HVAC PID Simulator: Physics, Control Loops and Energy Modeling

Abstract

This article explains the engineering model behind an interactive HVAC control simulator for Fan Coil Units and Air Handling Units. It develops the air-side and chilled-water heat-transfer equations, explains how valve position affects cooling capacity, describes the discrete PID controller used to regulate supply-air temperature, and connects the resulting thermal load to electrical demand, energy consumption, and operating cost.

A controller is useful only when its behavior is understood together with the process it is trying to control.

— Control engineering principle

Simulator link#

HVAC Lab Simulator

What the simulator represents#

The simulator models a simplified chilled-water HVAC system in which air passes through a cooling coil and is cooled before being supplied to a conditioned space.

The two equipment presets represent the same physical process at different scales.

A Fan Coil Unit, or FCU, typically serves a smaller zone and therefore operates with relatively low airflow and chilled-water flow. An Air Handling Unit, or AHU, represents a larger system with substantially higher airflow, water flow, and cooling capacity.

The basic control loop is

\text{Temperature setpoint} \rightarrow \text{PID controller} \rightarrow \text{chilled-water valve} \rightarrow \text{cooling coil} \rightarrow \text{supply-air temperature}.

The supply-air temperature is measured continuously and fed back to the controller.

If the air is warmer than the setpoint, the controller opens the chilled-water valve. More chilled water flows through the coil, the coil absorbs more heat, and the supply-air temperature decreases.

If the air becomes colder than required, the controller reduces the valve opening.

The air-side energy balance#

Cooling begins with the sensible heat carried by the air.

For a stream of air passing through the cooling coil, the sensible cooling rate can be approximated by

\dot{Q}_{air} = \dot{m}_{air} c_{p,air} \left( T_{inlet}-T_{supply} \right).

Here,

$\dot{Q}_{air}$ is the sensible cooling rate in kW,
$\dot{m}_{air}$ is the air mass-flow rate in kg/s,
$c_{p,air}$ is the specific heat of air,
$T_{inlet}$ is the temperature entering the coil,
$T_{supply}$ is the temperature leaving the coil.

The simulator converts volumetric airflow into mass flow using

\dot{m}_{air} = \rho_{air}\dot{V}_{air},

where the air density is approximated as

\rho_{air}\approx1.2\ \text{kg/m}^3.

The specific heat of air is approximated as

c_{p,air}\approx1.005\ \text{kJ/(kg·K)}.

Suppose an AHU moves

\dot{V}_{air}=2.5\ \text{m}^3/\text{s}

and cools air from

26^\circ C

16^\circ C.

The mass flow is approximately

\dot{m}_{air} = 1.2(2.5) = 3.0\ \text{kg/s}.

The cooling rate becomes

\dot{Q}_{air} = 3.0 \times 1.005 \times (26-16),

or approximately

30.2\ \text{kW}.

This illustrates why airflow has such a strong effect on coil capacity. Cooling the same temperature difference at twice the airflow requires approximately twice the sensible cooling rate.

The chilled-water side#

The cooling coil receives chilled water from a central plant or chiller.

The simulator represents the water-side thermal balance using

\dot{Q}_{CHW} = \dot{V}_{CHW} \rho_{water} c_{p,water} \left( T_{CHWR}-T_{CHWS} \right).

The terms are

$\dot{V}_{CHW}$ : chilled-water volumetric flow,
$\rho_{water}$ : water density,
$c_{p,water}$ : specific heat of water,
$T_{CHWS}$ : chilled-water supply temperature,
$T_{CHWR}$ : chilled-water return temperature.

Typical approximations are

\rho_{water} \approx 1000\ \text{kg/m}^3

and

c_{p,water} \approx 4.186\ \text{kJ/(kg·K)}.

Water can therefore transport a large amount of thermal energy with relatively modest flow.

For example, a water flow of

1\ \text{L/s}

with a temperature increase of

5^\circ C

carries approximately

\dot{Q} = 0.001 \times 1000 \times 4.186 \times 5,

which gives

\dot{Q} \approx 20.9\ \text{kW}.

This relationship is one reason chilled-water systems are so effective for transporting cooling energy through large buildings.

The control valve#

The chilled-water valve is the actuator controlled by the PID loop.

At its simplest, the simulator can represent water flow as

\dot{V}_{CHW} = \frac{u_{valve}}{100} \dot{V}_{max},

where

u_{valve} \in [0,100].

A valve command of 50% therefore produces approximately 50% of maximum flow when the selected valve characteristic is linear.

Real control valves are often less straightforward.

An equal-percentage valve provides relatively small flow changes at low openings and increasingly larger changes as the valve approaches full travel.

A simplified equal-percentage relationship can be written as

f(u) = \frac{R^{u}-1}{R-1},

where $u$ is normalized between 0 and 1 and $R$ represents the valve rangeability.

This difference has important control consequences.

A linear valve may provide relatively aggressive control at low load, while an equal-percentage valve can provide finer modulation near small openings and greater capacity as the valve opens.

Why the supply temperature does not change instantly#

If the controller moved the valve and the supply-air temperature changed immediately, the simulation would behave unrealistically.

Real HVAC systems contain thermal inertia.

The coil metal, water volume, air volume, sensors, and heat-exchanger surfaces require time to respond.

The simulator therefore uses a dynamic response rather than jumping directly to the theoretical steady-state temperature.

A common first-order representation is

\tau \frac{dT}{dt} + T = T_{target},

where $\tau$ is the thermal time constant.

In discrete form, the supply temperature can be updated approximately as

T_{k+1} = T_k + \alpha \left( T_{target}-T_k \right),

with

\alpha = 1-e^{-\Delta t/\tau}.

A large time constant produces a slow-moving coil.

A small time constant produces a faster response.

This apparently simple dynamic model is crucial because PID tuning only makes sense when the controlled process has dynamics of its own.

Coil effectiveness#

The cooling coil cannot normally bring the leaving-air temperature all the way to the chilled-water supply temperature.

Instead, the simulator calculates an effective leaving-air target based on the available cooling capacity and coil effectiveness.

A simplified conceptual model is

T_{target} = T_{inlet} - \varepsilon \left( T_{inlet}-T_{CHWS} \right),

where

0\leq\varepsilon\leq1

represents coil effectiveness.

\varepsilon=0,

the coil provides no cooling.

\varepsilon=1,

the theoretical leaving-air temperature approaches the entering chilled-water temperature.

In practice, effectiveness depends on several interacting variables, including airflow, water flow, coil geometry, heat-transfer coefficients, and the temperature difference between the two fluids.

The simulator combines these effects into a simplified dynamic model suitable for studying control behavior without requiring a full finite-volume heat-exchanger simulation.

The feedback error#

The objective of the controller is to maintain supply-air temperature near a user-selected setpoint.

The control error is defined as

e(t) = T_{supply}(t) - T_{setpoint}.

This sign convention is appropriate for direct-acting cooling control.

For example, if

T_{supply}=20^\circ C

and

T_{setpoint}=16^\circ C,

then

e=4^\circ C.

The air is too warm, so the controller should increase cooling.

T_{supply}=15^\circ C,

then

e=-1^\circ C.

The air is colder than required, so the controller should close the valve.

The PID controller#

The simulator implements a discrete PID controller.

The continuous conceptual form is

u(t) = K_p e(t) + K_i \int e(t)\,dt + K_d \frac{de(t)}{dt}.

The three terms perform different functions.

Proportional action#

The proportional contribution is

P=K_p e.

A larger error produces a larger valve command.

If $K_p$ is very small, the system reacts slowly.

If $K_p$ is too large, the system can become aggressive and oscillatory.

Integral action#

The integral contribution accumulates past error:

I = K_i \int e(t)\,dt.

Its main purpose is eliminating steady-state offset.

Suppose proportional control alone stabilizes the supply temperature at

16.8^\circ C

for a

16^\circ C

setpoint.

A small error remains.

The integral term continues accumulating this error and gradually increases cooling until the offset disappears.

Derivative action#

The derivative term responds to the rate of change of the error:

D = K_d \frac{de}{dt}.

It can be interpreted as anticipatory behavior.

If the temperature is approaching the setpoint very rapidly, derivative action can reduce the controller output before the system overshoots.

In practical HVAC systems, derivative action is often used conservatively because measurement noise can make it unstable or unnecessarily sensitive.

The discrete PID calculation#

Computers do not solve the PID equation continuously.

The simulation operates at discrete time intervals.

For each calculation step $\Delta t$ , the integral accumulator is updated approximately by

I_k = I_{k-1} + e_k\Delta t.

The derivative can be approximated by

\frac{de}{dt} \approx \frac{e_k-e_{k-1}}{\Delta t}.

The controller output becomes

u_k = K_p e_k + K_i I_k + K_d \frac{e_k-e_{k-1}}{\Delta t}.

The resulting command is limited to the physical valve range:

0\% \leq u_k \leq 100\%.

The simulator uses a physics step on the order of fractions of a second, allowing transient control behavior to remain smooth while still being computationally inexpensive in the browser.

Integral windup#

A practical PID controller must deal with actuator saturation.

Imagine the supply air is extremely warm and the controller calculates

u=180\%.

The physical valve cannot open beyond 100%.

If the integral term continues increasing while the valve is saturated, it can become very large.

Later, even after the temperature reaches the setpoint, the controller may remain at 100% output because the accumulated integral takes time to unwind.

This phenomenon is called integral windup.

The simulator prevents excessive windup by limiting or conditionally updating the integral accumulator while the controller output is saturated.

This is known as anti-windup.

Without anti-windup, step-response experiments can produce unrealistically large overshoot and long recovery times.

What happens after a disturbance#

The simulator includes a step-disturbance function because step changes are one of the most useful ways to study control systems.

Consider an abrupt

+5^\circ C

increase in entering-air temperature.

Immediately after the disturbance,

T_{inlet}\uparrow.

The coil cannot instantly compensate, so

T_{supply}\uparrow.

The error therefore increases:

e = T_{supply}-T_{setpoint} \uparrow.

The PID controller responds by increasing the valve output:

u_{valve}\uparrow.

Water flow increases:

\dot{V}_{CHW}\uparrow.

The coil absorbs more heat:

\dot{Q}_{coil}\uparrow.

Eventually the supply-air temperature moves back toward the setpoint.

The trend charts allow this entire sequence to be observed.

Reading the temperature chart#

The first chart compares

T_{supply}

with

T_{setpoint}.

A well-tuned system normally approaches the setpoint smoothly.

Important behaviors to observe include:

Rise or response time — how quickly the system reacts after a disturbance.

Overshoot — how far the controlled temperature moves beyond the target.

Settling time — how long the system takes to remain close to the setpoint.

Steady-state error — whether a persistent offset remains after the transient has disappeared.

These are standard control-system performance measures.

Reading the control chart#

The second trend shows controller error and valve position.

If the error is large and positive, the valve should generally move toward the open position.

As the temperature approaches the setpoint, the valve should settle at whatever intermediate position is required to balance the thermal load.

A valve remaining permanently at 100% can indicate that the requested operating point exceeds the modeled coil capacity.

A valve repeatedly cycling between high and low values can indicate overly aggressive PID tuning.

A slow valve response combined with persistent error may indicate insufficient proportional or integral action.

Reading the flow and power chart#

The third chart connects control behavior to energy behavior.

Valve movement changes chilled-water flow.

Chilled-water flow changes cooling capacity.

Cooling load influences chiller electrical power.

This creates an important engineering chain:

\text{PID output} \rightarrow \text{water flow} \rightarrow \text{thermal load} \rightarrow \text{chiller power}.

The controller is therefore doing more than controlling temperature.

It indirectly influences plant energy consumption.

Chiller COP#

The simulator converts cooling load into approximate electrical demand using the chiller coefficient of performance, or COP.

COP is defined as

COP = \frac{\dot{Q}_{thermal}}{P_{electrical}}.

Therefore,

P_{electrical} = \frac{\dot{Q}_{thermal}}{COP}.

Suppose the cooling coil requires

35\ \text{kW}

of thermal cooling and the chiller COP is

5.

The corresponding electrical power is approximately

P_{electrical} = \frac{35}{5} = 7\ \text{kW}.

A higher COP means more cooling is produced for each unit of electrical energy.

Energy accumulation#

Electrical power describes the instantaneous rate of energy use.

Energy consumption is power integrated over time.

For a simulation interval $\Delta t$ measured in seconds,

\Delta E = P_{electrical} \frac{\Delta t}{3600}.

The accumulated energy is

E = \sum P_{electrical} \frac{\Delta t}{3600}.

The result is expressed in

\text{kWh}.

This allows short-term control behavior to be connected directly to long-term energy consumption.

Monetary operating cost#

Once electrical energy is known, estimated cost is calculated from the electricity tariff.

If the tariff is

r \quad [\$/\text{kWh}],

then

Cost = E r.

For example, if the simulation accumulates

12\ \text{kWh}

and electricity costs

\$0.18/\text{kWh},

the estimated energy cost is

12\times0.18 = \$2.16.

This simple relationship helps connect control engineering with operating economics.

A controller that maintains temperature successfully but causes excessive oscillation may increase valve activity and cooling demand.

Control quality and energy performance are therefore related.

Chilled-water volume metering#

The simulator also integrates chilled-water flow.

The accumulated volume is

V_{total} = \sum \dot{V}_{CHW}\Delta t.

This value is displayed in cubic metres and can also be converted to gallons.

Unlike electrical energy, chilled-water volume is not necessarily a direct billing quantity in every building.

However, it is useful for understanding hydronic demand and comparing operating scenarios.

Converting cooling capacity to refrigeration tons#

Cooling capacity is often expressed in refrigeration tons in HVAC practice.

The approximate conversion is

1\ TR \approx 3.517\ \text{kW}_{thermal}.

Therefore,

TR = \frac{\dot{Q}_{thermal}}{3.517}.

For a thermal load of

35.17\ \text{kW},

the corresponding capacity is approximately

10\ TR.

The simulator displays both units so that users can move between SI engineering calculations and common HVAC industry terminology.

FCU versus AHU behavior#

Although the governing equations are the same, the presets produce noticeably different system responses.

An FCU typically uses smaller airflow and water-flow values.

Its thermal load is therefore comparatively small.

An AHU operates at larger scale.

Higher airflow means that each degree of temperature reduction requires considerably more cooling capacity:

\dot{Q}_{air} \propto \dot{V}_{air}.

Consequently, parameters that work well for the FCU are not guaranteed to produce identical dynamic behavior for the AHU.

This illustrates an important principle of control engineering:

PID tuning belongs to the process, not merely to the controller.

The same numerical gains can produce different behavior when the controlled equipment changes.

Why changing airflow changes the loop#

Increasing airflow affects the simulator in several ways.

For the same inlet and supply temperatures,

\dot{Q}_{air} = \rho \dot{V}_{air} c_p \Delta T.

Therefore,

\dot{V}_{air}\uparrow \Rightarrow \dot{Q}_{air}\uparrow.

The coil must remove more heat.

The controller may consequently require a larger valve opening.

If the maximum chilled-water flow is insufficient, the controller can reach

100\%

without achieving the desired setpoint.

This is an example of actuator saturation caused by insufficient plant capacity rather than poor PID tuning.

Why chilled-water supply temperature matters#

The cooling potential of a coil depends strongly on the temperature difference between the air and the chilled water.

If chilled-water supply temperature decreases,

T_{CHWS}\downarrow,

the driving temperature difference generally increases.

The coil can therefore produce more cooling for similar flow conditions.

Conversely, increasing chilled-water supply temperature can improve central plant efficiency in many real installations but may reduce available coil capacity.

This creates a system-level tradeoff between coil performance and chiller efficiency.

The simulator allows the thermal side of this interaction to be explored directly.

Why maximum water flow matters#

Maximum water flow defines the upper cooling capability available to the valve model.

If the valve reaches 100%, then

\dot{V}_{CHW} = \dot{V}_{max}.

If the required heat transfer still exceeds what the available water flow can support, the supply-air temperature cannot reach its target.

Increasing PID gains will not solve this condition.

The limitation is physical, not mathematical.

The Web Worker architecture#

The browser implementation separates the numerical simulation from the main React interface.

The architecture is approximately

React interface
      │
      │ parameters
      ▼
Web Worker
      │
      ├── thermal model
      ├── PID controller
      ├── hydronic calculations
      └── energy metering
      │
      │ telemetry
      ▼
React dashboard
      │
      ├── digital values
      └── Canvas trend charts

This separation is important because browser interfaces normally execute on the main JavaScript thread.

If intensive numerical calculations and chart rendering share that thread, interaction can become sluggish.

The Web Worker executes the simulation independently.

The user interface receives periodic telemetry messages and displays them without performing the physics calculations itself.

Physics timestep versus display timestep#

The simulator intentionally distinguishes between simulation frequency and display frequency.

The physics engine may calculate at approximately

\Delta t=0.2\ \text{s},

while telemetry is sent to the visible dashboard less frequently.

This means the mathematical process can retain reasonably fine time resolution without forcing React and the charts to redraw five times every second.

The concept is similar to a real building automation system.

A controller may execute its internal logic frequently even though an operator workstation records or plots data at a slower rate.

Why the charts use Canvas#

Earlier versions of the application used SVG-based chart rendering.

SVG is excellent for many dashboard applications, but continuously updating engineering trends can cause repeated DOM and path updates.

The current version draws the trend lines on HTML Canvas.

Canvas treats the chart primarily as a bitmap drawing surface rather than maintaining a large hierarchy of graphical DOM elements.

For a continuously running simulator this provides a simpler and more memory-efficient rendering strategy.

The history buffer is also deliberately bounded.

Once the maximum number of samples is reached, older samples are removed.

Therefore the amount of chart data remains approximately constant even if the simulator runs for several hours.

Simulation speed#

The simulator provides

1\times,\quad2\times,\quad5\times

speed modes.

Importantly, higher simulation speed does not need to mean that the browser interface redraws five times more frequently.

Instead, the worker advances simulated time faster while the visible telemetry rate remains controlled.

This preserves responsiveness and reduces unnecessary rendering work.

What the simulator is useful for#

The application is particularly useful for developing intuition about relationships that can otherwise appear abstract.

Users can experimentally observe how

proportional gain affects responsiveness,
integral gain removes residual offset,
excessive gains create oscillation,
thermal inertia creates delayed response,
airflow changes required cooling capacity,
water flow affects coil performance,
chilled-water temperature affects available cooling,
disturbances propagate through a feedback loop,
valve saturation reveals capacity limitations,
COP connects thermal demand to electrical demand,
and control decisions influence cumulative energy cost.

These relationships are difficult to appreciate from equations alone.

The simulator allows them to be changed one parameter at a time and observed in real time.

A suggested experiment: proportional control#

Begin with

K_i=0

and

K_d=0.

Use proportional control only.

Start with a low value of $K_p$ and observe the temperature response.

Gradually increase $K_p$ .

You should notice that the controller becomes more responsive.

At sufficiently aggressive settings, oscillation or overshoot may become more pronounced.

Also observe whether the final supply temperature settles exactly at the setpoint.

A residual error is common with proportional-only control.

A suggested experiment: adding integral action#

Next introduce a small integral gain.

Observe how the controller slowly corrects the remaining offset.

Increase $K_i$ carefully.

Too much integral action can create overshoot or repeated oscillation because the controller continues accumulating error during the transient.

This experiment illustrates why integral action is both powerful and potentially destabilizing.

A suggested experiment: step disturbance#

Allow the simulation to reach steady operation.

Then inject the temperature disturbance.

Observe the charts in this order:

inlet conditions change,
supply temperature moves away from setpoint,
controller error increases,
valve output increases,
chilled-water flow increases,
thermal load increases,
electrical demand increases,
supply temperature recovers.

This is a compact demonstration of closed-loop feedback.

A suggested experiment: insufficient capacity#

Reduce maximum chilled-water flow while keeping a demanding supply-air setpoint.

The valve may eventually reach

100\%.

If the temperature remains above the setpoint, increase $K_p$ significantly.

The result should demonstrate an important lesson:

the temperature still cannot reach the target if the available heat-transfer capacity is insufficient.

The problem is not PID tuning.

It is equipment capacity.

A suggested experiment: energy versus control quality#

Run two simulations with different PID gains.

In the first, tune the controller for a relatively smooth response.

In the second, use aggressive gains that produce repeated oscillation.

Compare

valve movement,
water flow,
instantaneous cooling demand,
accumulated electrical energy,
and final cost.

This experiment connects classical control performance to energy engineering.

What the model intentionally simplifies#

The simulator is designed primarily for control education and engineering intuition.

It is not a full psychrometric or equipment-selection program.

Several physical effects are simplified or omitted.

For example, the current model focuses mainly on sensible cooling and does not fully solve moisture condensation and latent heat transfer.

A detailed cooling-coil model would also consider

\text{humidity ratio},

\text{dew point},

\text{wet-coil effectiveness},

\text{condensate rate},

and

\text{enthalpy}.

Real hydronic systems may additionally require models of

pump curves,
differential pressure,
valve authority,
piping resistance,
variable-speed pumping,
coil fouling,
sensor noise,
actuator travel time,
transport delay,
mixed-air dampers,
economizer operation,
and chiller part-load efficiency.

These effects can be added progressively if greater fidelity is needed.

From simulation to building automation#

The same control concepts represented here appear directly in real Building Automation Systems.

A typical BAS implementation might contain

Supply Air Temperature Sensor
            │
            ▼
        PID Loop
            │
            ▼
Cooling Valve Command
            │
            ▼
      Chilled-Water Coil
            │
            ▼
Supply Air Temperature

The software implementation may reside in

an AHU controller,
an application-specific controller,
a programmable building controller,
a PLC,
or a supervisory automation platform.

The equipment may differ, but the mathematical structure remains recognizable.

Why understanding the physics improves tuning#

A PID controller is often taught as a purely mathematical object.

In HVAC practice, this can be misleading.

Suppose a temperature loop is oscillating.

Possible causes include

excessive proportional gain,
excessive integral gain,
an oversized valve,
poor valve authority,
sensor placement,
thermal delay,
rapidly changing airflow,
unstable chilled-water pressure,
or interactions with other control loops.

Changing PID gains without understanding the physical system can therefore hide rather than solve the underlying problem.

The simulator encourages the opposite approach:

understand the process first, then tune the controller.

Closing observation#

An HVAC control loop is a meeting point between thermodynamics and computation.

The air carries sensible heat.

The chilled water removes that heat.

The valve determines how much cooling capacity is available.

The coil introduces thermal dynamics.

The PID controller observes the resulting temperature error and continuously adjusts the process.

The chiller then converts this thermal demand into electrical demand, energy consumption, and operating cost.

The equations may be simple individually,

\dot{Q} = \dot{m}c_p\Delta T,

u = K_p e + K_i\int e\,dt + K_d\frac{de}{dt},

and

P_{electrical} = \frac{\dot{Q}_{thermal}}{COP},

but together they form a closed-loop dynamic system.

That interaction is the central purpose of the simulator: to make the relationship between physics, control, and energy visible rather than abstract.

References

ASHRAE. ASHRAE Handbook—Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers.
Åström, K. J., and Murray, R. M. Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press, 2008.
Åström, K. J., and Hägglund, T. PID Controllers: Theory, Design, and Tuning. Instrument Society of America.
McQuiston, F. C., Parker, J. D., and Spitler, J. D. Heating, Ventilating, and Air Conditioning: Analysis and Design. Wiley.
ASHRAE Guideline 36. High-Performance Sequences of Operation for HVAC Systems.