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CVaR‐based generation expansion planning of cascaded hydro‐photovoltaic‐pumped storage system with uncertain solar power considering flexibility constraints

2021; Institution of Engineering and Technology; Volume: 15; Issue: 21 Linguagem: Inglês

10.1049/gtd2.12232

1751-8695

Mingkun Du, Yuan Huang, Junyong Liu, Gang Wu, Shafqat Jawad,

Electric Power System Optimization

IET Generation, Transmission & DistributionVolume 15, Issue 21 p. 2953-2966 ORIGINAL RESEARCH PAPEROpen Access CVaR-based generation expansion planning of cascaded hydro-photovoltaic-pumped storage system with uncertain solar power considering flexibility constraints Mingkun Du, orcid.org/0000-0002-1917-5201 School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of ChinaSearch for more papers by this authorYuan Huang, Corresponding Author yuanhuang@scu.edu.cn School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of China Correspondence Yuan Huang, School of Electrical Engineering Information, Sichuan University, Chengdu 610065, People's Republic of China. Email: yuanhuang@scu.edu.cnSearch for more papers by this authorJunyong Liu, School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of ChinaSearch for more papers by this authorGang Wu, State Grid Sichuan Economic Research Institute, Chengdu, People's Republic of ChinaSearch for more papers by this authorShafqat Jawad, School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of ChinaSearch for more papers by this author Mingkun Du, orcid.org/0000-0002-1917-5201 School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of ChinaSearch for more papers by this authorYuan Huang, Corresponding Author yuanhuang@scu.edu.cn School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of China Correspondence Yuan Huang, School of Electrical Engineering Information, Sichuan University, Chengdu 610065, People's Republic of China. Email: yuanhuang@scu.edu.cnSearch for more papers by this authorJunyong Liu, School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of ChinaSearch for more papers by this authorGang Wu, State Grid Sichuan Economic Research Institute, Chengdu, People's Republic of ChinaSearch for more papers by this authorShafqat Jawad, School of Electrical Engineering Information, Sichuan University, Chengdu, People's Republic of ChinaSearch for more papers by this author First published: 17 June 2021 https://doi.org/10.1049/gtd2.12232AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract The development of a high solar energy penetrated power system requires considerable flexibility to hedge the risk of solar power curtailment and power shortage. This paper explores how the generation portfolio of cascaded hydro-photovoltaic-pumped storage (CH-PV-PS) generation system will be appropriately designed to balance the overall planning costs and operational flexibility constraints. The proposed study relies on the generation expansion planning (GEP) model of the CH-PV-PS system, considering a full set of flexibility constraints. An index designated ramp-capability reserve shortage (RCRS) based on the conditional value at risk (CVaR) method is introduced to quantify the risk of solar power and load uncertainty. The piecewise linearization method and triangle method are developed to accommodate the non-linear terms in the proposed model. Finally, the case studies are conducted to demonstrate the applicability and effectiveness of the proposed model. 1 INTRODUCTION Increasing the penetration of variable renewable energy is an inevitable way for the transition to a decarbonized power system, which is dominated by solar energy, wind energy, and hydroelectricity. The cascaded hydro-photovoltaic-pumped storage (CH-PV-PS) generation system concept originated in China aims to facilitate this transition [1-3]. However, the CH-PV-PS system always operates in an off-grid condition that escalates the challenges of meeting the need for flexibility to mitigate the mismatch between generation and consumption [4]. The long-term generation expansion planning (GEP) of the CH-PV-PS system considering flexibility constraints faces many difficulties and needs to be further studied. The flexibility constraints are introduced to ensure the system's adaptation to uncertainty and are of great significance to accommodate more renewable energies and improve the reliability of the power system connected with variable renewable energies. Ignoring the flexibility constraints might result in a significant underestimation of the curtailment of renewables and overall planning costs, leading to a sub-optimal planning result [5]. Belderbos and Delarue proposed a new system planning model on a power plant resolution using mixed-integer linear programming (MILP), taking into account technical flexibility constraints [6]. The results indicated that flexibility constraints have a critical impact on the optimal generation portfolio and cannot be ignored. Palmintier and Webster presented a generation expansion formulation for an 8760-hour chronological profile of load, demand, and wind [7]. It was solved as a single MILP optimization considering the operational flexibility constraints, pointing out that ignoring the flexibility constraints would create 35%–60% errors in the estimated carbon emissions with 20% renewables, resulting in an inability to meet tighter carbon emission regulations. The flexible resources planning to accommodate large-scale renewable energy has been considered in the current paper, including a battery energy storage system (BESS) [8], demand response (DR) [9], and integrated energy system (IES) [10]. The full set of flexibility constraints was introduced to the planning model mentioned above, and the results all illustrated effectively that the flexibility constraints cannot be ignored in the planning optimization. However, to the best of the author's knowledge, there has been no description in the literature introducing the flexibility constraints into the CH-PV-PS generation system expansion planning and providing a detailed analysis. There is a lack of consideration for flexibility constraints in the GEP model of the CH-PV-PS system. To deal with the uncertainty in the solar power output of a PV array, researchers have deployed a wide range of techniques. Stochastic optimization (SO) is an effective approach, where the power output of a PV array is assumed to follow a predetermined probability distribution [11]. The long-term planning model for the joint optimization of conventional thermal units, concentrating solar power (CSP), and storage devices have been proposed in [12], and the SO approach was employed to deal with the uncertainty of solar power. The robust optimization (RO) approach has the advantage that the uncertainties are represented by a robust set. The uncertain problem can thus be transformed into a deterministic problem under the worst-case scenarios [13]. The adaptive RO model was explored to reduce the uncertainty of power generation by integrating CSP plants with wind farms in [14]. However, the SO approach needs several scenarios predetermined by probability information, which may cause computational intractability. Besides, the RO approach can be immune to the worst cases of uncertainty realizations, so the planning results are conservative. Thus, the methods mentioned above are difficult to solve for the complex, large-scale CH-PV-PS planning problem. The conditional value-at-risk (CVaR) method as an effective method to cope with the uncertainty of renewable energy has been applied to the power system optimization problem [15-18]. In [15], the flexible look-ahead dispatch model based on the CVaR method was proposed, and an index CVaR-WP was introduced to evaluate the risk of wind power accommodation. The results indicated that the CVaR method could avoid the over-conservativeness of traditional RO. In [16], the CVaR method was used to express the uncertainties of wind power in an IES planning model and transformed the stochastic probability model of wind power to a determinate expression. Compared with the SO approach, the CVaR method did not require an accurate forecast power value and reduced the computational burden. However, the CVaR method is used to describe the uncertainty of single renewable energy (i.e. wind energy) in [15] and [16]. Consequently, the form of the CVaR method needs to alter if the uncertainty of both generations and loads is taken into account. In this paper, we introduce the index designated the ramp-capability reserve shortage (RCRS) based on the CVaR method to measure the risk of solar power curtailment and power shortage. Here the CVaR method is applied to model uncertainties of solar power and power load and has the advantage of reducing the conservatism of the conventional RO approach and the computational intractability of the SO approach. The GEP model of the CH-PV-PS system incorporating a full set of chronological flexibility constraints in operation is proposed to attempt a trade-off between planning cost and RCRS. The capacity and geographical allocation of the CH-PV-PS system can thus be optimized to mitigate the excess or shortage of solar power by utilizing our proposed model. The piecewise linearization method and triangle method are used in this paper to transform the non-linear model into a MILP model, which is easier to be solved. According to the above literature survey, the major contributions of this paper are as follows: Developing a GEP model of CH-PV-PS system considering flexibility constraints to determine the optimal newly built capacity and geographical allocation to achieve higher solar energy penetration. Modelling the uncertainties of solar power and power load based on the CVaR method avoids over-conservativeness and improves computing efficiency. Analysing how the optimal generation mix will change towards a higher penetration of solar power and the necessity of considering flexibility constraints. The rest of this paper is organized as follows. Section 2 introduces the RCRS based on the CVaR method, which signifies the uncertainties of solar power and power load. Section 3 presents the mathematical formulation of the GEP model of the CH-PV-PS system. Section 4 details the solution methodology of the proposed model. Section 5 illustrates some numerical examples, and optimization results are analysed. Finally, concluding remarks are drawn in Section 6. 2 CVAR-BASED RAMP-CAPABILITY RESERVE SHORTAGE CHARACTERIZATION 2.1 RCRS of power system The flexibility of the power system is defined here as the ability of a power system to deploy its flexible ramping resources to respond to changes in net load at a reasonable cost at a specific time scale [19]. The net load mainly consists of the power load and variable renewable generation (i.e. solar energy) in this paper. The time horizon is defined as the duration of net load change, equal to 1 hour in the following analysis. The RCRS of the power system corresponds to the deviation that the flexible system resources cannot meet the net load, as shown in Figure 1. FIGURE 1Open in figure viewerPowerPoint Schematic diagram of upward/downward RCRS In Figure 1, I represent the upward net load at time t, which is caused by solar power output, drastic reduction, and drastic load increase in a short time interval t→t + 1. The upward ramp-capability reserve provided by flexible resources is limited at time t. Therefore, when the upward net load exceeds the upward ramp-capability reserve, there will be an upward RCRS. On the contrary, when solar power output increases significantly or load reduces drastically in a short time interval t + 1→t + 2, the net load will increase rapidly, as shown by II in Figure 1. If the downward ramp-capability reserve provided by flexible resources is insufficient at that time, the downward RCRS will arise. 2.2 Mathematical formulation of RCRS Value at risk (VaR) refers to the maximum possible loss of an investment portfolio under a given confidence level in the market. However, numerous researches have shown that there are some defects with VaR. For instance, VaR is not a consistent risk measurement method and cannot provide an adequate picture of risks reflected in the extreme tail, which results in the underestimate of the investment risk. Therefore, based on VaR, Rockafellar and Uryasev [20] proposed CVaR, which refers to the conditional expectation of loss exceeding VaR for all conditions. As a common risk measurement method, CVaR calculated the probability of system loss to measure the operational risk of the system accurately. This paper adopts the CVaR method to deal with the uncertainty of solar power and load in planning optimization. The uncertainty caused by solar power and power load can be statistically described using the normal distribution [11, 21]. However, the normal probability density function (PDF) generally uses − ∞ and + ∞ as the lower limits and upper limits, respectively, which is not suitable for the statistical description of solar power and power load due to specific lower and upper limits. Thus, the uncertainty related to solar generation and power load is modelled by normal PDF with specific upper and lower limits in this paper. The diagram of the PDF of solar power and power load is shown in Figure 2. FIGURE 2Open in figure viewerPowerPoint Diagram of the PDF (a) solar power, (b) power load The PDF of solar power is shown in Figure 2(a). μ s is the expectation value of solar power output, while P s L L and P s U L denote the lower limits and upper limits of probability interval for solar power. The probability interval [ P s L L , P s U L ] can be optimized through the whole model or fixed in a certain ratio [16]. For the given node, if the actual value of solar power output is within [ P s L L , P s U L ], the solar power can be accommodated without introducing any risk. Next, if the actual value of solar power exceeds P s L L , the system will be suffered the risk of solar power curtailment, which may result in an upward RCRS. Furthermore, if the actual value of solar power is less than P s L L , the total generation outputs will be insufficient, and the system will face the risk of power shortage, resulting in the downward RCRS. A similar analysis is applied to Figure 2(b). Consider the case where there is upward RCRS in the system. In this case, the VaR values of solar power and load are P s U L and P d L L , respectively. The CVaR is the conditional expected value that the risk loss exceeds the VaR. Thus, for a given node, the upper CVaR ϕ u p ( P s , m U L , P d , n L L ) corresponding to the upward net load consisting of solar power and load demand can be calculated by (1): ϕ u p ( P s , m U L , P d , n L L ) = ∑ m ∈ Ω s ϕ s u p ( P s , m U L ) + ∑ n ∈ Ω d ϕ d u p ( P d , n L L ) ϕ s u p ( P s , m U L ) = ∫ 0 ≤ P s , m − P s , m U L ≤ P s , m m a x − P s , m U L P s , m − P s , m U L Pr ( P s , m ) d P s , m ϕ d u p ( P d , n L L ) = ∫ 0 ≤ P d , n L L − P d , n ≤ P d , n L L − P d , n m i n P d , n L L − P d , n Pr ( P d , n ) d P d , n (1) Similarly, for the given node, the lower CVaR ϕ d o w n ( P s , m L L , P d , n U L ) can be calculated by (2): ϕ d o w n ( P s , m L L , P d , n U L ) = ∑ s ∈ Ω w ϕ s d o w n ( P s , m L L ) + ∑ n ∈ Ω d ϕ d d o w n ( P d , n U L ) ϕ s d o w n ( P s , m L L ) = ∫ 0 ≤ P s , m L L − P s , m ≤ P s , m L L − P s , m m i n P s , m L L − P s , m Pr ( P s , m ) d P s , m ϕ d d o w n ( P d , n U L ) = ∫ 0 ≤ P d , n − P d , n U L ≤ P d , n m a x − P d , n U L P d , n − P d , n U L Pr ( P d , n ) d P d , n (2) Therefore, in this case, the upward/downward RCRS for the given node can be calculated by (3a) and (3b), respectively. R C R u p and R C R d o w n denote the up/down ramp-capability reserves scheduled, respectively, as detailed in Section 3.5: R C R S u p = max ϕ u p ( P s , m U L , P d , n L L ) − R C R u p , 0 (3a) R C R S d o w n = max ϕ d o w n ( P s , m L L , P d , n U L ) − R C R d o w n , 0 (3b) 3 MODELLING FORMULATION The structure of the CH-PV-PS generation system [1], which comprises cascaded hydro units (CHUs), PV unit, and PHES unit is shown in Figure 3. It is a typical structure that CHUs, PV and PHES are linked to the AC buses and send their electricity to the utility grids. The CHUs consist of hydroelectric units within a river basin. The PV unit is composed of a PV array and a DC/AC converter. The PHES unit consists of a pump turbine, a synchronous motor, and a full-size converter (FSC), which is used to transmit the energy between the stator and the utility grids. In addition, the control centre samples PV output power processes the operation data of the CH-PV-PS system and sends control instructions to CHUs and PHES. FIGURE 3Open in figure viewerPowerPoint Structure of the CH-PV-PS generation system 3.1 Objective function The objective function minimizes the overall system cost by minimizing annualized investment cost C C H − P V − P S i n v of CH-PV-PS system, annual system operation cost C o p , and CVaR related to RCRS C R C R S : min C o s t = min C C H − P V − P S i n v + C o p + C R C R S (4)such that, C C H − P V − P S i n v = γ C R F ∑ i ∈ Ω C H U i n v c C H U , i i n v C a p C H U , i i n v x i i n v + ∑ j ∈ Ω P H E S i n v c P H E S , j i n v C a p P H E S , j i n v x j i n v (5) C o p = ∑ t ∈ T ∑ u ∈ Ω T h c u o n S U u ( t ) + c u P u ( t ) Δ τ + c u u p R C R u u p ( t ) + c u d o w n R C R u d o w n ( t ) + ∑ t ∈ T ∑ k ∈ Ω C H U c k o n S U k ( t ) + c k o f f S D k ( t ) + c k u p R C R k u p ( t ) + c k d o w n R C R k d o w n ( t ) + ∑ t ∈ T ∑ z ∈ Ω P H E S c g , z o n S U g , z ( t ) + c p , z o n S U p , z ( t ) + c g , z u p R C R g , z u p ( t ) + c g , z d o w n R C R g , z d o w n ( t ) + c p , z u p R C R p , z u p ( t ) + c p , z d o w n R C R p , z d o w n ( t ) (6) C R C R S = ∑ t ∈ T ξ u p R C R S u p ( t ) + ξ d o w n R C R S d o w n ( t ) (7) The investment cost of the CH-PV-PS system, formulated in (5), includes annualized investments of newly built cascaded hydro units and PHES units. The total annual operation cost includes start-up costs, fuel costs, and upward/downward ramp-capability reserve costs of existing thermal units in the first line, start-up/shut-down costs and upward/downward ramp-capability reserve costs of cascaded hydro units in the second line, and start-up costs and upward/downward ramp-capability reserve costs of PHES units in the third line, as shown in (6). The CVaR related to RCRS, formulated in (7), is calculated by multiplying the upward/downward RCRS in (3) with the price for RCRS. 3.2 Cascaded hydro units operation constraints P k ( t ) = c 1 , k V k ( t ) 2 + c 2 , k Q k d i s ( t ) 2 + c 3 , k V k ( t ) Q k d i s ( t ) + c 4 , k V k ( t ) + c 5 , k Q k d i s ( t ) + c 6 , k , ∀ k ∈ Ω C H U , ∀ t ∈ T (8) P ̲ k ≤ P k ( t ) ≤ P ¯ k ≤ C a p C H U , k , ∀ k ∈ Ω C H U , ∀ t ∈ T (9) V k ( t + 1 ) = V k ( t ) + Q k i n ( t ) − Q k d i s ( t ) − Q k s p ( t ) · Δ τ · Δ s Q k i n ( t ) = Q k n a ( t ) + ∑ u p s ∈ Ω U C H U Q u p s d i s ( t ) + Q u p s s p ( t ) ∀ k ∈ Ω C H U , ∀ t ∈ T (10) V ̲ k ≤ V k ( t ) ≤ V ¯ k , ∀ k ∈ Ω C H U , ∀ t ∈ T (11) Q ̲ k d i s ≤ Q k d i s ( t ) ≤ Q ¯ k d i s , ∀ k ∈ Ω C H U , ∀ t ∈ T (12) Q ̲ k O ≤ Q k d i s ( t ) + Q k s p ( t ) ≤ Q ¯ k O , ∀ k ∈ Ω C H U , ∀ t ∈ T (13) P k ( t ) − P k ( t − 1 ) ≤ 1 − S U k ( t ) R U k , ∀ k ∈ Ω C H U , ∀ t ∈ T (14) P k ( t − 1 ) − P k ( t ) ≤ 1 − S D k ( t ) R D k , ∀ k ∈ Ω C H U , ∀ t ∈ T (15) The power output [22] of the kth cascaded hydro unit and its limit is presented in (8) and (9), respectively. The water balance constraint, formulated in constraint (10), ensures the water balance in the time dimension of a single cascaded hydro unit and the space dimension of upstream and downstream cascaded hydro units. Reservoir volume, water discharge, and water outflow for the kth cascaded hydro unit are limited by constraints (11)–(13), respectively. Finally, the ramping constraint, formulated in (14) and (15), ensures that upward/downward flexibility provided by cascaded hydro units should not interrupt the ramp-capability of units. 3.3 PHES units operation constraints 0 ≤ x g , z ( t ) + x p , z ( t ) ≤ 1 , ∀ z ∈ Ω P H E S , ∀ t ∈ T (16) 0 ≤ P g , z ( t ) ≤ x g , z ( t ) P ¯ g , z ≤ x g , z ( t ) C a p P H E S , z , ∀ z ∈ Ω P H E S , ∀ t ∈ T (17) 0 ≤ P p , z ( t ) ≤ x p , z ( t ) P ¯ p , z ≤ x p , z ( t ) C a p P H E S , z , ∀ z ∈ Ω P H E S , ∀ t ∈ T (18) ∑ t ∈ T P p , z ( t ) · η z − P g , z ( t ) P g , z ( t ) η z η z = 0 , ∀ z ∈ Ω P H E S (19) The PHES unis can provide sufficient ramp-capability reserves with the advantage of short start-up/shut-down times and no minimum up/downtimes. As formulated in (16), the generating-pumping mutual exclusion constraint ensures that PHES units cannot be in the generating and pumping status simultaneously. The power output of PHES units is limited by constraints (17) and (18). Constraint (19) refers to the energy balance of a single PHES unit. 3.4 Energy balance constraints ∑ u ∈ b P u ( t ) + ∑ k ∈ b P k ( t ) + ∑ z ∈ b P g , z ( t ) − P p , z ( t ) + ∑ l ∈ L b P l ( t ) + ∑ m ∈ b P s , m U L ( t ) ≥ ∑ n ∈ b P d , n L L ( t ) , ∀ t ∈ T , ∀ b ∈ N b (20a) ∑ u ∈ b P u ( t ) + ∑ k ∈ b P k ( t ) + ∑ z ∈ b P g , z ( t ) − P p , z ( t ) + ∑ l ∈ L b P l ( t ) + ∑ m ∈ b P s , m L L ( t ) ≤ ∑ n ∈ b P d , n U L ( t ) , ∀ t ∈ T , ∀ b ∈ N b (20b) P l ( t ) = θ b ( t ) − θ b ′ ( t ) θ b ( t ) − θ b ′ ( t ) X l X l , ∀ t ∈ T , ∀ l ∈ L b (21) − P l max ≤ P l ( t ) ≤ P l max , ∀ t ∈ T , ∀ l ∈ L b (22) The power loads should be matched with total generators net output from cascaded hydro units, PHES units, PV units, and the energy balance is guaranteed in (20). The DC power flow equation and the power flow constraint are guaranteed in (21) and (22). 3.5 Flexibility constraints R C R u p ( t ) = ∑ u ∈ Ω T h R C R u u p ( t ) + ∑ k ∈ Ω C H U R C R k u p ( t ) + ∑ z ∈ Ω P H E S R C R g , z u p ( t ) + R C R p , z u p ( t ) , ∀ t ∈ T (23) R C R d o w n ( t ) = ∑ u ∈ Ω T h R C R u d o w n ( t ) + ∑ k ∈ Ω C H U R C R k d o w n ( t ) + ∑ z ∈ Ω P H E S R C R g , z d o w n ( t ) + R C R p , z d o w n ( t ) , ∀ t ∈ T (24) R C R u u p ( t ) ≤ min R U u , P ¯ u − P u ( t ) , ∀ u ∈ Ω T h , ∀ t ∈ T R C R u d o w n ( t ) ≤ min R D u , P u ( t ) − P ̲ u , ∀ u ∈ Ω T h , ∀ t ∈ T (25) R C R k u p ( t ) ≤ min R U k , P ¯ k − P k ( t ) , ∀ k ∈ Ω C H U , ∀ t ∈ T R C R k d o w n ( t ) ≤ min R D k , P k ( t ) − P ̲ k , ∀ k ∈ Ω C H U , ∀ t ∈ T (26) R C R g , z u p ( t ) ≤ P ¯ g , z x g , z ( t ) − P g , z ( t ) , ∀ z ∈ Ω P H E S , ∀ t ∈ T R C R g , z d o w n ( t ) ≤ P g , z ( t ) , ∀ z ∈ Ω P H E S , ∀ t ∈ T R C R p , z u p ( t ) ≤ P p , z ( t ) , ∀ z ∈ Ω P H E S , ∀ t ∈ T R C R g , z d o w n ( t ) ≤ P ¯ p , z x p , z ( t ) − P p , z ( t ) , ∀ z ∈ Ω P H E S , ∀ t ∈ T (27) The flexible up/down ramp-capability reserves, as formulated in (23) and (24), are provided by existing thermal units, newly built cascaded hydro units, and newly built PHES units. The ramp-capability reserve of thermal units is constrained by ramping limit, maximum power output, and minimum power output of thermal units, as shown in (25). The ramp-capability reserve of cascaded hydro units is formulated in (26) and is limited by ramping limit, maximum power output, and minimum power output of cascaded hydro units. In this paper, we assume that the upward/downward ramp ratio of PHES is very high, and no start-up and shut down times [23]. Therefore, the ramp-capability reserve of PHES units is only constrained by the upper/lower power output limit, as formulated in (27). 4 SOLUTION METHODS 4.1 Piecewise linearization method for linearizing the objective function The non-linear integral terms in the penalty cost C R C R S related to RCRS for the curtailment of solar power and power load, as shown in (7), are challenging to find an effective solution. The piecewise linearization method [24] deals with the non-linear integral terms in this paper. The piecewise linearization method can transform the non-linear problem into a MILP problem. Figure 4 shows the piecewise linearization of the non-linear terms ϕ s u p ( P s , m U L ) and ϕ d u p ( P d , n L L ) corresponding to the upward RCRS. FIGURE 4Open in figure viewerPowerPoint Diagram of the piecewise linearization (a) ϕ s u p ( P s , m U L ) , (b) ϕ d u p ( P d , n L L ) The piecewise linearization process of ϕ s u p ( P s , m U L ) is illustrated in Figure 4(a) . First, we introduce a number O + 1 of sampling coordinates { M s , 1 , …, M s , O + 1 }, where the sampling coordinates are defined as breakpoints on the P s , m U L axis. Thus, the P s , m U L axis is divided into O intervals by these breakpoints. Then, let us introduce a continuous variable α s , m , o associated with the interval [ M s , o , M s , o + 1 ] (o = 1, 2, …, O). To use the above techniques in a MILP solver, it is necessary to include in the binary variables β s , m , o associated with the oth interval [ M s , o , M s , o + 1 ] (o = 1, 2, …, O). The approximate value of ϕ s u p ( P s , m U L ) can then be obtained by imposing the constraints (28) and (29): ϕ s u p ( P s , m U L ) = ∑ o ∈ O a o , o + 1 s α s , m , o + b o , o + 1 s β s , m , o , ∀ m ∈ Ω s (28) ∑ o ∈ O α s , m , o = P s , m U L , ∀ m ∈ Ω s ∑ o ∈ O β s , m , o = 1 , ∀ m ∈ Ω s β s , m , o · ρ s , m , o ≤ α s , m , o ≤ β s , m , o · ρ s , m , o + 1 , ∀ m ∈ Ω s , ∀ o ∈ O (29) Similarly, the approximate value of ϕ s u p ( P s , m U L ) in Figure 4(b) can be transformed to (30) and (31): ϕ d u p ( P d , n L L ) = ∑ o ∈ O a o , o + 1 d α d , n , o + b o , o + 1 d β d , n , o , ∀ n ∈ Ω d (30) ∑ o ∈ O α d , n , o = P d , n U L , ∀ n ∈ Ω d ∑ o ∈ O α d , n , o = P d , n U L , ∀ n ∈ Ω d β d , n , o · ρ d , n , o ≤ α d , n , o ≤ β d , n , o · ρ d , n , o + 1 , ∀ n ∈ Ω d , ∀ o ∈ O (31) Similarly, the non-linear terms ϕ s d o w n ( P s , m L L ) and ϕ d d o w n ( P d , n U L ) corresponding to the downward RCRS can be piecewise linearized using the same above techniques. 4.2 Triangle method for linearizing the two variables constraints In the proposed model, two continuous variables are included in the non-linear constraint (8). To cope with this non-linear constraint, the triangle method [25] is proposed to accommodate the two variables in (8). The triangle method is still a piecewise linearization approximation of functions of two variables. Constraint (8) indicates that the power output of the cascaded hydro units is related to th