x_{5n} ) \le L_{\text{Bn}} \quad n = 1{ \sim }N $$
    (9)
  2. 9.

    Upper-Lower constraints of new unit capacity

    $$ \Delta X_{{\min_{\text{in}} }} \le \Delta x_{\text{in}} \le \Delta X_{{\max_{\text{in}} }} \quad i = 1{ \sim }{\text{NG}},\,n = 1{ \sim }N $$
    (10)

where, \( X_{{\max_{\text{in}} }} \) and \( X_{{\max_{\text{in}} }} \) are the minimum and the maximum capacity of new units at year n respectively.

  1. 10.

    CO2 constraint

    $$ \sum\limits_{i = 1}^{\text{NG}} {\xi_{i} y_{i} \le {\text{CO}}_{{2{\text{MAX}}_{n} }} } $$
    (11)

where

ξ i :

Fuel consumption rate of the ith unit (Ton/MWh)

\( {\text{CO}}_{{2{\text{MAX}}_{n} }} \) :

Maximum quantity of CO2 permitted in year n (Ton/year)

  1. 11.

    SOX constraint

    $$ \sum\limits_{i = 1}^{\text{NG}} {{\text{SO}}_{{X_{\text{in}} }} \xi_{i} y_{n} \le {\text{SO}}_{{{\text{XMAX}}_{n} }} } $$
    (12)

where

\( {\text{SO}}_{{X_{\text{in}} }} \) :

SOX density of the i-unit in year n (ppm/Ton)

\( {\text{SO}}_{{{\text{XMAX}}_{n} }} \) :

Maximum quantity of SOX permitted in year n (ppm/year)

  1. 12.

    NOX constraint

    $$ \sum\limits_{i = 1}^{\text{NG}} {{\text{NO}}_{{X_{\text{in}} }} \xi_{i} y_{n} \le {\text{NO}}_{{{\text{XMAX}}_{n} }} } $$
    (13)

where

\( {\text{NO}}_{{{\text{X}}_{\text{in}} }} \) :

NOX density of the i-unit in year n (ppm/Ton)

\( {\text{NO}}_{{{\text{XMAX}}_{n} }} \) :

Maximum quantity of NOX permitted in year n (ppm/year)

4 The Fuzzy LP Formulation of Flexible Generation Mix

4.1 The Optimal Decision Theory by Fuzzy Set Theory

The fuzzy decision D resulting from q fuzzy goals G 1, …, G q and p fuzzy constraints C 1, …, C p is the intersection of them;

$$ D = \left(\mathop \bigcap\limits_{i = 1}^{q} G_{i} \right) \bigcap \left(\mathop \bigcap \limits_{j = 1}^{p} C_{i} \right) $$
(14)

And also its membership function μ D resulting from fuzzy goals and constraints is defined by

$$ \mu_{D} (x) = \hbox{min} \left[ {\mathop {\hbox{min} }\limits_{{i = 1{ \sim }p}} \mu_{{G_{i} }} ,\mathop {\hbox{min} }\limits_{{j = 1{ \sim }q}} \mu_{{C_{i} }} } \right] $$
(15)

where, min is an abbreviation of minimum. The fuzzy mathematical programming problem consists of finding the maximum of the fuzzy decision D.

$$ \mu_{D} (x^{*} ) = \hbox{max} \mu_{D} (x) $$
(16)

Where x * is the optimal decision solution max is an abbreviation of maximum. The vector Eq. (16) can be rewritten as the Eq. (17).

$$ \mu_{D} (x_{1}^{*} ,x_{2}^{*} , \ldots ,x_{N}^{*} ) = \mathop {\hbox{max} }\limits_{{x_{1} \cdot \cdot \cdot x_{N} }} \mu_{D} (x_{1} ,x_{2} , \ldots ,x_{N} ) $$
(17)

Figure 4 shows the concept for optimal decision on fuzzy sets, which is proposed by Bellman Zadeh.

Fig. 4
figure 4

Concept for optimal decision on fuzzy sets

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4.2 The Function of Fuzzy Linear Programming

Fuzziness of Cost: Z ⪍ Z 0

The Satisfaction Level in the best generation mix is flexible because the total cost is unjustly.

Fuzziness of Reliability criterion: R n  ⪎ R 0n

The reliability criterion in the long term based best generation mix may be flexible criterion than entirely strict. Using parameter λ, therefore, which means satisfaction level physically, best generation mix problem can be formulated as following.

Objective functions: Maximize λ

Constraints Subject to F (Δx in , y in) + ΔZ 0λ = Z 0 + ΔZ 0

$$ \begin{aligned} & \left( {\left( {\sum\limits_{i = 1}^{\text{NG}} {(x_{{{\text{in}} - 1}} + \Delta x_{\text{in}} ) + {\text{HYD}}_{n} } } \right) - L_{\text{pn}} } \right)/L_{\text{pn}} - \Delta R_{o} \lambda = R_{o} - \Delta R_{o} \\ & \sum\limits_{i = 1}^{\text{NG}} {y_{\text{in}} \ge (L_{\text{pn}} + L_{\text{Bn}} ) \times 8760/2} + V_{n} - {\text{HYD}} \times 8760 \times {\text{CF}}_{H} \\ \end{aligned} $$

  • y in ≤ (x in-1 + Δx in) × 8760 × CF i

  • x i0 = EX i

  • x in = x in-1 + Δx in

  • y 3n  ≥ LEPminn /ρ3

  • y 5n  = η g xVn

  • (x 1n  − x 5n ) ≤ LBn

  • Δx in ≤ \( {\text{DX}}_{{{ \hbox{max} }_{\text{in}} }} \)

$$ \begin{aligned} & \sum\limits_{i = 1}^{\text{NG}} {\xi_{i} y_{i} \le {\text{CO}}_{{2{\text{MAX}}_{n} }} } \\ & \sum\limits_{i = 1}^{\text{NG}} {{\text{SO}}_{{X_{\text{in}} }} \xi_{i} y_{n} \le {\text{SO}}_{{{\text{XMAX}}_{n} }} } \\ & \sum\limits_{i = 1}^{\text{NG}} {{\text{NO}}_{{X_{\text{in}} }} \xi_{i} y_{n} \le {\text{NO}}_{{{\text{XMAX}}_{n} }} } \\ \end{aligned} $$

5 The Fuzzy LP Formulation of Flexible Generation Mix

Budget for long-term generation expansion planning is given unjustly and usually in exact. As it is, it is fuzzy. The optimal decision theory by fuzzy set theory is a reasonable methodology for solving the optimal problem including the multiple objectives and ambiguity. The theory is introduced in detail in Appendix. Aspiration level for decision makers may be expressed by (18). In addition, the reliability criterion in the long term based best generation mix may be a flexible criterion rather than entirely strict as in (19).

$$ \text{Fuzziness}\,\text{of}\,\text{Cost}:{Z}\,\underline{\lesssim}\,{Z}_{0} $$
(18)
$$ \text{Fuzziness}\,\text{of}\,\text{Reliability}\,\text{criterion}:{R}_{n}\, \underline{\gtrsim }\, {R}_{{0{n}}} $$
(19)

where

Z 0 :

Aspiration level of budget

R 0n :

Aspiration level of reliability criterion in n-year

Using parameter λ, which refers to the term satisfaction level, the best generation mix problem can be formulated as follows.

Objective functions: Maximize λ

Constraints Subject to F(Δx in , y in) + ΔZ 0λ = Z 0 + ΔZ 0

$$ \begin{aligned} & \left( {\left( {\sum\limits_{i = 1}^{\text{NG}} {(x_{{{\text{in}} - 1}} + \Delta x_{\text{in}} ) + {\text{HYD}}_{n} } } \right) - L_{\text{pn}} } \right)/L_{\text{pn}} - \Delta R_{o} \lambda = R_{o} - \Delta R_{o} \\ & \sum\limits_{i = 1}^{{{\text{NG}}}} {y_{\text{in}} \ge (L_{\text{pn}} + L_{\text{Bn}} ) \times 8760/2} + V_{n} - {\text{HYD}} \times 8760 \times {\text{CF}}_{H} \\ \end{aligned} $$

  • y in ≤ (x in-1 + Δx in) × 8760xCF i

  • x i0 = EX i

  • x in = x in-1 + Δx in

  • y 3n  ≥ LEPminn /ρ3

  • y 5n  = η g  × Vn

  • (x 1n  − x 5n ) ≤ LBn

  • Δx in ≤ \( {\text{DX}}_{{{ \hbox{max} }_{\text{in}} }} \)

$$ \begin{aligned} & \sum\limits_{i = 1}^{\text{NG}} {\xi_{i} y_{i} \le {\text{CO}}_{{2{\text{MAX}}_{n} }} } \\ & \sum\limits_{i = 1}^{\text{NG}} {{\text{SO}}_{{X_{\text{in}} }} \xi_{i} y_{n} \le {\text{SO}}_{{{\text{XMAX}}_{n} }} } \\ & \sum\limits_{i = 1}^{\text{NG}} {{\text{NO}}_{{X_{\text{in}} }} \xi_{i} y_{n} \le {\text{NO}}_{{{\text{XMAX}}_{n} }} } \\ \end{aligned} $$

6 Case Studies

The generation mix was performed on Korean power system to demonstrate the effectiveness of the proposed method. The test system consists of 8 types generators over 22 years (2008–2030) with the initial year being 2008. The maximum, minimum load and the hydro capacity in standard years are listed in Table 1. The characteristics and economic data are summarized in Table 2.

Table 1 Maximum load, minimum load, and hydro plant at standard years
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Table 2 The characteristics and economic data
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Maximum new construction capacity is limited by limitation of human and equipment resources, etc. It is assumed, as shown in Table 3, that the maximum and minimum capacities per a stage year of new generator are same for all stage/year in this study. Table 4 shows maximum permissible limitation of air pollution emission input data.

Table 3 Maximum and minimum of capacity per a stage year of new generators (MW)
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Table 4 Maximum permissible limitation of air pollution emission (CO2:[106 Ton/year], SOX and NOX:109[ppm/year])
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6.1 Test Results

The proposed fuzzy set theory based best generation mixes were simulated considering WTG and SCG and air pollution criterion. Aspiration level (Z 0) of total cost (budget) was specified as 27,500 Billion Won. The BGM result is shown in Table 5. The satisfaction level for the BGM was found to be 0.832. The result indicates an increase in the mix of nuclear power plants and renewable energy generators increase and a decrease in that of the coal power plants. Figure 5 is a graphical representation of the percent ratio results.

Table 5 Best generation mix considering renewable energy generators using the proposed method (%)
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Fig. 5
figure 5

Best generation mix by the proposed method (Case F1)

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6.2 Sensitivity Analysis

6.2.1 Capacity Factor of WTG and SCG

The CF of WTG and SCG in BGM is important in order to determine the mix of WTG and SCG because they can be evaluated more reasonable by higher accurate CF of the WTG and SCG. The CF of WTG and SCG depend on the resources which are wind speed and solar radiation respectively. The capacity factor includes ambiguity. The capacity factors of WTG and SCG are usually in the ranges 8–16 % and 14–18 % respectively. Tables 6 and 7 show the best generation mix results at CFWTG = 12 % and CFWTG = 14 % respectively. It is noted that the WTG is competitive with SCG at CFWTG = 14 % as shown in Table 7. Tables 8 and 9 show the cost evaluation results for the various CF of WTG and SCG.

Table 6 Best generation mix for CFWTG = 12 % using fuzzy set theory (%)
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Table 7 Best generation mix for CFWTG = 14 % Using fuzzy set theory (%)
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Table 8 Cost variations corresponding to changes in the CF of WTG (Billion won)
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Table 9 Cost variations corresponding to changes in the CF of SCG (Billion won)
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From the results, as the CF of the renewable energy generators increases, the total cost decreases because the operating cost of conventional generator decreases relatively even if the construction cost increases. The total costs per unit CF decreased from the higher CFs of WTG and SCG yield about 20 Billion Won/% and 27 Billion Won/% respectively. The result describes the economic credit of renewable energy resource in BGM problem. Finally, the higher CF makes the satisfaction level to increase because the total cost is decreased. Equation (2) Apparent escalation rate (e c ) of construction materials of WTG and SCG

It is expected that the apparent escalation rate (CER, e c ) of construction material cost of the WTG and SCG will be decreased as a result of technology development that has been taking place. Therefore, the negative CER may be considered. Specially, the CER of SCG will be decreased more rapidly rather than the CER of the WTG. For example, Tables 10 and 11 show the best generation mix result at e cSCG = −7.0 % and e cSCG = −5.5 %. Tables 12 and 13 show the cost results for the various CER of WTG and SCG. As the CER of renewable generators increases, the total cost decreases because the construction cost of renewable energy generator decreases regardless of whether the operating cost of conventional generators increases or stays fixed. It is noted that the WTG and SCG can win conventional generators in mix from e cWTG = −4.0 % and e cSCG = −6.0 % respectively as shown in Tables 10, 11, 12 and 13. The total costs per unit ER decreased by the lower CER of WTG and SCG yield about 10 Billion Won/% and 50 Billion Won/%.

Table 10 Best generation mix for e cSCG = −7.0 % using the proposed approach (%)
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Table 11 Best generation mix for e cSCG = −5.5 % using the proposed approach (%)
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Table 12 Total cost evaluation corresponding to changes in the CER of WTG (Billion Won)
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Table 13 Total cost evaluation corresponding in changes in the CER of SCG (Billion Won)
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The results describe the economic credits of renewable energy resources in view point of construction material cost. The lower escalation rate makes the satisfaction level increase because total cost is decreased.

7 Conclusions

This chapter has proposed a fuzzy LP-based approach for the long-term best generation mix with multi-criteria considering renewable energy generator such as WTGs and SCGs and air pollution constraints. The proposed method can accommodate sensitivity analysis of CF and CER of WTG and SCG including uncertainty.

The effectiveness of the proposed approach is demonstrated by applying it to the best multi-year generation mix problem of the Korean power system for year 2030. The test system contains nuclear, coal, LNG, oil, pumped-storage hydro, WTG and SCG plants. The economic credits of renewable energy resource in best generation mix problem were proposed in view point of CF and CER of WTG and SCG. It is noted that the WTG and SCG are competitive generator types in the mix and the WTG and SCG can win the conventional generators in mix from e cWTG = −4.0 % and e cSCG = −6.0 % respectively for the Korean power system. Work a technique for a more accurate evaluation of the CF is under going and results will be reported as soon as they become available. The proposed model and method are useful tools for various economics evaluation in BGM considering renewable energy generators and emission constraints.