Once upon a time, in a small village nestled deep in the heart of a dense forest, lived a witch. This witch was not like the ones seen in fairy tales with pointed hats and broomsticks, but rather a mysterious and powerful sorceress. She had long black hair, piercing green eyes, and a presence that could send chills down anyone's spine. Despite her fearsome appearance, the witch had a gentle heart and desired to use her powers for good. However, destiny had different plans for her. The villagers, filled with fear and superstition, saw her as a threat and an embodiment of evil.
The villagers, filled with fear and superstition, saw her as a threat and an embodiment of evil. They whispered tales of her wickedness, spreading rumors and lies about her supposed curses and malevolence. Constrained by the villagers' prejudices, the witch withdrew into seclusion, living in a dilapidated cottage deep within the woods.
WITCH documentation
The system integration module of WITCH is dedicated to modelling the integration of variable renewable energies (VRE) - specifically wind and solar PV - into the electrical grid. A detailed description of this modeling is available in (Carrara and Marangoni 2016) .
Apart from the implicit constraint represented by the CES structure, the limitation to VRE penetration into the electrical grid is modelled through two explicit constraints, based on (Sullivan, Krey, and Riahi 2013) .
- A constraint on the flexibility of the power generation fleet
- A constraint on the installed capacity of the power generation fleet
4.5.1 Flexibility constraint
The flexibility constraint requires that the annual average energy production be sufficiently flexible to be borne by the grid and to be able to follow the load. All energy technologies are assigned a value from -1 to 1 accounting for their grade of flexibility. Negative values are assigned to inflexible, variable technologies (i.e. VREs). Zero is assigned to those technologies which are not inflexible, but, due to technical constraints, cannot assure flexibility to follow the load (e.g. nuclear and concentrated solar power, CSP, which we assume coupled with a thermal storage which guarantees some dispatchability). Higher and higher positive coefficients are instead assigned to the progressively more flexible technologies, up to 1 which characterises storage, that by definition provides full flexibility. Gas is assigned 0.5 because in WITCH we only model Combined Cycles: Combustion Turbines would be characterised by full flexibility (1). A negative value (-0.1) is also assigned to the overall demand, in order to account for the fact that the grid itself requires some flexibility to meet changes and uncertainty in the load.
| Power technology | Flexibility coefficient |
|---|---|
| Load | -0.1 |
| Wind | -0.08 |
| PV | -0.05 |
| CSP | 0 |
| Nuclear | 0 |
| Coal | 0.15 |
| Oil | 0.3 |
| Biomass | 0.3 |
| Gas | 0.5 |
| Hydro | 0.5 |
| Storage | 1 |
The constraint is then formulated so that the sum of the energy generated by the different technologies weighted on the corresponding flexibility coefficients, which can be called flexible generation, result higher or equal than zero.
\[ \sum_ Q_(jel,t,n) \times f(jel) + Q_('el',t,n) \times f(LOAD) \geq 0 \]
Storage is not supposed to actually generate useful electricity, being essentially adopted for flexibility purposes (indeed, it could be thought as a flexibility measure not only on the generation side, but also for the demand side management). Within this constraint, the equivalent electricity contribution from storage is obtained by multiplying its capacity by a fixed value of 2000 h/yr.
4.5.2 Capacity constraint
The capacity constraint guarantees that sufficient capacity is built to meet the instantaneous peak electricity demand. In particular, the so called firm capacity must be at least 1.5-2 times (depending on the region) as the yearly average load, the latter being simply calculated as the yearly energy demand divided by the yearly hours (8760). The firm capacity represents the capacity that is considered guaranteed. For non-variable technologies it is simply the nameplate capacity. For variable technologies, the firm capacity is calculated multiplying the installed capacity by two parameters: the capacity factor and the capacity value. The capacity factor is the ratio between the actual energy output over a time period and the maximum theoretical output which would be achievable by running the plant at the nameplate capacity over the same time period. It substantially indicates the average capacity in normalised terms. The capacity value is a factor decreasing with increasing penetration in the electricity mix (starting from 0.9 with no penetration) which indicates that for variable technologies not only is the average capacity not always guaranteed (thus the 0.9 even with no penetration), but this fact becomes more and more critical with increasing levels of VRE penetration. Storage capacity is multiplied by a capacity value as well, fixed to 0.85, which takes into account the reduction of its contribution at high shares of VRE penetration.
4.5.3 Electrical grid
In order to take into account the investment in the electrical grid needed, we use a stylized model based only on grid capital:
The grid capital stock is adjusted to power capacity, taking into account a linear relationship between grid capacity and the capacity of traditional power generation technologies (indicated as standard in the formula). Moreover, it includes an additional grid stock requirement for i) connecting wind and solar plants located far from load centres or shore, and ii) building a wider interconnection for the integration of VREs (curtailment reduction, dispatchability increase, etc.), which increases exponentially with VRE penetration.
References
Carrara, Samuel, and Giacomo Marangoni. 2016. “Including System Integration of Variable Renewable Energies in a Constant Elasticity of Substitution Framework: The Case of the Witch Model.”
Sullivan, Patrick, Volker Krey, and Keywan Riahi. 2013. “Impacts of Considering Electric Sector Variability and Reliability in the MESSAGE Model.” Energy Strategy Reviews 1 (3): 157–63.
In order to take into account the investment in the electrical grid needed, we use a stylized model based only on grid capital:
She spent her days tending to mystical herbs and concocting potions that had the power to heal and protect. Yet, the villagers continued to shun her and live in ignorance of her true nature. One fateful day, a young girl named Amelia fell gravely ill. Desperate for a cure, her parents turned to the witch, seeking her help despite their own apprehension. With kindness in her heart, the witch agreed to aid the young girl and ventured into the village, disregarding the wary eyes that followed her every step. In Amelia's home, the witch brewed a potent remedy, combining various herbs and incantations known only to her. As Amelia consumed the potion, her fever subsided, and color returned to her cheeks. The grateful parents and the village witnessed the witch's magic firsthand and realized the error of their ways. Word spread quickly about the witch's healing powers, and soon, villagers plagued by ailments and misfortunes flocked to her cottage seeking solace. With each healing, the witch's reputation began to change, slowly dismantling the constraints placed on her by their prejudice. As the villagers grew to know the witch, they discovered her true benevolence and the incredible extent of her power. She used her abilities to protect the village from external threats, ensuring that harmony reigned within their midst. The once-feared sorceress became an integral part of their community, someone to be revered and loved. In the end, it was not the witch who remained constrained, but the villagers themselves. Through their prejudices and fear, they had limited their own growth and the potential for harmony. The witch had merely been a vessel for their self-reflection, allowing them to break free from their narrow-minded beliefs. From that day forward, the villagers saw the witch for who she truly was - a beacon of light in a world filled with darkness. And as for the witch, she continued to use her powers for good, ensuring that love and acceptance prevailed over ignorance and fear..
Reviews for "The Enigma of the Constrained Witch"
1. Sarah - 2 stars - I was really disappointed with "The Witch Constrained". The story was predictable and the characters felt flat and uninteresting. The plot was slow-paced and I found myself struggling to stay engaged. Overall, it lacked the thrill and suspense that I was expecting from a supernatural thriller. I wouldn't recommend it.
2. John - 1 star - "The Witch Constrained" was a complete waste of time for me. The writing was terrible and the dialogue felt forced and unnatural. The story seemed to drag on with unnecessary details and the ending was anticlimactic. The characters were poorly developed and I found it hard to care about their fates. I regret picking up this book and would caution others to avoid it.
3. Emily - 2 stars - I can't say I enjoyed "The Witch Constrained". While the premise had potential, the execution fell flat for me. The pacing was uneven, with long stretches of dull exposition and rushed action scenes. The romance subplot felt forced and irrelevant to the main plot. Overall, it lacked depth and failed to captivate me as a reader.
4. Mark - 2 stars - "The Witch Constrained" had an interesting concept, but it failed to deliver. The writing style was convoluted and confusing, making it hard to follow the story. The characters lacked substance and their motivations were unclear. The plot had potential, but it was poorly executed, leaving me unsatisfied and disappointed. I wouldn't recommend this book to others.