Insights from a Comprehensive Capacity Expansion Planning Modeling on the Operation and Value of Hydropower Plants under High Renewable Penetrations

Abstract

This paper presents a quantitative assessment of the value of hydroelectric power plants (HPPs) in power systems with a significant penetration of variable renewable energy sources (VRESs). Through a capacity expansion planning (CEP) model that incorporates a detailed representation of HPP operating principles, the study investigates the construction and application of HPP rule curves essential for seasonal operation. A comparative analysis is also conducted between the proposed rule curve formulation and alternative modeling techniques from the literature. The CEP model optimizes installed capacities per technology to achieve predefined VRES penetration targets, considering hourly granularity and separate rule curves for each HPP. A case study involving twelve reservoir hydropower stations and two open-loop pumped hydro stations is examined, accounting for standalone plants and cascaded hydro systems across six river basins. The study evaluates the additional generation and storage required to replace the hydropower fleet under high VRES penetration levels, assessing the resulting increases in total system cost emanating from introducing such new investments. Furthermore, the study approximates the storage capabilities of HPPs and investigates the impact of simplified HPP modeling on system operation and investment decisions. Overall, the findings underscore the importance of reevaluating hydro rule curves for future high VRES penetration conditions and highlight the significance of HPPs in the energy transition towards carbon neutrality.

1. Introduction

To determine the optimum generation system development when targeting very high variable renewable energy source (VRES) penetration levels, an adequately developed capacity expansion planning (CEP) model needs to be applied. The CEP is generally formulated as a mathematical programming problem [1,2,3,4], aiming to identify the optimum configuration of generation and storage assets that minimize a total cost function, including both fixed and variable operating costs over an annual or multi-year horizon, subject to constraints or targets related to VRES penetration levels or a CO₂ emissions reduction to be achieved [5,6]. CEP results offer valuable insights into both the renewable and storage mix required to support the transition to deep decarbonization [7] and the operation of system assets subject to seasonal management rules, including long-duration storage and hydroelectric power plants (HPP) with large capacity reservoirs. The latter effectively falls within the broader storage category [8,9], as their water inflows exhibit seasonal variation, allowing the reservoirs to perform energy arbitrage over the year [10,11].

The value of HPPs has been discussed in the literature, with several papers highlighting the beneficial impact of hydroelectric production on power system operation. More specifically, in [12], the value of hydro generation is measured as the additional cost of electricity to be borne by end consumers during extreme drought events, assuming that HPPs are missing from the generation fleet. The analysis of [12] shows that, in the absence of HPP production, the electricity price can be twice as high as the average marginal price of the baseline scenarios where hydropower generation is present. Besides energy arbitrage, other quantifiable system-level benefits stemming from the presence of HPPs in the energy mix include contributions to resource adequacy [13], inertial response [11], reactive power support [13], fast-frequency and balancing services [11], active power reserves provision [10], black-start functionality [13], as well grid resilience enhancement [14] during extreme events [13,14]. Multi-purpose HPPs also provide significant value in applications other than power generation, including irrigation, water supply, protection of the eco-cultural environment, flood control and navigation [15], which determine, to some extent, the availability of water reserves to be used for generation purposes.

The optimal utilization of HPP water inflows, in conjunction with the fulfillment of water discharge obligations originating from environmental and other non-energy-related considerations, requires the coordinated scheduling of each hydro plant operation throughout the year. To this end, the construction of a rule curve for each HPP, determining the trendline that HPP energy reserves should follow throughout the year, becomes necessary. This allows for the management of long-duration HPP storage assets for power supply purposes and, most importantly, for securing the required water reserves for other purposes. Properly constructed rule curves are utilized as instruments to approximate the operation of HPPs in unit-commitment and economic dispatch optimization algorithms [16], while they also provide a baseline in resource adequacy studies to estimate the level of hydro plant contribution to resource adequacy [17]. Such rule curves are constructed based on several criteria, generally applied independently, including the maximization of hydropower production and the fulfillment of water supply, irrigation, and ecologic discharge obligations downstream of the plant [18,19].

The established practice for managing HPP reservoirs in real-world systems involves the use of predefined rule curves, derived from historical data and assumptions regarding the hydraulic conditions of the year under examination [20]. However, it is questionable whether such an approach is valid for future power systems dominated by VRES and storage, considering that HPP rule curves reflect limited VRES conditions of the past [21]. A related topic that is examined in the literature is the impact of climate change, with studies showing that inevitable future variations in natural inflows play a significant role in the formation of reliable rule curves [22,23,24,25]. These studies conclude that the indiscriminate adoption of existing rule curves may compromise water supply, electrification, and environmental obligations, especially during prolonged drought periods in the future.

The appropriate representation of hydropower plants, including the treatment of inflows and mandatory water discharge in the CEP, has received very limited attention in the relevant literature. Specifically, the production of HPPs is often introduced in the CEP via fixed, ex-ante calculated time series, or through simplifying assumptions regarding the seasonal operation of each plant. For instance, in [26], HPPs with large reservoirs are forced to operate constantly close to their maximum output, while the rest of the hydro fleet generates a specific amount of energy daily, with limited flexibility regarding the dispatch of this energy on the daily load curve. A similar approach is proposed in [27], where HPPs operate freely within the examined year, targeting a predetermined annual capacity factor for the plant. The approaches of [26,27] do not explicitly consider the temporal correlation of natural inflows and mandatory outflows of HPPs within the CEP, which allows for a more flexible operation of the hydro plants, but also results in an unrealistic representation of the reservoir rule curve. Other studies foresee clustering all available HPPs into a single station [28,29], ignoring the management principles governing cascaded reservoirs. A few studies incorporate a detailed representation of HPP operation [30,31], which, however, resort to typical daily profiles to represent the annual operation of system assets, at the cost of not capturing the seasonal profiles of individual hydro plants and, therefore, lacking the capability to determine the annual rule curve of each plant.

It is, therefore, clear that a detailed representation of HPPs in a high-fidelity CEP model is essential when examining high-VRES penetration conditions of future power systems, for two reasons:

• HPPs constitute renewable generation and large storage facilities at the same time [9]. Proper modeling, considering reservoir capacities and interdependencies between cascaded plants, natural inflows, and mandatory outflows, including their temporal characteristics, is essential to inform the CEP algorithm into determining a VRES and storage mix that is both optimal and feasible.
• A proper time-domain CEP model that adopts an annual horizon with an hourly timestep, without resorting to temporal clustering techniques, will allow the formation of rule curves suitable for managing individual and cascaded hydroelectric plants beyond the typical daily market cycle, in future high VRES penetration conditions, provided that natural inflows and water discharge for usages other than power generation are well-defined beforehand.

In this paper, a CEP model is developed incorporating a detailed representation of HPPs, aiming at determining the optimal portfolio of conventional generation, storage, and intermittent renewables to attain a predefined VRES penetration target. The model is built upon the linear programming (LP) mathematical optimization technique and adopts an annual optimization horizon with an hourly timestep. HPPs are treated individually, considering the natural inflows and mandatory outflows of each reservoir, as well as the relations between cascaded reservoirs belonging to the same river basin. The model is applied to a power system resembling the future development of the Greek system, targeting annual VRES penetration levels of around 90%. Overall, the study aims to achieve the following:

• Highlight the significance of the proper HPP fleet representation in CEP problems and emphasize the compromises of other simplified approaches with regard to the operation of hydro assets and the accuracy of the CEP investment decisions.
• Utilize the CEP model to construct rule curves of high fidelity for the HPPs, accounting for the natural inflows and mandatory outflow obligations per reservoir stemming from water usage in sectors other than electrification.
• Quantify the value of HPPs in the energy mix in terms of additional generation (renewable and conventional) and storage needed to replace the hydropower fleet and to estimate the respective impact of such an absence on system costs.
• Estimate the required renewable and storage installed capacities for the Greek power system to reach VRES penetrations over 90% at a minimum cost.

The remainder of this paper is organized as follows: In Section 2 the mathematical formulation of the CΕP model is presented. Section 3 describes the case study and the assumptions of the analysis. The performance of the capacity expansion algorithm is evaluated in Section 4, along with the quantification of the value of HPPs, while Section 5 demonstrates the value of the detailed representation of HPPs in the CEP. Finally, the conclusions of this study are summarized in Section 6.

2. CEP Mathematical Formulation

The CEP problem implemented in this paper involves a cost minimization objective, along with several technical and operational constraints imposed by the generation fleet and the system. The output of the model includes the installed capacities of VRES per technology, alternative storage technologies, i.e., Li-ion battery energy storage units (BES), closed-loop pumped hydro storage (PHS), and power-to-gas facilities (P2G), as well as natural-gas-fired conventional units. Further, the model can furnish hourly operation information for all system assets, allowing for an accurate representation of the seasonal operation of individual HPPs and the construction of respective rule curves with high fidelity.

The problem formulation addresses a greenfield power system development, except for hydropower plants, whose capacity is predetermined instead of being decided within the CEP process. The optimization of HPP capacity within the CEP is also possible; however, we opted not to enable such an option to ensure that the results accurately reflect the specifics of the case study. The underlying assumption is that the existing hydropower fleet has occupied all available locations, and no space has been left for new establishments. The HPP fleet might include plants with natural inflows and reservoirs, with possible pumping capabilities (open-loop PHS stations), as well as run-of-the-river (RoR) small hydro plants.

2.1. Objective Function

The objective function of the CEP problem, (1), comprises several cost components aimed at capturing the total annualized cost associated with system generation and storage facilities. Firstly, the annualized fixed cost of all system assets, represented by ${\mathrm{C}}_{\mathrm{tot}}^{\mathrm{inv}}$ and calculated in (2), is included. Secondly, the annual variable operating costs of each asset (${\mathrm{C}}_{\mathrm{tot}}^{\mathrm{var}}$), calculated in (3), are incorporated. Lastly, a cost attributed to slack variables of the problem (${\mathrm{C}}_{\mathrm{tot}}^{\mathrm{slack}}$), described in (4), is introduced.

The annualized fixed cost of the total system considers the annualized fixed costs of individual components, calculated as in (5), multiplied by their respective installed capacities, including installed RES (N_res), thermal (N_th), and storage capacities, with the latter being calculated separately for the storage power (${\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}$) and storage energy (${\mathrm{N}}_{\mathrm{sto}}^{\mathrm{en}\mathrm{e}}$) components. The annualized fixed cost per asset includes the overnight investment cost, further analyzed in (6), and the fixed operation and maintenance cost (7). More precisely, the overnight investment cost is determined using the formula of (6), which handles the capital expenditure (CAPEX_res/th/sto), the discount rate (i), and the lifetime (L_res/th/sto) of each technology. Meanwhile, the fixed operation and maintenance cost is represented as a percentage (OPEX_res/th/sto) of the CAPEX for each technology.

Furthermore, the annual variable operating costs encompasses the operating cost of thermal facilities (${\mathrm{c}}_{\mathrm{th}}^{\mathrm{var}}$) along with their respective output power (P_th,t), hydro generation (${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}$) with the operating cost of hydro (${\mathrm{c}}_{\mathrm{h}}^{\mathrm{var}}$), storage discharge (${\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{dis}}$) with the corresponding storage operating cost (${\mathrm{c}}_{\mathrm{sto}}^{\mathrm{var}}$), and renewable generation (P_res,t) with its respective operating cost (${\mathrm{c}}_{\mathrm{res}}^{\mathrm{var}}$).

Finally, the cost associated with slack variables is required to guarantee the feasibility of the solution regarding the management of hydroelectric power plants when natural inflows exceed the available storage capacity of a reservoir and the successful fulfillment of reserves’ requirements of the system. Overflows, denoted by ${\mathrm{sl}}_{\mathrm{h},\mathrm{t}}^{\mathrm{ovrfl}}$, incur a penalizing cost (${\mathrm{c}}_{\mathrm{ovrfl}}^{\mathrm{slack}}$), while potential unserved reserve requirements, (${\mathrm{sl}}_{\mathrm{r}.\mathrm{t}}$), are charged with ${\mathrm{c}}_{\mathrm{reserve}}^{\mathrm{slack}}$.

$$\mathrm{obj}=\min ({\mathrm{C}}_{\mathrm{tot}}^{\mathrm{inv}}+{\mathrm{C}}_{\mathrm{tot}}^{\mathrm{var}}+{\mathrm{C}}_{\mathrm{tot}}^{\mathrm{slack}})$$

(1)

$${\mathrm{C}}_{\mathrm{tot}}^{\mathrm{inv}}={\mathrm{N}}_{\mathrm{res}}·{\mathrm{c}}_{\mathrm{res}}^{\mathrm{inv}}+{\mathrm{N}}_{\mathrm{th}}·{\mathrm{c}}_{\mathrm{th}}^{\mathrm{inv}}+{\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}·{\mathrm{c}}_{\mathrm{sto}}^{\mathrm{pow}}+{\mathrm{N}}_{\mathrm{sto}}^{\mathrm{ene}}·{\mathrm{c}}_{\mathrm{sto}}^{\mathrm{ene}}$$

(2)

$${\mathrm{C}}_{\mathrm{tot}}^{\mathrm{var}}={\sum }_{\mathrm{th},\mathrm{t}}{\mathrm{P}}_{\mathrm{th},\mathrm{t}}·{\mathrm{c}}_{\mathrm{th}}^{\mathrm{var}}+{\sum }_{\mathrm{h},\mathrm{t}}{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}·{\mathrm{c}}_{\mathrm{h}}^{\mathrm{var}}+{\sum }_{\mathrm{sto},\mathrm{t}}{\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{dis}}·{\mathrm{c}}_{\mathrm{sto}}^{\mathrm{var}}+{\sum }_{\mathrm{res},\mathrm{t}}{\mathrm{P}}_{\mathrm{res},\mathrm{t}}·{\mathrm{c}}_{\mathrm{res}}^{\mathrm{var}}$$

(3)

$${\mathrm{C}}_{\mathrm{tot}}^{\mathrm{slack}}={\sum }_{\mathrm{h},\mathrm{t}}\mathrm{s}{\mathrm{l}}_{\mathrm{h},\mathrm{t}}^{\mathrm{ovrfl}}·{\mathrm{c}}_{\mathrm{ovrfl}}^{\mathrm{slack}}+{\sum }_{\mathrm{r},\mathrm{t}}{\mathrm{sl}}_{\mathrm{r},\mathrm{t}}·{\mathrm{c}}_{\mathrm{reserve}}^{\mathrm{slack}}$$

(4)

$${\mathrm{c}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}^{\mathrm{inv}}={\mathrm{c}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}^{\mathrm{overnight}}+\mathrm{O}\&{\mathrm{M}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}\forall \mathrm{res},\mathrm{th},\mathrm{sto}$$

(5)

$${\mathrm{c}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}^{\mathrm{overnight}}=\mathrm{CAPE}{\mathrm{X}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}·\frac{\mathrm{i}}{1-{\left(1+\mathrm{i}\right)}^{-{\mathrm{L}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}}}\forall \mathrm{res}, \mathrm{th}, \mathrm{sto}$$

(6)

$$\mathrm{O}\&{\mathrm{M}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}=\mathrm{CAPE}{\mathrm{X}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}·\mathrm{OPE}{\mathrm{X}}_{\mathrm{res}/\mathrm{th}/\mathrm{sto}}\forall \mathrm{res}, \mathrm{th}, \mathrm{sto}$$

(7)

2.2. Fundamental Problem Constraints

The energy equilibrium constraint is described in (8), where the aggregate production of thermal units (${\mathrm{P}}_{\mathrm{th},\mathrm{t}}$), the hydro fleet generation (${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}$), the discharge of storage technologies (${\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{dis}}$), and the production of RES units (${\mathrm{P}}_{\mathrm{res},\mathrm{t}}$) should meet the system’s demand. The latter encompasses the inelastic load demand requirements (${\mathrm{D}}_{\mathrm{t}}$), the pumping of open-loop pumped hydro stations (${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{pump}}$), and the charging of storages (${\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{ch}}$). Constraint (9) represents the reserves’ equilibria per time interval, t, and reserve type, r, distributing reserves along thermal (${\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{th}}$), hydro (${\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{h}}$), storage (${\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{sto}}$), and VRES technologies (${\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{res}}$) and accounting for a slack variable (${\mathrm{sl}}_{\mathrm{r},\mathrm{t}}$) to ensure the fulfilment of reserves requirements (${\mathrm{RR}}_{\mathrm{r},\mathrm{t}}$). Equation (9) applies individually for six types of reserves: the frequency containment reserves (FCR), and the automatic and manual frequency restoration reserves (aFRR and mFRR), in both directions, upwards and downwards. The reserves requirements of the rightmost part of (9) are computed according to the fundamentals of [32], suitably adjusted to serve the purposes of the case study examined in this paper.

$${\sum }_{\mathrm{th}}{\mathrm{P}}_{\mathrm{th},\mathrm{t}}+{\mathrm{P}}_{\mathrm{res},\mathrm{t}}+{\sum }_{\mathrm{h}}{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}+{\sum }_{\mathrm{sto}}{\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{dis}}={\mathrm{D}}_{\mathrm{t}}+{\sum }_{\mathrm{h}}{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{pump}}+{\sum }_{\mathrm{sto}}{\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{ch}}\forall \mathrm{t}$$

(8)

$${\sum }_{\mathrm{th}}{\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{th}}+{\sum }_{\mathrm{h}}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{h}}+{\sum }_{\mathrm{sto}}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{sto}}+\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{res}}+\mathrm{s}{\mathrm{l}}_{\mathrm{r},\mathrm{t}}=\mathrm{R}{\mathrm{R}}_{\mathrm{r},\mathrm{t}}\forall \mathrm{r}, \mathrm{t}$$

(9)

Constraint (10) defines the renewable production that participates in the energy mix at t, restricted by the production of the small RoR plants (${\mathrm{RoR}}_{\mathrm{t}}$), the available energy of each VRES technology (${\mathrm{A}}_{\mathrm{res},\mathrm{t}}$), and the respective installed capacity selected by the algorithm for each VRES technology (${\mathrm{N}}_{\mathrm{res}}$).

$${\mathrm{P}}_{\mathrm{res},\mathrm{t}}\le {\mathrm{N}}_{\mathrm{res}}·{\mathrm{A}}_{\mathrm{res},\mathrm{t}}+\mathrm{Ro}{\mathrm{R}}_{\mathrm{t}}\forall \mathrm{t}$$

(10)

Constraints (11) and (12) limit the production of thermal units accounting for the maximum installed capacity of this category (${\mathrm{N}}_{\mathrm{th}}$) and the allocated positive and negative spinning reserves. Similarly, constraint (13) delimits the charging and discharging power of each storage unit according to the installed power capacity of the technology (${\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}$), as determined by the CEP algorithm.

$${\mathrm{P}}_{\mathrm{th},\mathrm{t}}+{\sum }_{{\mathrm{r}}^{+}}{\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{th}} \le {\mathrm{N}}_{\mathrm{th}}\forall \mathrm{th}, \mathrm{t}$$

(11)

$${\mathrm{P}}_{\mathrm{th},\mathrm{t}}-{\sum }_{{\mathrm{r}}^{-}}{\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{th}} \ge 0 \forall \mathrm{th}, \mathrm{t}$$

(12)

$${\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{ch}/\mathrm{dis}}\le {\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}\forall \mathrm{sto}, \mathrm{t}$$

(13)

Constraints (14) to (17) deal with the allocation of reserves to storage stations. Specifically, BES stations are assumed to be capable of rapid switches between charging and discharging modes, as expressed by (14) and (15), allowing for reserves provision (${\mathrm{RP}}_{\mathrm{r},\mathrm{t}}^{\mathrm{Li}-\mathrm{Ion}}$) up to twice their installed capacity [33]. For the rest of the storage technologies, (16) and (17) apply [34], implying that upward reserves can be allocated by increasing the power output of the station at discharge. In contrast, downward reserves are provided by further charging when the station absorbs energy from the grid.

$${\mathrm{P}}_{\mathrm{Li}-\mathrm{Ion},\mathrm{t}}^{\mathrm{dis}}+{\sum }_{{\mathrm{r}}^{+}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{Li}-\mathrm{Ion}}\le {\mathrm{N}}_{\mathrm{Li}-\mathrm{Ion}}^{\mathrm{pow}}+{\mathrm{P}}_{\mathrm{Li}-\mathrm{Ion},\mathrm{t}}^{\mathrm{ch}}\forall \mathrm{t}$$

(14)

$${\mathrm{P}}_{\mathrm{Li}-\mathrm{Ion},\mathrm{t}}^{\mathrm{ch}}+{\sum }_{{\mathrm{r}}^{\mathrm{i}}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{Li}-\mathrm{Ion}}\le {\mathrm{N}}_{\mathrm{Li}-\mathrm{Ion}}^{\mathrm{pow}}+{\mathrm{P}}_{\mathrm{Li}-\mathrm{Ion},\mathrm{t}}^{\mathrm{dis}}\forall \mathrm{t}$$

(15)

$${\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{dis}}+{\sum }_{{\mathrm{r}}^{+}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{sto}}\le {\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}\forall \mathrm{t}, \mathrm{sto}\in \{\mathrm{P}2\mathrm{G}, \mathrm{PHS}\}$$

(16)

$${\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{ch}}+{\sum }_{{\mathrm{r}}^{-}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{sto}}\le {\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}\forall \mathrm{t}, \mathrm{sto}\in \{\mathrm{P}2\mathrm{G}, \mathrm{PHS}\}$$

(17)

Constraint (18) determines the state of charge of each storage technology (${\mathrm{SoC}}_{\mathrm{t}}^{\mathrm{sto}}$), considering the state of charge during the previous time interval (${\mathrm{SoC}}_{\mathrm{t}-1}^{\mathrm{sto}}$) and the charging and discharging quantities at t, accounting for the roundtrip efficiency of each technology (${\mathrm{n}}_{\mathrm{sto}}$). Constraints (19) and (20) further limit storage operation to guarantee that enough space is left for charging when storage provides downward reserves or that enough energy is stored for the storage to provide upward reserves. These constraints are designed to maintain each reserve type’s conservation period (ΔT).

$${\mathrm{SoC}}_{\mathrm{t}}^{\mathrm{sto}}=\mathrm{So}{\mathrm{C}}_{\mathrm{t}-1}^{\mathrm{sto}}+\sqrt{{\mathrm{n}}_{\mathrm{sto}}}·{\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{ch}}-\frac{{\mathrm{P}}_{\mathrm{sto},\mathrm{t}}^{\mathrm{dis}}}{\sqrt{{\mathrm{n}}_{\mathrm{sto}}}}\forall \mathrm{sto}, \mathrm{t}$$

(18)

$${\mathrm{SoC}}_{\mathrm{t}}^{\mathrm{sto}}\le {\mathrm{N}}_{\mathrm{sto}}^{\mathrm{ene}}-\sqrt{{\mathrm{n}}_{\mathrm{sto}}}·{\sum }_{r^{-}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{s}\mathrm{t}\mathrm{o}}·\Delta {\mathrm{T}}_{\mathrm{r}}\forall \mathrm{sto}, \mathrm{t}$$

(19)

$${\mathrm{SoC}}_{\mathrm{t}}^{\mathrm{sto}}\ge \left[{\sum }_{r^{+}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{s}\mathrm{t}\mathrm{o}}·\Delta {\mathrm{T}}_{\mathrm{r}}\right]/\sqrt{{\mathrm{n}}_{\mathrm{sto}}}\forall \mathrm{sto}, \mathrm{t}$$

(20)

Constraints (21) to (23) delimit the reserve provision of each system asset, accounting for its maximum capability to contribute to each reserve type (φ). Notably, not all generating units or storage facilities contribute equally to each reserve category due to limitations related to the fast-response specificities of each technology, their technical capabilities, and the timeframe within which allocated reserves should be released to alleviate disturbances [35,36]. For instance, aFRR should be deployed by assets incorporating fast-response capabilities to follow net-load fluctuations in almost real time while being able to follow dispatch orders from an automatic generation control infrastructure.

Constraint (24) refers to the allocation of downward mFRR by the system VRES units. This paper assumes that VRESs can provide downward mFRR as a percentage of their available generation (here, assumed to be 20%), indicating that potential renewable curtailments could be perceived as balancing services. On the contrary, the provision of faster downward reserve types, such as aFRR, requiring increased dispatchability for reliable deployment, is not allowed for VRES technologies in this formulation.

$${\mathrm{RP}}_{\mathrm{r}.\mathrm{t}}^{\mathrm{th}}\le {\mathsf{\phi }}_{\mathrm{r},\mathrm{th}}·{\mathrm{N}}_{\mathrm{th}}\forall \mathrm{th},\mathrm{r}, \mathrm{t}$$

(21)

$${\mathrm{RP}}_{\mathrm{r}.\mathrm{t}}^{\mathrm{sto}}\le {\mathsf{\phi }}_{\mathrm{r},\mathrm{sto}}·{\mathrm{N}}_{\mathrm{sto}}^{\mathrm{pow}}\forall \mathrm{sto},\mathrm{r}, \mathrm{t}$$

(22)

$${\mathrm{RP}}_{\mathrm{r}.\mathrm{t}}^{\mathrm{h}}\le {\mathsf{\phi }}_{\mathrm{r},\mathrm{h}}·\overline{{\mathrm{P}}_{\mathrm{h}}^{\mathrm{gen}}}\forall \mathrm{h},\mathrm{r}, \mathrm{t}$$

(23)

$${\mathrm{RP}}_{\mathrm{mFRR}-\mathrm{down}.\mathrm{t}}^{\mathrm{res}}\le 20\%·{\mathrm{P}}_{\mathrm{res},\mathrm{t}}\forall \mathrm{t}$$

(24)

Constraint (25) presents the renewable penetration goal, λ, defined complementarily as the maximum permissible penetration of conventional thermal energy. Finally, constraint (26) allows for introducing a maximum VRES curtailment level in the problem formulation, if required.

$${\sum }_{\mathrm{th},\mathrm{t}}{\mathrm{P}}_{\mathrm{th},\mathrm{t}} \le (1 - \mathsf{\lambda })·{\sum }_{\mathrm{t}}{\mathrm{D}}_{\mathrm{t}}$$

(25)

$${\sum }_{\mathrm{res},\mathrm{t}}{\mathrm{A}}_{\mathrm{res},\mathrm{t}}·{\mathrm{N}}_{\mathrm{res}}-{\sum }_{\mathrm{t}}{\mathrm{P}}_{\mathrm{res},\mathrm{t}}\le ({\sum }_{\mathrm{res},\mathrm{t}}{\mathrm{A}}_{\mathrm{res},\mathrm{t}}·{\mathrm{N}}_{\mathrm{res}})·\mathrm{cuts}$$

(26)

2.3. Hydropower Plant Incorporation in the CEP

In the following, we intend to develop a CEP methodology incorporating cascaded hydro plants along a river basin. The proposed formulation manages the volume of water stored in each reservoir, rather than the eventual energy output of the plant, better adhering to the fundamentals governing the operation of HPPs.

Constraint (27) describes the water equilibrium at the reservoir of each hydropower plant, including the natural inflows (${\mathrm{I}}_{\mathrm{h},\mathrm{t}}$), water discharge (${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}$), water pumping (${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{pump}}$), and water overflows (${\mathrm{sl}}_{\mathrm{h},\mathrm{t}}^{\mathrm{ovrfl}}$) when the reservoir’s storage capacity is exceeded. The water volume stored in each HPP reservoir (${\mathrm{V}}_{\mathrm{t}}^{\mathrm{h}}$), estimated in cubic meters, establishes the rule curve of the plant.

$${\mathrm{V}}_{\mathrm{t}}^{\mathrm{h}}={\mathrm{V}}_{\mathrm{t}-1}^{\mathrm{h}}+{\mathrm{I}}_{\mathrm{h},\mathrm{t}}+{\sum }_{\mathrm{hpp}}\left({\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{pump}}·{\mathrm{n}}_{\mathrm{h}}·{\mathsf{\gamma }}_{\mathrm{h}} - {\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}·\frac{1}{{\mathrm{n}}_{\mathrm{h}}}·{\mathsf{\gamma }}_{\mathrm{h}}\right)·{\mathrm{m}}_{\mathrm{h},\mathrm{hpp}}-\mathrm{s}{\mathrm{l}}_{\mathrm{h},\mathrm{t}}^{\mathrm{ovrfl}}\forall \mathrm{h}, \mathrm{t}$$

(27)

Absorbed (via pumping) and produced HPP energies are converted to water values using the average head of the plant (${\mathrm{H}}_{\mathrm{h}}$) and the water density (${\mathsf{\gamma }}_{\mathrm{h}}$), (28). The water volume stored in each HPP reservoir is upper-bounded by the maximum capacity of the reservoir ($\overline{{\mathrm{V}}^{\mathrm{h}}}$), (29). Constraint (30) imposes the water volume of the HPP at the end of the year to equal the corresponding volume at the beginning of the year, implicitly assuming that each plant will discharge over the year only its inflows, either natural or resulting from the discharge of upstream plants.

$${\mathsf{\gamma }}_{\mathrm{h}}=\frac{3.6·{10}^5}{{\mathrm{H}}_{\mathrm{h}}·\mathrm{g}}$$

(28)

$${\mathrm{V}}_{\mathrm{t}}^{\mathrm{h}}\le \overline{{\mathrm{V}}^{\mathrm{h}}}\forall \mathrm{h}, \mathrm{t}$$

(29)

$${\mathrm{V}}_{8760}^{\mathrm{h}}={\mathrm{V}}_1^{\mathrm{h}}$$

(30)

Constraints (31) and (32) bound the energy production of each HPP, accounting for the levels of allocated reserves per time interval. Similarly, constraint (33) bounds the pumping capacity of open-loop pumped hydro storage plants with natural inflows. Finally, (34) guarantees that the mandatory production of each HPP ($\underset{\_}{{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}}$) will be fulfilled for each day of the year. It should be stressed that a minimum mandatory water discharge is required daily for purposes other than the generation of electricity, including water supply to the nearby regions, irrigation, and ecological water flow requirements downstream the plant, [37].

$${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}+{\sum }_{{\mathrm{r}}^{+}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{h}}\le \overline{{\mathrm{P}}_{\mathrm{h}}^{\mathrm{gen}}}\forall \mathrm{h}, \mathrm{t}$$

(31)

$${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}- {\sum }_{{\mathrm{r}}^{-}}\mathrm{R}{\mathrm{P}}_{\mathrm{r},\mathrm{t}}^{\mathrm{h}}\ge 0 \forall \mathrm{h}, \mathrm{t}$$

(32)

$${\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{pump}}\le \overline{{\mathrm{P}}_{\mathrm{h}}^{\mathrm{pump}}}\forall \mathrm{h}, \mathrm{t}$$

(33)

$${\sum }_{\mathrm{t}}^{\mathrm{t}+23}{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}\ge \underset{\_}{{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}}\forall \mathrm{h}, \mathrm{t}\in [\mathrm{mod}\left(\mathrm{t},24\right)=1]$$

(34)

To model cascaded plants, we mapped the relations between reservoirs, as in Figure 1. Notably, the mapping parameter m_h,hpp of (27), whose elements take the values {−1, 0, 1}, signifies the hydraulic connection between cascaded upstream and downstream plants of the river basin, as presented in Figure 1 and

Table 1. Mapping the relations between the cascades and individual HPPs of Figure 1.

Hydropower Station	HPP.1	HPP.2	HPP.3	HPP.4	HPP.5	HPP.6	HPP.7
HPP.1	1	0	0	0	0	0	0
HPP.2	−1	1	0	0	0	0	0
HPP.3	0	−1	1	0	0	0	0
HPP.4	0	0	0	1	0	0	0
HPP.5	0	0	0	0	1	0	0
HPP.6	0	0	0	0	0	1	0
HPP.7	0	0	0	0	−1	−1	1

. More specifically, Table 1 reads as follows:

• Both rows and columns represent the available HPP stations of the power system.
• Let A(x,y) be a matrix element with the value ‘1’. This is only feasible if the locations x (row) and y (column) refer to the same HPP. It should be noted that, if the HPPs are placed in the same order in the rows and columns of the matrix, all elements along the main diagonal of the matrix will equal ‘1’.
• Let B(z,y) be a matrix element with the value ‘−1’. This signifies that the HPP in row z is downstream of the HPP located in column y.
• When no physical connection between two reservoirs exists, the corresponding matrix element equals zero.

Figure 1 illustrates the cascaded reservoirs for the example Table 1, referring to a power system with two sets of cascaded HPPs and one standalone plant.

It should be noted that the proposed formulation does not account for the relations between power output, water discharge, and reservoir head (the well-known Hill chart) of each HPP due to the nonlinearity of the relevant constraints. The linearization of the Hill chart is possible within a MILP formulation, yet it introduces several binary variables in the problem [38], substantially increasing the size and the computational complexity of the algorithm.

3. Case Study and Assumptions

The case study’s power system has a peak load demand of 13 GW with a load factor of 63% and a total of 14 HPPs with a cumulative power capacity of 3.1 GW. VRES technologies eligible for installation are onshore and offshore wind farms (WFs), with respective capacity factors of 28.3% and 53.8%, and solar photovoltaics (PVs) with an annual yield of 1556 kWh/kW. The time series of annual demand and VRES available energy introduced in the CEP algorithm are presented in Figure A2 of the Appendix B.

The structure of the cascaded and individual HPPs is illustrated in Figure A1 of the Appendix A, while their mapping, as implemented in the LP problem via parameter m_h,hpp, is presented in Figure A1. Additionally, the problem assumed three types of thermal units: combined-cycle gas turbines (CCGTs), CCGTs with carbon capture and storage (CCGTs–CCS), and open-cycle gas turbines (OCGTs). The techno-economic characteristics of each category are given in the appendixes. Three types of storage were anticipated: battery energy storage of Li-ion technology, pumped hydro storage, and power-to-gas storage, whose technical characteristics are given in

Table A2. Main technical characteristics of generation and storage technologies considered.

	(Roundtrip) Efficiency	Emissions Rate	Reference
BES	90%	-	[43]
PHS	75%	-	[44]
P2G	48%	-	[45,46]
CCGT	61%	0.202 tn/MWhth	[47]
CCGT-CCS	55%	0.028 tn/MWhth	[48]
OCGT	40%	0.202 tn/MWhth	[47]

.

The investment and operating costs for the production and storage facilities are presented in

Table A4. Investment cost, O&M, and lifetime of generation and storage technologies considered.

	CAPEX [EUR/kW(h)]	Fixed O&M [% CAPEX]	Lifetime [years]	Annualized Overnight Cost [EUR/kW(h)/y]	Variable O&M [EUR/MWh]	Reference
CCGT	800	2.50%	30	71.06	Based on fuel and carbon price	[49]
CCGT-CCS	1450	2.50%	30	128.80		[50]
OCGT	350	2.50%	30	35.53		[4,41,51,52]
solar PV	550	1.50%	25	51.52	2.00	[4]
Onshore WF	950	2.50%	25	88.99	2.00	[49]
Offshore WF	1825	3.00%	25	170.96	2.00	[49]
Hydropower	2000	2.00%	80	160.33	2.00	[53,54]
BES (Power)	165	2.50%	10	24.59	6.00	[43]
BES (Energy)	75	2.50%	10	11.45		[43]
PHS (Power)	800	2.00%	50	65.39		[55] (Own estimates according to the data published for PCI PHS projects by ENTSO-E.)
PHS (Energy)	20	2.00%	50	1.64		[55]
P2G (Power)	1000	2.00%	22.5	97.21		[47]
P2G (Energy)	4.5	1.50%	22.5	0.44		[41]

. Natural gas prices and carbon emissions rights were set to EUR 35/MWh_th [39,40] and EUR 100/MWh_th, respectively.

Regarding the quantification of reserves requirements, a dynamic model was utilized, accounting for the hourly variabilities and assumed forecasting errors of the corresponding time series introduced in the problem. The energy reserves required for storage stations to provide upwards and downwards FCR, aFRR, and mFRR were set to 15 min, 30 min, and 120 min, respectively. The reserves’ provision capabilities per reserve type and storage technology (φ) are shown in

Table A3. Assumed maximum reserves provision capabilities of generation and storage assets.

	FCR	aFRR	mFRR Up	mFRR Down
CCGT	7%	45%	100%	100%
CCGT CCS	7%	45%	100%	100%
OCGT	7%	45%	100%	100%
PVs	0%	0%	0%	20%
Onshore WPs
Offshore WPs
Hydro	0%	75%	100%	100%
BES	200%	200%	200%	200%
PHS	0%	75%	100%	100%
P2G	100%	100%	100%	100%

.

The characteristics of the case study’s power system are similar to those of the Greek system, but not necessarily identical [41].

Finally, the planning model was implemented in the GAMS environment (version 24.1.3) using the CPLEX optimization solver (version 12.5.1.0). All simulations were conducted on a DELL Inspiron 5593 equipped with an Intel Core i7 processor running at 1.5 GHz, with 16 GB of RAM.

4. CEP Algorithm Results for the Base Case Scenario: Assessing the Value of HPPs

4.1. Generation and Storage Mix

A base case scenario was investigated to assess the performance of the proposed CEP algorithm under very high VRES penetration levels, to the order of 90%. To achieve this target, the CEP algorithm selected the VRES and thermal production development presented in Figure 2, and the storage capacities of Figure 3. The generation mix consisted primarily of VRES technologies, with a total installed capacity exceeding 34 GW, whose operation is supported by 7.42 GW of storage power capacity with a cumulative energy capacity ca. 180 GWh. Existing hydropower plants, whose capacity is not optimized within the CEP, are also presented in Figure 2. Additionally, a total of 2.57 GW of thermal units was also required, with 0.53 GW being CCGTs with CCS. A negligible OCGT capacity of only 20 MW was also selected by the CEP, serving as a peaking unit and operating only for a few hours (~60 h) throughout the year.

Solar PVs dominated the energy mix, as their selected capacity is twice as high as that of onshore WFs and almost sevenfold the capacity of offshore WFs. Apparently, solar PVs constitute the most lucrative option for the CEP, given their significantly lower fixed cost per unit of capacity factor (rt_res) compared to the rest of the available VRES technologies. The rt_res ratio, defined in (35), serves as a metric to quantify the “capital efficiency” of each VRES technology, i.e., its annualized investment cost per unit of primary available electricity output. This metric is similar in nature to the LCOE of each technology and is referenced in this discussion because it best represents how the production cost is reflected in the CEP model. The lower the value of the rt_res metric, the more cost-efficient the respective VRES generation technology and, therefore, more suitable for selection in the CEP problem. In the examined case study, solar PVs had an rt_res of EUR 336.39/kW, while onshore and offshore WFs presented rt_res values of EUR 397.86/kW and EUR 419.63/kW, respectively.

$$\mathrm{r}{\mathrm{t}}_{\mathrm{res}}=\frac{{\mathrm{c}}_{\mathrm{res}}^{\mathrm{inv}}}{{\sum }_{\mathrm{t}}{\mathrm{A}}_{\mathrm{res},\mathrm{t}}/8760}\forall \mathrm{res}$$

(35)

The remainder of the required generation capacity consisted of CCGT units, with and without CCS technology (Figure 2), as well as the hydropower plants whose capacity was predetermined. Simple CCGT units were preferred over CCS facilities due to the increased investment cost of the latter, and the relatively low CO₂ emission rights cost assumed in the study. Should the CO₂ emissions rights increase, the balance between CCGT variants would change in favor of CCGT-CCS.

Regarding storage, BES and PHS technologies share almost the same power capacity of ~3.3 GW, while the power capacity of P2G technology is significantly lower, at 0.75 GW. PHS and P2G come with increased energy capacities, at ~90 GWh and ~78 GWh, respectively, while BES stations are accompanied by a substantially lower aggregate energy capacity of 10.75 GWh. The respective energy-to-power ratios of each storage technology are illustrated in Figure 3c. Installed capacity results of storages are largely dictated by the investment cost assumptions of each technology. BES stations are more costly in terms of energy capacity compared to the PHS and P2G alternatives. This leads to the deployment of BES facilities with a substantially lower duration compared to the other technologies. On the other hand, the power cost components of BES facilities are less costly, allowing the deployment of amplified battery power capacity, which is mainly utilized for reserves provision rather than energy arbitrage, as will arise later in this section.

The system annual cost was estimated at around EUR 5.48B, or EUR 76.6/MWh of load demand served. The system cost components included the annualized investment costs of VRES, storage, and thermal units (Figure 4), the variable operating costs corresponding to the fuel and CO₂ emission rights cost of thermal production, and the variable costs of HPP, VRES, and storage, according to the assumptions of Table A4 in the Appendix C. Given that the CEP optimization of this study did not make investment decisions on HPPs, the relevant fixed costs for these existing assets were ex-post included in the calculations. Apparently, the capital cost of VRES is the dominant system cost component, given that a bulk introduction of renewables is required to achieve the 90% penetration target (see Figure 2).

4.2. Indicative Operation of Generation and Storage Assets

Figure 5 presents the hourly generation mix of the system over a period of two consecutive weeks. The detailed operation of storage facilities during this period is shown in Figure 6, including their output power (positive/negative values denote discharging/charging) and state of charge (SoC). BES units in Figure 6a perform a regular daily cycle, charging during midday hours of high PV availability and discharging in the evening peak of the demand.

On the other hand, the operation of the P2G system in Figure 6c follows a different pattern, presenting biweekly rather than daily circularity without being imposed by the repeatability of the solar PV production. In particular, the cycling of the P2G system cannot be directly associated with a single parameter, as is the case with the BES and PV production. Instead, the first high-level conclusion drawn from Figure 5 and Figure 6c is that P2G storage facilities continuously charge the days that there is an excess of VRES production (the first 84 h of the examined two-week period), where the load is supplied almost solely with renewable energy throughout the day, and gradually discharged during the peak demand periods of the days with lower wind production.

Finally, the operation of the PHS stations combines several features of BES and P2G stations. More specifically, the long duration of the PHS allows for the performance of a daily cycle related to PV generation, superimposed over a longer one, which is similar to that of the P2G system. This is directly related to the techno-economic characteristics of the PHS, especially the high roundtrip efficiency and the low cost of energy capacity, which enable both the performance of energy arbitrage over the daily load curve with limited losses and the investment in longer storage duration to support system needs during low-wind-power events.

The observed operating patterns are directly related to the techno-economic characteristics of storage technologies and specifically to their roundtrip efficiency (Table A2) and energy capacity (Figure 3c). P2G cycling is the lowest due to its low efficiency; however, due to its high capacity, it is best suited to manage VRES excess production sustained over prolonged intervals. High-efficiency BES systems, on the other hand, are preferred to do the daily arbitrage of PV over-generation, while PHS, combining a reasonable roundtrip efficiency and a high energy capacity, enables both daily and longer-duration energy arbitrage.

Figure 5 also shows that VRES curtailments are inevitable and primarily occur during midday hours of simultaneously high solar PV and wind availability, when the storage absorbing capability has been depleted. Annual VRES curtailments reach ca. 12.5% of available energy in the base case scenario, while instantaneous renewable energy spillage can be as high as twice the system demand requirements during the examined time interval (e.g., in the first days of Figure 5).

Regarding reserves, upward and downward FCR are provided almost exclusively by BES and P2G stations (Figure 7). FRR requirements are covered by all assets, including VRES for the provision of downward mFRR. PHS stations are assigned more frequently to provide upward mFRR, which requires increased energy reserves (2 h at the allocated reserves power), while the low duration BES stations mostly provide FCR and aFRR.

4.3. Annual Operation of Hydropower Plants

Figure 8 and Figure 9 present the annual reservoir filling rate, the inflows (both natural inflows and water discharged by the upstream HPPs belonging to the same cascade), and the mandatory outflows of individual HPPs with large and small reservoirs, respectively. Figure 8 summarizes the operation of the master HPPs with large reservoirs (over 200 h storage capacity), generally the first in order along the river basin, except for HPP_3.2 and HPP_5.2, which are second in order, yet with increased natural inflows. Figure 9 shows the HPP stations incorporating smaller storage capacities, below 50 h, which are usually downstream the main reservoir of the cascaded complex. For HPPs with large reservoirs, in Figure 8, besides the variation of water level in their reservoirs (operation curve), an indicative rule curve is also presented, constructed using the reservoir level at the end of each month to filter out the daily variability.

HPP stations with large reservoirs operate seasonally, performing a single charge–discharge cycle over the course of a year. The profile of stored energy in each HPP reservoir is shaped by several factors, including the timing of natural inflows and water volumes released by upstream plants, the mandatory water discharge requirements, the capacity of the power plant, as well as by power system operation and resource adequacy needs. For this reason, the shape of rule curves may differ considerably between plants.

To shed light on the rationale driving the operation of large HPP plants, HPP_1.1 power station is an indicative example, which operates alone along the River 1 basin (Figure A1). The reservoir of HPP_1.1 gradually fills up in the winter season up to April, exploiting the large natural inflows and low mandatory discharges of that period, thus storing enough water to cover the increased mandatory outflows in the summer period (April to August). On the contrary, the operating profile of the HPP_3.1 station, having no mandatory water injections, is mainly subject to its natural water inflows. The absence of mandatory water injections throughout the year enables the plant to fully exploit its natural inflows during the first three months of the year for power production, preferring to discharge rather than store. From then on, the station practically stores all its inflows until mid-September, when its storage capacity is depleted. The need for the HPP_3.1 to store energy during the summer period instead of generating electricity is mainly imposed by the mandatory water requirements of the downstream HPP_3.2 and HPP_3.3 stations of the same cascaded complex, which are facing increased mandatory water injections in the last few months of the year (September to December). To this end, the large, first-in-order reservoir of the complex chooses to maintain enough energy reserves to support the operation of the dependent downstream stations when required.

Figure 9 refers to HPP stations with smaller storage capacities, below 50 h, which are usually located downstream of the main reservoir in each cascaded complex. These plants are not subject to seasonal management, due to their small storage capacity and the fact that their principal inflows originate from the discharging of upstream master reservoirs. Their charge level fluctuates considerably, in certain cases even on a daily basis; for this reason, there is no point in deriving rule curves for such plants.

Rule curves allow optimal reservoir management to simultaneously serve multiple purposes, including power system operation and reserve adequacy needs. As expected, such curves are characteristic of system operating conditions regarding VRES integration and, specifically, the seasonality of VRES energy availability. To demonstrate this, the rule curves of four large HPPs derived for the base case scenario of 90% VRES penetration (solid lines) are compared in Figure 10 with the respective curves for conditions of 35% annual VRES penetration (dashed lines). The optimal operation of the same hydro plants may differ significantly depending on the renewables penetration level. While the rule curve of HPP_1.1 is not affected by the increase in VRES penetration, the other three plants are operated in a markedly different manner, with their seasonal water storage functionality being much more pronounced at higher VRES system penetrations.

4.4. Calculating the Value of HPPs

To highlight the significance of hydropower production for the power system and quantify its value under high VRES penetration, a CEP scenario was investigated that is the same as the base case scenario presented in Section 4, but without any HPP in the mix. Figure 11 and Figure 12 show the differences in the installed capacities of VRES, thermal units, and storage technologies when the CEP is executed with and without the HPP fleet, targeting a minimum 90% VRES penetration in both cases.

In the absence of HPPs, the power system needs to produce an additional 4.3 TWh of renewable energy, which corresponds to the energy content of natural inflows. This energy is covered by deploying an additional 5 GW of VRES generation, specifically 1.8 GW of PVs and 3.2 GW of onshore wind farms (Figure 11a). The inherent capability of HPPs to store inflows’ energy into their large reservoirs and inject it later into the grid, when required, proves to be quite significant for the computation of the necessary storage stations for the power system. Hence, the lack of HPPs imposes an extra 1.6 GW/60 GWh storage capacity. The newly introduced storage mix is dominated by PHS stations (1.2 GW/47 GWh), while BES and P2G systems equally share the remainder (approx. 400 MW each). HPP stations also contribute significantly to the system’s upward and downward FRR requirements. In their absence, the gap in the respective reserves equilibria is covered by the additional 1.6 GW of storage and 2 GW of new gas turbines (Figure 11b), whose sole purpose is to contribute to the fulfillment of system’s FRR requirements.

What is most important is that, in the absence of HPPs, the aggregate annual system cost increased by 4.76%, from EUR 5.48B to EUR 5.74B (Figure 13). The increase was mainly due to the additional investment in generation and storage assets. However, an increase of EUR 100M in the variable generation cost was also noted due to the introduction of additional natural-gas-based generation.

Overall, it is evident that HPPs are essential elements of the generation mix, playing a significant role in the energy transition and supporting the cost-effective decarbonization of the power sector. The 3.1 GW of installed HPP capacity are equivalent to 5 GW of additional VRES technologies, 1.6 GW/60 GWh of new storage, and 2 GW of gas-fired stations, while, at the same time, they are saving EUR 260 M per annum from system costs.

To determine the value of HPP storage capacity alone, regardless of their clean energy contribution and the power capacity services they provide, we examined a scenario where the same HPP capacity was available as in the base case scenario, however, all plants were devoid of their reservoirs, i.e., they were converted into run-of-river plants, not having the capability to store inflows for more than one day. The necessary investments in new storage facilities to replace the missing HPP reservoirs, under the same VRES and thermal generation investment decisions, are shown in Figure 14. The addition of 680 GWh of storage is noted in Figure 14b, mainly in the form of P2G (~650 GWh) and secondarily in PHS (~30 GWh). The massive increase in P2G facilities comes with an increment of ~1.7 GW in their capacity, which leads to a slight decrease of required investments in BES systems, in the range of ~0.5 GW/2 GWh.

5. Evaluation of HPP Modeling in the CEP

In this section, we investigate the impact of alternative modeling principles used widely in the CEP optimization against the proposed modeling to explore the pros and cons of each case. More specifically, the following four cases are compared:

• Detailed HPP representation (base case approach adopted in this paper): In this case, the detailed representation of the cascaded HPPs of the power system is introduced in the CEP, accounting for the natural inflows of each reservoir with a daily granularity, the mandatory outflows for reasons other than electrification, the discharge water volumes of the upstream plants, etc. The formulation of this approach is as proposed in this paper.
• Simplified HPP representation, as in [28,42]: This problem variant simplifies the previous one, supposing that all system HPPs are aggregated into two plants, a reservoir HPP and an open-loop HPP. In this case, the daily natural inflows of all HPPs and the mandatory water discharges are accounted for; however, the correlation between the cascaded reservoirs is omitted. In this formulation, the open-loop HPP has no operating restrictions regarding the pumping limitations the downstream reservoirs would impose if the cascaded hydro plants’ chain had been accurately represented in the problem. In the LP-formulated CEP, Constraint (36) should be introduced replacing (27) of the detailed formulation.

$${\mathrm{V}}_{\mathrm{t}}^{\mathrm{h}}={\mathrm{V}}_{\mathrm{t}-1}^{\mathrm{h}}+{\mathrm{I}}_{\mathrm{h},\mathrm{t}}+\left({\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{pump}}·{\mathrm{n}}_{\mathrm{h}} - \frac{{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}}{{\mathrm{n}}_{\mathrm{h}}}\right)·{\mathsf{\gamma }}_{\mathrm{h}} - \mathrm{s}{\mathrm{l}}_{\mathrm{h},\mathrm{t}}^{\mathrm{ovrfl}}\forall \mathrm{h}, \mathrm{t}$$

(36)

• Annual capacity factor approach, as in [3,27]: All HPPs in the CEP are subject to a collective annual capacity factor, corresponding to the energy content of natural inflows of all plants. In this case, the scheduling of HPPs does not account for the temporal characteristics of natural inflows, while mandatory outflows are entirely ignored. Open-loop PHS plants are generally neglected in this formulation. Constraint (37) is introduced in the CEP.

$${\sum }_{\mathrm{t}=1}^{8760}{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}=\mathrm{C}{\mathrm{F}}_{\mathrm{hydro}}$$

(37)

• Daily capacity factor approach, as in [26]: This approach is conceptually similar to the previous one, yet, now, system HPPs are subject to a daily capacity factor within the CEP. This improvement increases the level of detail on the HPPs modeling, resorting to the principle that the capacity factor of HPPs differs per day according to the prevailing hydraulic conditions, which, in turn, determine the available natural inflows. The relevant constraint in the CEP is formulated as in (38).

$${\sum }_{\mathrm{t}}{\mathrm{P}}_{\mathrm{h},\mathrm{t}}^{\mathrm{gen}}=\mathrm{C}{\mathrm{F}}_{\mathrm{hydro}}^{\mathrm{day}}\forall \mathrm{t}\in \mathrm{day}$$

(38)

Figure 15a presents the CEP results with regard to the installed capacity of VRES technologies obtained by employing the alternative HPP modeling approaches evaluated in this section. In Figure 15b, the annual energy yield of all HPP assets is presented per case. The CEP results with regard to storage are shown in Figure 16.

With the simplified HPP representation, PV and onshore wind farm capacities are respectively reduced by 1 GW and 0.2 GW compared to the base case scenario, to achieve the same 90% annual VRES penetration target. At the same time, the energy capacity of P2G facilities is heavily underestimated (reduced almost by half in Figure 16), in favor of the increased exploitation of open-loop PHS facilities (Figure 15b). As in the simplified HPPs representation approach, the spatial relations between the cascaded reservoirs are omitted, the actual limitations imposed on the small, usually last-in-order, open-loop PHS of the system due to the fulfillment of their reservoirs from the water discharges of the upstream larger dams are entirely ignored. This leaves almost the entire available open-loop PHS storage capacity to be explored for arbitrage, eventually reducing the system’s standalone storage energy capacity needs. The most profound reduction is obtained in the P2G needs, which are the least efficient storage assets, with the standalone PHS to follow, while the BES systems remain practically unchanged. It should be noted that PHS power capacity slightly increases in order for the system to procure the required reserves, part of which had been provided by the P2G system in the detailed formulation.

On the other hand, formulations utilizing capacity factors are not able to model the pumping capability of HPPs. These approaches also result in lower total installed RES capacity compared to the detailed formulation, presenting differences regarding the distribution of renewable mix as well, i.e., the annual CF approach underestimates the onshore WFs capacity, while the daily CF approach overestimates it. Lack of pumping in scenarios with annual and daily capacity factors are replaced by a major deployment of PHS with an extra capacity of 0.6 GW/12 GWh and 0.22 GW/30 GWh, respectively. The BES capacity follows the deployment of PVs, being reduced by 0.5 GW/1 GWh and 1.5 GW/7 GWh, respectively. Regarding the daily CF modeling, restricting HPPs from storing water for periods longer than a day increases P2G by ~0.75 GW/114 GWh.

By definition, operating rule curves cannot be produced when annual or monthly capacity factor approaches are chosen, as no energy balance of HPP reservoirs is accounted for in the algorithm. On the contrary, the detailed base case scenario can determine the rule curve of each reservoir independently, as demonstrated earlier. On the other hand, the simplified approach can provide information regarding the rule curve of the equivalent HPP of the power system, aggregating all individual plants.

As obtained by Figure 17a, the overall stored energy of available HPPs is comparable between the simplified and the detailed case (black dashed and yellow lines, respectively), with minor differences to be attributed to the modeling simplification of the former. However, as has already been shown, the individual HPP’s operation differs significantly, being subject to a unique seasonal operating schedule that cannot be extracted by a single panoramic view of the total hydro management.

To further highlight the simplifications to which the annual and daily CF modeling approaches are resorting, we illustrate in Figure 17b the total monthly production of system HPPs in all examined cases. Apparently, the annual and daily CF alternatives present a significantly diverse operating pattern for system HPPs compared to the detailed and simplified methods, which are quite similar. More specifically, the annual capacity factor approach (red line) leaves a consecutive six-month period, from April to September, with practically zero hydroelectric production, while the daily capacity factor case (blue line) presents considerable operating variations compared to the base case scenario. Obviously, the annual/daily capacity factor cases oversimplify the operation of the HPPs. In these approaches, the CEP attributes either increased or no flexibility to HPPs in generating electricity to the extent and at the time needed, without considering the complexity of HPPs’ cascaded operation and, most importantly, the temporal relationship among natural inflows, mandatory outflows and level of reservoirs’ fulfillment, which restrict the final outcome.

Overall, the detailed HPP representation proves to be a comprehensive, yet not excessively complex solution, permitting the inclusion of all constraints essential for the management of system hydropower assets to represent the operation of individual plants and support more realistic planning decisions. The simplified HPP representation in the CEP overestimates the operation of open-loop PHS plants and severely underestimates the required capacity of new closed-loop PHS and P2G investments, while it cannot provide information regarding the operating profiles of individual HPPs; on the other hand, it significantly reduces the execution time of the CEP algorithm (by almost 40% -see

Table 2. Number of variables and execution time for the different HPP modeling approaches in the CEP.

	Number of Variables	Execution Time
Annual CF approach	5,920,051	5 min & 32 s
Daily CF approach	5,920,051	15 min & 51 s
Simplified HPP representation	6,647,131	28 min & 47 s
Detailed HPP representation	14,855,251	47 min & 12 s

). The annual and daily capacity factor approaches are characterized by even faster execution times (reduced by 70% compared to the detailed representation); however, they suffer from inherent inefficiencies to represent seasonal reservoir management, leading to unrealistic operating profiles and inaccurate CEP results.

Finally, with regard to the potential of the alternative approaches to produce rule curves, it is clear that the annual or daily capacity factor approaches are entirely unsuitable. With the detailed HPP representation, rule curves can be produced for individual reservoirs, as demonstrated earlier in Section 4.3, while the simplified HPP representation approach can only provide rule curve information for an equivalent HPP, aggregating all individual plants of the system.

6. Conclusions

In this paper, a comprehensive representation of hydropower plants is developed for inclusion in CEP problems built upon the LP mathematical optimization. The proposed model allows proper representation of the operating constraints of individual hydropower assets and cascades, captures their seasonal operation, and enables the derivation of optimal annual management patterns to serve multiple objectives, such as supporting high VRES penetrations and adapting to the availability of renewable energy.

A case study system that includes fourteen large HPPs with reservoirs, two of them being open-loop pumped storage facilities, resembling electrical characteristics of the Greek power system was selected as representative. The proposed CEP algorithm was applied to determine the optimal expansion of the system and the operation of hydroelectric facilities, targeting a future VRES penetration to the order of 90% and aiming at minimizing system investment and variable operating costs. Eventually, to move towards such a promising decarbonization objective, the power system should be characterized by high solar and wind energy penetration, while increased amounts of electricity storage will be required through three different technologies: batteries, pumped hydro, and P2G.

The CEP algorithm also supplied significant evidence regarding the annual operation of hydropower plants. Rule curves were derived for individual HPPs, simultaneously addressing the multiple objectives served by large HPPs (system operational needs, resource adequacy, VRES penetration support, irrigation/water supply, and environmental constraints). The analysis showed that the proposed CEP model could form suitable rule curves for the HPPs with larger reservoirs, which are usually first in order in the river basin they belong to. For the smaller plants, which are generally below the larger ones and in the middle or last in the cascaded complex, the establishment of rule curves proved to be unfeasible, as their operation is highly affected by the water discharges of the upwards larger reservoirs. Additionally, the analysis showed that the rule curves of the HPPs are significantly affected by the prevailing VRES penetration conditions, indicating that they should be periodically reevaluated as renewable production is gradually amplified in the energy mix.

To further evaluate the proposed HPP modeling methodology, a comparison with three alternative approaches available in the literature was performed. Simplified approaches already available in the literature, which resort to daily and annual capacity factors for the operation of HPPs or assume the aggregate representation of hydro production in the CEP, appear to be compromised in several respects. More specifically, they either fail to capture the complex operation of cascaded hydropower systems or suffer from inefficiencies in representing seasonal reservoir management, leading to unrealistic operating profiles and debatable CEP results. On the other hand, the proposed modeling exhibits significantly higher execution times, which might be tenfold the time required by the most simplified methods, rendering its application less appealing.

The value of hydropower in a decarbonized future power system has also been quantified in the paper as the additional installed capacity of other system assets required to replace the entire HPPs fleet. Specifically, to compensate for the lack of 3.1 GW of hydroelectric power plants, it is necessary to install 5 GW of VRES, 1.6 GW/60 GWh new storage facilities, mainly of PHS technology, and an additional 2 GW of natural-gas-fired turbines. The analysis demonstrated the value of HPPs with large reservoirs as long-duration storage facilities and highlighted their significance as flexibility resources, primarily to provide power reserves to the system, effectively substituting thermal units from this task. As a result, the replacement of HPPs with increased capacities of other generation and storage assets led to an increase of almost 5% in electricity cost.

Finally, future work should explore the inclusion of a broader range of climatic conditions in the analysis. While the hourly granularity employed in the analysis yields reliable results, it is crucial to acknowledge that climatic conditions can impact the outcome of capacity expansion planning, particularly when investment decisions must withstand extreme climate events such as prolonged droughts or periods of limited wind availability. Therefore, expanding the current model to adequately address the stochastic nature of RES availability, inflows, and load demand in future studies is imperative. This extension will provide a more comprehensive understanding of system dynamics, enabling the development of more robust decision-making processes in grid optimization strategies. Additionally, it will enhance understanding of hydropower plant rule curve formation.