Understanding Systems and Their Mathematical Models
What is a System?
A system is a set of active elements (e.g., objects, operations, processes) separated from and interacting with the environment. These elements are ordered, deliberately selected, and possess specific properties and relationships designed to meet a particular purpose or fulfill tasks within the environment.
Difference Between a System and a Set
A system is distinguished from a simple set by its active elements, their interactions with the environment, and their organization towards a specific goal.
Energy System
An energy system is a sequence of energy transformations, separated from and interacting with the environment. It involves interconnected installations that implement these transformations, linked by streams of substances and energy. The system has specific properties aimed at covering the demand for useful energy in the environment at a given place and time.
Examples: Boiler, power plant.
National Gas System
(No specific definition provided in the original text, but it can be inferred as a system for the production, transmission, and distribution of natural gas.)
Autonomous System
An autonomous system, during its operation, neither takes from nor sends to the environment substance or energy streams.
Coherent System
A coherent system has interrelated functional elements. A change in any element causes changes in all others. In a coherent system:
- All functional elements are active.
- All elements are interconnected by active relationships.
Examples: Steam (block) power plant, turbojet engine.
Independent System
An independent system has unrelated functional elements. A change in one does not affect others. Independence is the opposite of coherence. Full independence is rare in energy systems.
Example: Thermal collector systems.
Centralization in a System
Centralization occurs when a component or subsystem plays a steering role in relation to other system elements. It involves increasing the relationship of subsystems to one element, which plays a decisive role in the entire system’s operation.
Example: Small systems with current monitoring, control room of chemical installations.
Stable System
A stable system maintains variables determining its purposeful functioning within imposed limits.
Example: Pressure in a steam boiler, temperature in a refrigerating chamber.
Stable systems have functional elements that stabilize the values of variables crucial for proper operation and purpose fulfillment.
Physical Model
A physical model is an abstract representation that simplifies a real system while retaining its most important features. Physical modeling requires adherence to the laws of physics in all parts of the model.
Examples: Black body, ideal liquid, perfect gas, Bohr atom model.
Mathematical Models of Systems
Mathematical models use mathematical language to describe system behavior. They consist of mathematical relationships expressing:
- General physical and chemical laws.
- Technological characteristics of processes, including technical, ecological, economic, and other constraints.
- Definitions of quantities used in the description of system elements.
System Structure
System structure is a network of relations between system elements, distinguished by type or importance. It can be represented graphically (using a graph), in matrix form, or alphanumerically. Establishing the structure allows for recording flows of substances, energy, and information and determining relationships between elements.
Optimization Models
Optimization models have two parts: a criterion function (goal function) determining system operation quality and optimization constraints. They are used to find extremes of the criterion function (mathematical programming) or select the optimal function for changing system parameters (dynamic programming).
Dynamic Models
Dynamic models describe changes in system parameters over time. They are used to describe transient states caused by changes in the system’s interaction with its environment. They usually contain differential or integral relationships.
Deterministic and Statistical Models
In deterministic models, all parameters are known. Statistical models involve random variables with indefinite statistical distributions, requiring mathematical statistics methods for their solution.
Thermodynamic Functions in Mathematical Models
Thermodynamic functions are used in mathematical models through empirical equations approximating their values (e.g., h=h(p,t), v=v(p,t), s=s(p,t), etc.). These equations are based on:
- Calculations using tabled data.
- Approximating equations based on discrete empirical results.
- Differentiation of other functions, such as the equation approximating the value of Helmholtz free energy.
They are divided into two categories:
- For scientific purposes (high accuracy).
- For general technical purposes (less accuracy, simpler construction).
National Power System
The national power system can be described through its subsystems:
- Generation:
- Steam thermal power plants.
- Steam or gas-steam combined heat and power (CHP) plants using hard coal, natural gas, or oil (electricity production depends on heat demand).
- Hydroelectric power plants.
- Wind turbines.
- Photovoltaic cells.
- Transmission.
- Distribution.
- Trade and Sales.
Transmission and distribution are coordinated by the Transmission System Operator, the central unit responsible for ensuring electricity quality, supply security, and economically optimal system operation.